1 Introduction↩︎

Quantum groups [1], [2] represent a generalization of classical symmetry groups in quantum systems and provide a powerful mathematical framework for describing quantum phenomena. Their duals are algebraic structures emerging from the deformation of Lie algebras, and appear in various areas of theoretical physics and mathematics [3], [4]. In quantum systems, quantum groups offer a method to describe symmetries that are not captured by traditional group theory. They incorporate noncommutative geometry and quantum deformations, allowing the exploration of quantum systems with complex symmetry properties.

Among the deformations of symmetry algebras, the most widely studied is the standard or \(q\)-deformation, introduced in the context of quantum groups by Drinfel’d and Jimbo in the 1980s [2], [5]. This construction relies on the introduction of a parameter \(q\), which modifies both the commutation relations and the Hopf algebra structure, and can be formulated for a broad class of Lie algebras. In this work, we focus on the case of \(\mathcal{U}(\mathfrak{sl}(2, \mathbb{R}))\), leading to the quantum algebra \(\mathcal{U}_{q}( \mathfrak{sl}(2, \mathbb{R}))\). The \(q\)-deformation has a rich interpretation in representation theory and has found applications in integrable systems and spin models in statistical physics [6]–[9]. In the limit \(q\rightarrow 1\), the undeformed algebra is recovered. For \(q\neq 1\), however, the Clebsch–Gordan coefficients are replaced by their \(q\)-analogs, which are expressed in terms of basic hypergeometric series and \(q\)-factorials. Consequently, the construction of collective states is affected [10], [11]. In addition to this standard deformation, there also exists a non-standard or \(h\)-deformation, known as the Jordanian deformation [12], [13]. This construction relies on the introduction of a parameter \(h\), which modifies the Hopf algebra structure but not the commutation relations. In the limit \(h \rightarrow 0\) the undeformed algebra is also recovered. Unlike the standard \(q\)-deformation, which is governed by a quasitriangular \(R\)-matrix, the parameter \(h\) in this non-standard case is introduced through a purely triangular \(R\)-matrix. This structure corresponds to a nilpotent solution of the classical Yang–Baxter equation, typically generated via a Drinfeld twist [14]. Consequently, the structure of \(\mathcal{U}_{h} (\mathfrak{sl}(2, \mathbb{R}))\) exhibits coproducts that are essentially different from those of the \(q\)-deformation.

A comparative study between standard and non-standard deformations is of great interest, as both provide alternative algebraic frameworks to explore how the symmetry of a system controls the properties of quantum states. In particular, while \(q\)-deformations are well established in the theory of quantum groups and their physical applications, \(h\)-deformations have shown great potential as both a mathematical and physical laboratory for generating new classes of quantum states [15], with tunable properties through the deformation parameter \(h\). Focusing our analysis on the fundamental spin-1/2 representation, both deformations allow for the systematic generation of new classes of multi-qubit quantum states. The algebraic construction of the \(q\)-deformed and \(h\)-deformed states finds its natural physical realization as the energy eigenstates of the deformed Kittel–Shore (KS) model. The undeformed Kittel–Shore Hamiltonian is given by [16] \[\hat{H}_{KS}=- I \sum_{i<j}^{N}\Vec{J}_{i}\cdot\Vec{J}_{j}, \label{eq:hamiltonian1}\tag{1}\] where \(I\) is the interaction constant between the spins and \(\Vec{J}_{i}\) represents the total angular momentum operator of the \(i\)-th site. The KS model describes a system of \(N\) qubits characterized by long-range all-to-all interactions where every site is coupled identically to all others. This globally connected architecture is governed by an underlying \(\mathcal{U}(\mathfrak{sl}(2, \mathbb{R}))\) coalgebra symmetry, which dictates the dynamical evolution of the system. The \(q\)-KS model has been described in [17]. Here, we present the \(h\)-KS model for the first time and discuss both deformations, including the differences between their corresponding quantum states.

Due to its ability to describe systems where spins exhibit isotropic exchange owing to constant long-range coupling, the undeformed KS model is particularly suited to characterize small clusters of equidistant spins (dimers, equilateral triangles, regular tetrahedra), allowing the study of the magnetic properties of organic molecules containing embedded paramagnetic ions [18], the melting behavior of small clusters in porous media [19], as well as the collective properties emerging from individual magnetic moments in ultra-small spin systems [20], [21]. In the field of condensed matter and material science, the KS model is used to model high-temperature superconductors [22], to study magnetization in layers where spins are identically coupled [23]–[25], and to thermodynamically analyze multiferroic thermomagnetic generators [26] (combining the efficiency model described in [27] with the model of Curie point suppression of [28]) to characterize transition behaviors in nickel-based nanosolids. These applications show how the model connects microscopic spin dynamics and macroscopic thermodynamic efficiency. Moreover, the KS model is also currently used in quantum information processing and the study of non-equilibrium phases of matter. Its structure provides a natural description for qubits implemented in ammonia-based molecular gates [29] or electronic spins in semiconductor quantum dots [30], where few-body interactions must be carefully tuned to circumvent decoherence and multi-body effects [31]. It is also a candidate for the realization of discrete-time crystals in superconducting circuits and cold atom traps [32]. In this paper, we compare the experimental states with those theoretically obtained from the deformed models, showing that both of them can resemble some features of the errors and noise of the state constructions.

Quantum fidelity serves as a fundamental statistical measure used to quantify the similarity between two quantum states, effectively functioning as a metric for information preservation in quantum communication and computation [33]. For pure states, represented by state vectors \(|\psi\rangle\) and \(|\phi\rangle\), the fidelity is defined as the squared modulus of their inner product, \(F(|\psi\rangle, |\phi\rangle) = |\langle\psi|\phi\rangle|^2\), which ranges from unity for identical states to zero for those that are perfectly orthogonal. In the more general context of mixed states described by density operators \(\rho\) and \(\sigma\), the fidelity is generalized [34] through \(F(\rho, \sigma) = (\text{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})^2\), providing a geometric interpretation of the distance between states within the Hilbert space while accounting for statistical uncertainty. This measure is essential for benchmarking the accuracy of quantum gates and state preparation, as it characterizes the deviation of an experimental result from its theoretical target due to environmental decoherence or operational errors. In this work, we study the fidelity to compare the results for some undeformed and deformed states, obtained from deformed \(q\)- and \(h\)-KS Hamiltonians. Particular attention will be focused on Dicke states [35], which are a prominent class of quantum states characterized as the fully symmetric eigenstates of total spin operators in SU(2) models. Their primary mathematical property is their symmetry, as they remain unchanged (invariant) under the operation of the permutation group. For a system of \(N\) qubits with \(k\) excitations, the corresponding Dicke state \(\ket{D_N^{(k)}}\) in the undeformed theory reads \[\vert D^{(k)}_{N}\rangle = \frac{1}{\sqrt{\binom{N}{k}}}\sum_{S \in \mathcal{B}_{N,k}} |S\rangle, \label{eq:dicke95def}\tag{2}\] where \(\mathcal{B}_{N,k}\) represents the set of all binary words of length \(N\) and Hamming weight \(k\) (that is, all possible states with exactly \(k\) spins \(\uparrow\) and \(N-k\) spins \(\downarrow\)) [36]. In quantum computing, Dicke states are highly prized [37]–[39] due to their symmetry, which allows them to simultaneously offer strong multipartite entanglement and protection against particle deletion. Furthermore, they are essential for achieving metrological enhancement in sensitive measurements [40]–[43].We focus our attention on Dicke states, yet the analysis is not restricted to them. The quantum algebraic approach used in this work also identifies states within other representation, labeled here as \(M\), \(V\), \(T\), \(R\), and \(Q\) representations. These states are of significant interest as they manifest intrinsically different algebraic symmetries. The fundamental distinction between the \(q\) and \(h\) deformations manifests in their asymptotic structural impact on the state space. On one hand, the \(q\)-deformation operates as a smooth algebraic perturbation, ensuring that the deformed states retain a non-vanishing residual fidelity for large values of \(q\). On the other hand, the Jordanian \(h\)-deformation triggers a severe, state-dependent orthogonalization governed entirely by the internal combinatorics of the spin lattice. Remarkably, for \(q\)- and \(h\)-deformed Dicke states of \(N\) qubits, the fidelity per site in the thermodynamic limit is computed, showing that a \(N^{-1}\) rescaling in the deformation parameter is needed to avoid orthogonality catastrophe. This rescaling was also obtained in [17] in order to assure spectral and thermodynamic stability of the \(q\)-KS model.

The remainder of this paper is organized as follows. Sec. 2 delves into the mathematical structure of both undeformed and deformed \(\mathcal{U}_{q}(\mathfrak{sl}(2, \mathbb{R}))\) and \(\mathcal{U}_{h}(\mathfrak{sl}(2, \mathbb{R}))\) algebras, examining their commutation relations, coproducts, and representations. It also addresses the construction of Clebsch–Gordan coefficients to develop multi-particle deformed quantum states. In Sec. 3, we introduce the Kittel–Shore model and construct the corresponding superintegrable deformations. In Sec. 4, we compare these theoretical predictions with recent experimental results on quantum state preparation. Sec. 5 offers a comparative analysis of the fidelity for undeformed, \(q\)-deformed, and \(h\)-deformed quantum states of qubits. Finally, we conclude with some insights and future directions in Sec. 6.

2 Revisiting \(\mathcal{U}(\mathfrak{sl}(2, \mathbb{R}))\), \(\mathcal{U}_{q} (\mathfrak{sl}(2, \mathbb{R}))\) and \(\mathcal{U}_{h}(\mathfrak{sl}(2, \mathbb{R} ))\) Hopf algebras↩︎

The Lie algebra \(\mathfrak{sl}(2, \mathbb{R})\) is a fundamental component in the description of quantum systems with rotational symmetry, such as spins. Its algebraic properties are defined by the following commutation relations of the generators \(\{J_{z}, J_+, J_-\}\): \[[J_{z},J_{\pm}]= \pm J_{\pm},\qquad [J_{+},J_{-}]= 2 J_{z}, \label{eq:su2Lie}\tag{3}\] where \[J_{x}= \frac{1}{2}\sigma_{x}, \qquad J_{y}= \frac{1}{2}\sigma_{y}, \qquad J_{z}= \frac{1}{2}\sigma_{z}, \qquad J_{\pm}= J_{x}\pm i J_{y},\] gives the fundamental \(2 \times 2\) finite-dimensional irreducible representation in terms of the Pauli matrices.

To describe an \(N\)-particle system, the single-spin algebraic structure must be extended through a formal composition rule that preserves the underlying symmetry across the many-body Hilbert space. This framework is provided by the Hopf algebraic structure of the universal enveloping algebra \(\mathcal{U}(\mathfrak{sl}(2, \mathbb{R}))\), which defines a consistent mapping for collective operators. Beyond the standard Lie algebra axioms, the Hopf structure includes a coproduct (\(\Delta\)), counit (\(\epsilon\)), and antipode (\(S\)). These operations are indispensable for consistently defining and manipulating tensor products across representation spaces. Coproduct, in particular, describes how the algebra’s generators act on a composite state. For \(\mathcal{U}(\mathfrak{sl}(2, \mathbb{R}))\), it is defined as: \[\Delta (J_{i})=J_{i}\otimes 1+1\otimes J_{i}.\] This relationship allows the calculation of the collective properties of a particle system from the individual properties of its components.

The decomposition of a composite state into irreducible representations is critical in quantum mechanics [44], [45]. This task is accomplished using Clebsch–Gordan coefficients, which act as the expansion coefficients in a linear combination. They act as scalars in a linear combination that transforms the tensor product basis of two (or more) irreducible representations into a new basis that decomposes the space into a direct sum of irreducible representations: \[\begin{align} \label{boots1} \ket{j,m} =\sum_{m=m_1+m_2} \mathcal{C}^{j_1,j_2,j}_{m_1,m_2,m} \ket{j_1,m_1}\otimes \ket{j_2, m_2}, \end{align}\tag{4}\] where \(j,j_1,j_2\) are non-negative integers or half-integers such that \(j=(j_{1}+ j_{2})\,\dots,|j_{1}-j_{2}|\), and \(m_{i}=-j,...,j\). \(\mathcal{C}^{j_1,j_2,j}_{m_1,m_2,m}\) are the associated Clebsch–Gordan coefficients, which can be written as

\[\begin{align} &&\label{boots2} \mathcal{C}^{j_1,j_2,j}_{m_1,m_2,m}=\delta_{m_1+m_2,m} \left(\frac{ (j_{1}+j_{2}-j)! (j_{1}-j_{2}+j)! (-j_{1}+j_{2}+j)! }{(j_{1}+j_{2}+j+1)!} \right)^{1/2} \nonumber\\ &&\times \left((j_1+m_1)!(j_1-m_1)!(j_2+m_2)!(j_2-m_2)!(j+m)!(j-m)!(2j+1) \right)^{1/2} \nonumber\\ &&\times \sum_k \frac{(-1)^{k}}{k!(j_{1}+j_{2}-j-k)!(j_{1}-m_{1}-k)!(j_{2}+m_{2}-k)!(j-j_{2}+m_{1}+k)! (j-j_{1}-m_{2}+k)!},\nonumber\\ \end{align}\tag{5}\] with \(\delta_{a,b}\) the Kronecker delta.

2.1 The standard deformation↩︎

The Hopf algebra \(\mathcal{U}_{q}(\mathfrak{sl}(2, \mathbb{R}))\) is generated by \(\{L_{z},L_{+},L_{-}\}\) and can be defined in terms of the following relations [46] where \(q\in \mathbb{C} \setminus \{ e^{i \frac{2\pi \, r}{n}} : n \in \mathbb{N}, r = 0, 1, \dots, n-1 \}\): \[[L_{z},L_{\pm}]= \pm L_{\pm},\qquad [L_{+},L_{-}]= \left[ 2 L_{z}\right]_{q} = \frac{q^{L_z}-q^{-L_z}}{q^{1/2}-q^{-1/2}}. \label{suq2}\tag{6}\] Above, we have made use of the symbol for \(q\)-numbers \[[\ell]_{q}\mathrel{\vcenter{:}}= \frac{q^{\ell/2}-q^{-\ell/2}}{q^{1/2}-q^{-1/2}}=\frac{\sinh(\eta \ell/2)}{\sinh(\eta/2)}, \quad \ell\in\mathbb{Z} \label{eq:qnumber},\tag{7}\] where the usual convention[2] \(q=e^\eta\) has been used. The coalgebra structure for the \(\mathcal{U}_{q}(\mathfrak{sl}(2, \mathbb{R}))\) algebra is generated by the deformed coproduct map \[\Delta_{q}(L_{\pm})= q^{-L_z/2}\otimes L_{\pm}+ L_{\pm}\otimes q^{L_z/2}, \qquad \Delta_{q}(L_{z})= 1 \otimes L_{z}+ L_{z}\otimes 1. \label{qcop}\tag{8}\] Irreducible components in the tensor product (given by the deformed coproduct map 8 ) of two \(\mathcal{U}_{q}(\mathfrak{sl}(2, \mathbb{R}))\) representations are again obtained in terms of 4 , where the \(q\)-Clebsch–Gordan coefficients (see [47] and references therein) are used: \[\begin{align} {\cal C}_{m_1, m_2, m}^{j_1, j_2, j}(q)\,=&\,\delta_{m,m_1+m_2}q^{\frac{1}{2} (j_1 m_2-j_2 m_1)-\frac{1}{4} (-j+j_1+j_2) (j+j_1+j_2+1)}\\ &\sqrt{\frac{[2 j+1]_{q}[j+m]_{q}! [j_{2}-m_{2}]_{q}![j+j_{1}-j_{2}]_{q}! [-j+j_{1}+j_{2}]_{q}! [j+j_{1}+j_{2}+1]_{q}! }{[j-m]_{q}! [j_{1}-m_{1}]_{q}! [j_{1}+m_{1}]_{q}! [j_{2}+m_{2}]_{q}! [j-j_{1}+j_{2}]_{q}! }}\\ &\sum_{n=0}^{\min (-j+j_1+j_2,j_2-m_2)}\frac{ [2 j_{2}-n]_{q}! (-1)^{-j+j_1+j_2+n}q^{\frac{1}{2} n (j_1+m_1)}[j_{1}+j_{2}-m-n]_{q}! }{ [n]_{q}! [j_{2}-m_{2}-n]_{q}! [-j+j_{1}+j_{2}-n]_{q}! [j+j_{1}+j_{2}-n+1]_{q}! }\, . \end{align} \label{eq:qcg}\tag{9}\] Using these coefficients, the quantum states are obtained. While cases for up to 3 qubits can be found in [17], [48], the newly derived 4-qubit states, along with the previous ones, are collected in Appendix 7.

2.2 The non-standard deformation↩︎

The Hopf algebra \(\mathcal{U}_h(\mathfrak{sl}(2, \mathbb{R}))\) is generated by \(\{H, Z_+, Z_-\}\) together with the usual commutators for \(\mathfrak{sl} (2, \mathbb{R})\) of Eq. 3 , providing \(H=2J_{z}, \, Z_{\pm}= J_{\pm}\). The generators \(\{H, Z_{\pm}\}\) act on the computational basis exactly the same as the generators of \(\mathcal{U}(\mathfrak{sl}(2, \mathbb{R}))\) [49]. Therefore, the entire effect of the deformation is encoded by the action on multipartite states.

The \(\mathcal{U}_{h}(\mathfrak{sl}(2, \mathbb{R}))\) co-algebra structure is given by the coproduct map, which reads [50] \[\begin{align} \Delta_h(H) &= & H\otimes 1 + 1 \otimes H + 2H \otimes \sum_{n=1}^\infty \left(\frac{hZ_{+}}{ 2} \right)^n + \sum_{n=1}^\infty \left(- \frac{hZ_{+}}{ 2} \right)^n \otimes 2H,\nonumber\\ \Delta_h(Z_+) &= & (1\otimes Z_+ + Z_+ \otimes 1)\left(\sum_{n=0}^\infty \left(-\frac{h^{2}}{4}\right)^n \; Z_+^n \otimes Z_+^n \right),\nonumber\\ \Delta_h(Z_-) &= & Z_- \otimes \sum_{n=0}^{\infty} (n+1) \left(\frac{h Z_{+}}{2} \right)^n + \sum_{n=0}^{\infty}\; (n+1) \left(-\frac{h Z_{+}}{2} \right)^n \otimes Z_- \nonumber \\ & &+ h\left(C_h-{\frac{H^{2}}{ 4}} \right) \otimes \sum_{n=1}^{\infty} n \left(\frac{h Z_{+}}{2} \right)^n - \sum_{n=1}^{\infty} n \left( -\frac{h Z_{+}}{2} \right)^n \otimes h \left(C_h-{\frac{H^{2}}{ 4}}\right) \nonumber \\ & &+ \left({\frac{h}{ 2}}\right)^2 Z_+Z_-Z_+ \otimes \sum_{n=2}^{\infty} (n-1) \left(\frac{h Z_{+}}{2} \right)^n + \sum_{n=2}^{\infty} (n-1) \left(-\frac{h Z_{+}}{2} \right)^n \otimes \left({\frac{h}{ 2} }\right)^2 Z_+Z_-Z_+,\nonumber\\ \label{eq:coproduct95j} \end{align}\tag{10}\] where \(C_{h}= Z_{+}Z_{-}+ H^{2}/4 -H/2\) is the Casimir element. In the limit \(h \to 0\) we recover the undeformed coproduct.

The Clebsch–Gordan coefficients for \(\mathcal{U}_{h}(\mathfrak{sl}(2, \mathbb{R}))\) (denoted by \({\cal C}^{j_1,j_2,j} _{m_1,m_2,m}(h)\)) can be derived from those of \(\mathcal{U}(\mathfrak{sl}(2, \mathbb{R}))\) by means of [49] \[\begin{align} \label{program1} &&{\cal C}^{j_1,j_2,j}_{m_1,m_2,m}(h) = \sum_{m_1+m_2=m} \mathcal{C}^{j_1,j_2,j}_{m_1,m_2,m} A^{m_1,m_2}_{n_1-m_1,n_2-m_2}, \end{align}\tag{11}\] where \[\begin{align} \label{program2} && A^{m_1,m_2}_{r,l} = a^{m_1,m_2}_{r,l} \frac{\alpha_{j_1,m_1+r}\, \alpha_{j_2,m_2+l}}{ \alpha_{j_1,m_1}\alpha_{j_2,m_2}},\nonumber\\ && \alpha_{j,m}= \sqrt{\frac{(j+m)!}{(j-m)!}},\nonumber \\ && a_{r,l}^{m_1,m_2}= (-1)^r 2^{-r-l} h^{r+l} (b_{r,l}^{m_1,m_2}-b_{r-1,l-1}^{m_1,m_2}), \\ &&b^{m_1,m_2}_{r,l} = \left\{ \begin{array}{ll} \frac{(-2m_{1}-r)_{l}(-2m_{2}-l)_{r}}{ r!\,l!} & if r\geq 0 andl\geq 0 , \nonumber \\[2mm] 0 & otherwise, \end{array} \right. \end{align}\tag{12}\] with \((a)_b\) the Pochhammer symbol. Note that the above \(h\)-Clebsch–Gordan coefficients generally yield unnormalized states. In light of the results in [15], we provide the explicit form of the quantum states in Appendix 8.

3 Kittel–Shore model↩︎

The algebraic construction of the aforementioned \(q\)-deformed and \(h\)-deformed states (see Appendices 7 and 8) finds its natural physical realization as the energy eigenstates of the deformed KS Hamiltonians.

A fundamental feature of this model is its maximal superintegrability. In the context of many-body physics, a system with \(N\) degrees of freedom is considered maximally superintegrable if it possesses \(2N-1\) independent integrals of motion (conserved quantities) that commute with the Hamiltonian, forming a non-abelian symmetry algebra [51], [52]. Unlike many-body systems that require the Bethe-ansatz [53] for their analytical resolution, the high degree of symmetry in the Kittel–Shore model allows for the energy spectrum to be determined exactly through purely algebraic methods. The theoretical justification for extending this framework to the quantum-deformed regime rests on the fundamental results established in [54]–[56]. These works prove that the integrability properties of a Hamiltonian system are rigorously preserved under a deformation process, provided that the underlying coalgebra symmetry remains intact. Since the \(q\)- and \(h\)- states are derived specifically from the deformed coproduct structures of \(\mathcal{U}_{q}(\mathfrak{sl}(2, \mathbb{R}))\) and \(\mathcal{U}_{h}(\mathfrak{sl}(2, \mathbb{R}))\), it is guaranteed that the resulting physical system retains its complete set of conserved quantities, ensuring that the deformation alters the correlation structure without breaking the solvable nature of the model.

To construct the deformed Kittel–Shore model, the Hamiltonian 1 is expressed in terms of the Casimir operators of the underlying algebra. Since the Casimir operator commutes with all generators of the algebra—thereby belonging to its center—it serves as the fundamental building block for a consistent deformation that preserves the integrability of the system. The Kittel–Shore Hamiltonian is thus reformulated as \[\hat{H}_{KS}= -\frac{I}{2}\left(\Delta^{(N)}(C)-\sum_{i=1}^{N}C^{(i)}\right),\] where \(\Delta^{(N)}\) denotes the \(N\)-th order coproduct: \[\Delta^{(N)}=(1^{\,\otimes (N-2)}\otimes \Delta^{(2)})\circ\Delta^{(N-1)}= (\Delta^{(2)}\otimes 1^{\,\otimes (N-2)})\circ\Delta^{(N-1)}. \label{fl}\tag{13}\] By replacing the undeformed Casimir operator with its \(q\)- or \(h\)-deformed counterpart, we derive the corresponding deformed versions of the model. It is worth noticing that the coproduct of the Casimir element is the Casimir of the coproduct.

3.1 The Standard \(q\)-Deformation↩︎

The Casimir operator of \(\mathcal{U} _{q}(\mathfrak{sl}(2, \mathbb{R}))\) is given by: \[C_{q}= L_{-}L_{+}+ [L_{z}]_{q}[L_{z}+\mathbf{I}]_{q}=L_{+}L_{-}+ [L_{z}]_{q} [L_{z}-\mathbf{I}]_{q}\, , \label{qcas}\tag{14}\] For a system of \(N\) qubits, each local site carries the spin-\(1/2\) representation, resulting in a local Casimir eigenvalue of \([1/2]_{q}[3/2]_{q}\). Consequently, the additive constant in the Hamiltonian effectively scales with the number of particles and the deformation parameter (\(\sum_{i=1}^{N}C_{q}^{(i)}=N[1/2 ]_{q}[3/2]_{q}\)).

In the absence of an external magnetic field, the energy spectrum of the undeformed KS model is given by \[E(j_{\alpha_i})=-\frac{I}{2}\left( j_{\alpha_i}(j_{\alpha_i}+1)-\frac{3N}{4} \right), \quad i=1,\ldots,r , \label{eigenvaluesKS}\tag{15}\] where \(j_{\alpha_i}\) denotes the total spin of the \(i\)-th irreducible component appearing in the decomposition of the \(N\)-fold tensor product of spin-\(1/2\) representations, \[\left(D^{1/2}\right)^{\otimes N} = D^{j_{\alpha_1}} \oplus D^{j_{\alpha_2}} \oplus \cdots \oplus D^{j_{\alpha_{r}}}.\] The label \(\alpha_i\) distinguishes the different irreducible components, including possible multiple copies with the same value of the total spin. For \(N\) spin-\(1/2\) representations, the total number of such components is \[r=\binom{N}{\lfloor N/2\rfloor}.\] Thus, different labels \(\alpha_i\) and \(\alpha_j\) may correspond to the same spin value, \(j_{\alpha_i}=j_{\alpha_j}\), reflecting the multiplicity of that irreducible representation. Since the KS Hamiltonian depends only on the total spin Casimir, the energy is determined by the value of \(j_{\alpha_i}\).

On the other hand, the energy spectrum of the \(q\)-KS model is determined by \[E_q(j_{\alpha_i})=-\frac{I}{2}\left( [j_{\alpha_i}]_{q}\,[j_{\alpha_i}+1]_{q} -N\left[\frac{1}{2}\right]_{q} \left[\frac{3}{2}\right]_{q} \right), \quad i=1,\ldots,r . \label{eigenvaluesqKS}\tag{16}\]

Concerning the associated eigenvectors, those of the \(q\)-deformed KS Hamiltonian remain orthogonal, as in the non-deformed case, but their structure is modified by the deformation, which reweights the components in the computational basis through the parameter \(q\). This is reflected in the explicit form of the Clebsch–Gordan coefficients of \(\mathcal{U}_{q}(\mathfrak{sl}(2, \mathbb{R}))\) in Eq. (9 ) and in the states given in Appendix 7. At the spectral level, the deformation alters the energy eigenvalues and modifies the probability amplitudes of the computational basis states, thereby reshaping the entanglement structure across the full Hilbert space.

3.2 The Jordanian \(h\)-Deformation↩︎

The Jordanian Casimir operator \(C_{h}\) is the central element of the algebra \(\mathcal{U}_{h}(\mathfrak{sl}(2, \mathbb{R}))\). In the basis \(\{Z_{+}, Z_{-}, H\}\), which is related to the classical generators (\(\{J_{+} ,J_{-},J_{z}\}\)) through a non-linear map [15], [50], the Casimir preserves its functional form relative to the undeformed case: \[C_{h}= Z_{+}Z_{-}+ \frac{H}{2}\left(\frac{H}{2}- \mathbf{I}\right). \label{hcas}\tag{17}\] According to [50] for finite-dimensional irreducible representations, the generators act on the standard basis \(|j, m\rangle\) as follows: \[\begin{align} H |j, m\rangle & = 2m |j, m\rangle, \nonumber \\ Z_{+}|j, m\rangle & = |j, m+1\rangle, \nonumber \\ Z_{-}|j, m\rangle & = (j+m)(j-m+1) |j, m-1\rangle. \end{align}\] Although this action seems to mimic the classical one, the structure of the coproduct \(\Delta_{h}\) is what distinguishes the system. The Jordanian deformation breaks hermiticity of the Hamiltonian because it introduces an inherently asymmetric coproduct for \(Z_+, Z_-\) (Eq.@eq:eq:coproduct95j ) that is fundamentally incompatible with the standard involution (\(*\)-structure) of the \(\mathfrak{sl}(2, \mathbb{R})\) algebra (which defines the adjoint relation \(Z_\pm^\dagger=Z_\mp\) ). To build a multi-site or many-body Hamiltonian, one must iteratively apply this asymmetric coproduct (Eq.@eq:fl ). Because the \(h\)-deformed Hamiltonian is non-Hermitian, one loses the unique orthonormal basis guaranteed by the spectral theorem, introducing an inherent arbitrariness into the eigenvectors. To resolve this geometric ambiguity, one can leverage the underlying algebraic symmetry: constructing a basis using h-Clebsch–Gordan coefficients uniquely privileges a specific set of states letting the quantum group’s structure naturally dictate the physical basis of the system. Indeed, as can be seen in Appendix 8, the eigenvectors of the \(h\)-KS model are smooth \(h\)- deformations of the usual undeformed Dicke states that include qubit states with \(k\) extra excitations endowed with weights of the type \(h^k\). Moreover, the spectrum remains invariant because the total Hamiltonian retains a triangular block structure in the uncoupled basis given by the \(h\)-Clebsch–Gordan coefficients, meaning the deformation alters only the off-diagonal coupling without shifting the bare energy levels. Consequently, the energy levels of the \(h\)-KS model are identical to those of the undeformed Kittel–Shore model for any value of \(h\). This distinguishes the Jordanian case from the \(q\)-KS model, where the energy levels are explicitly shifted by the \(q\)-deformation.

Spectral invariance has physical implications. From a thermodynamic perspective, the partition function \(Z = \text{Tr}(e^{-\beta H})\) and all derived observables—such as specific heat, internal energy, and magnetic susceptibility—remain unaffected by the \(h\)-deformation. Thus, the \(h\)-KS model is thermodynamically indistinguishable from the classical Kittel–Shore system.

However, the \(h\)-deformation is not physically trivial. While the eigenvalues are preserved, the eigenvectors are deformed, as seen in Eq. (11 ) and Appendix 8. These “Jordanian eigenstates” are no longer the standard states but rather complex superpositions that break the exchange symmetry of the computational basis. This makes the \(h\)-KS model a highly interesting framework for quantum information theory. The deformation parameter \(h\) modulates the quantum correlations without changing the energy of the states, providing a unique mechanism to control the correlation structure of a qubit network while keeping its thermal properties constant.

Identifying the specific physical system is vital for a meaningful interpretation of the results, as it allows the deformation parameters \(q\) and \(h\) to be viewed not merely as abstract mathematical parameters, but as physical coupling constants that induce anisotropy and modulate the correlation structure within a superintegrable qubit network. This physical modulation manifests as a redistribution of the state weights in both deformed cases, with the h-deformation uniquely introducing novel quantum states that are fundamentally absent in the undeformed model. From a physical point of view, these deformations allow us to parameterize experimental non-idealities like anisotropic noise or environmental coupling, giving the fidelities in Sec. 5 a distinct phenomenological relevance. By revealing how breaking or deforming the underlying symmetry reshapes the distribution and stability of quantum information across the eigenstates, the deformed Kittel–Shore model provides a concrete framework for entanglement engineering in next-generation quantum devices.

4 Comparison with experimental results↩︎

Before proceeding with a detailed characterization of the fidelities, it is interesting to establish a connection between the deformed eigenstates and the states typically targeted in quantum information experiments. This section presents a comparison between the experimental results and the theoretical predictions obtained using the \(q\)- and \(h\)-deformation models. To demonstrate the suitability of both deformations with respect to their potential to describe experimental output, we focus on the results of two papers, one for each deformation.

The deformed states considered in our approach are constructed algebraically by coupling spin-\(1/2\) representations through the corresponding deformed Clebsch–Gordan coefficients. On the experimental side, the target states are prepared by means of quantum circuits specifically designed for Dicke-type states; in particular, when mitigated data are provided, readout errors are reduced by calibrating the measurement response of the device and correcting the observed probability distributions before the comparison with theory.

As previously stated in the above Section, the \(q\)-deformation weights the states in a \(q\)-dependent manner without introducing any new state. However, \(h\)-deformation introduces new states proportional to \(h\) to the original states. Therefore, these deformations can be applied to different experimental situations, depending on whether new states appear.

We use \(q\)-deformation to describe the states of [39] and \(h\)-deformation for those of [57]. This, as we will see, is motivated by the fact that the former states present a negligible contribution to the new states, whereas in the latter, a significant contribution appears.

The choice of these two experimental works is guided by the availability and compatibility of the reported data. For the present comparison, it is not sufficient that a given work prepares a multipartite entangled state; the experimental probability distribution, or enough information to reconstruct it, must be explicitly provided, and the target states must be compatible with the families of states generated by the deformed algebraic construction. To the best of our knowledge, among the available experimental implementations with sufficiently detailed reported data, the results reported in [39] and [57] provide a particularly suitable benchmark, since they include experimentally accessible data for Dicke-type states and allow for a direct comparison with the theoretical probabilities predicted by the \(q\)- and \(h\)-deformed models.

4.1 Comparison of experimental results with \(q\)-deformation↩︎

This subsection presents a detailed comparison between the experimental results of a IBM Q Montreal device [39] (both for raw states directly obtained from the device and for error-mitigated states) and our theoretical predictions using \(q\)-deformation models. Specifically, our analysis focuses on a direct comparison between the states \(\ket{D_2^0}\), \(\ket{D_3^{-1/2}}\), \(\ket{D_4^{-1}}\), and \(\ket{D_4^{0}}\) studied in their paper and the corresponding deformed states obtained from our theoretical framework \(\ket{D_2^0}_q\), \(\ket{D_3^{-1/2}}_q\), \(\ket{D_4^{-1}}_q\), and \(\ket{D_4^{0}}_q\). Using the data available in [39], we obtain the optimal value of \(q\) by a least-squares fit of the probability of each component of the target states in the computational basis \[\mathcal{B}_N= \left\{ \ket{s_1s_2\cdots s_N}\;:\; s_\ell\in\{\uparrow,\downarrow\},\;\ell=1,\ldots,N \right\}, \label{eq22}\tag{18}\] where \[\ket{s_1s_2\cdots s_N} = \ket{s_1}\otimes\ket{s_2}\otimes\cdots\otimes\ket{s_N}.\] Results are shown in Figs. 1, 2 as well as in Tables 1, 2, 3, and 4. In all the tables, the columns specify the considered quantum state, the experimental probability directly measured from the device (Device), the theoretical probability from the \(q\)-deformation fitted to the device data (\(q\)-Device), the probability after applying error mitigation (Mitigated), and the theoretical probability from the \(q\)-deformation fitted to the mitigated data (\(q\)-Mitigated).

It can be seen that among all possible values, the best fit to the experimental data is consistently achieved for \(q\) close to 1 (with the maximum and minimum fitted values being 1.0638 and 0.9684, respectively). Moreover, our \(q\)-deformed model exhibits a significantly better agreement with the error-mitigated states than with the raw states directly obtained from the device because the latter are not properly normalized within the computational subspace considered in the fit. Since the theoretical \(q\)-deformed probabilities form a normalized distribution, measurement-error mitigation restores the total probability closer to one. This indicates that the deformation successfully captures the idealized target state once experimental noise is filtered out.

4.2 Comparison of experimental results with \(h\)-deformation↩︎

We will discuss the experimental results for the states \(\ket{D_3^{-1}}\) and \(\ket{D_4^{-2}}\) obtained in [57] through the IBM Q platform on the Internet. In this case, because new states emerge that differ from those of the undeformed model, the use of the \(h\)-deformation allows for a better description. Again, the optimal value of \(h\) is obtained by comparing the experimental data with our theoretical predictions and performing a least-squares adjustment of the probability of each component of the target states in the computational basis. The results are shown in Figs. 3 and 4, as well as in Tables 5 and ¿tbl:table:w43?. In all tables, the columns detail the considered quantum state (State), the experimentally obtained probability (Exp.), and the corresponding theoretical probability derived from the \(h\)-deformation model (\(h\)-def.). Note that in [57] only raw, unmitigated device data are reported.

Notice that in the theoretical model the \(h\)-deformed states deviate more significantly from the undeformed counterparts than in the \(q\)-deformation. This wider discrepancy arises because the \(h\)-deformed states include specific computational-basis components with non-vanishing probability, despite being completely absent in the undeformed case. Figs. 3 and 4 show this fact. However, one can also see that some of the computational basis components observed experimentally are not present in the theoretical \(h\)-deformed states although the overall structure of the theoretical distributions still resembles the experimental ones. These differences are mainly related to the relative weights of the allowed components. In particular, the \(h\)-deformation assigns the same probability to one-excitation terms, whereas the experimental data show an asymmetric distribution among them. This indicates that the \(h\)-deformation captures the appearance of additional components, but not the full redistribution of weights observed experimentally. Therefore, a more accurate description of these data may require a hybrid \((q,h)\)-deformation, since the observed deviations combine two effects: the appearance of additional computational-basis components, naturally captured by the \(h\)-deformation, and a redistribution of the weights of the expected components, which is characteristic of the \(q\)-deformation.

5 Fidelities of deformed states↩︎

The aim of this section is to study the fidelities of the \(q\)- and \(h\)-deformed states presented in [17], [58] and [15], respectively, with their corresponding undeformed counterparts.

5.1 Deformed states with two qubits↩︎

The fidelity of the \(q\)-deformed states \(\lvert M_{2}^{0}\rangle_{q}\) and \(\lvert D_{2}^{0}\rangle_{q}\) is given by \[F(\lvert M_{2}^{0}\rangle_{q=1},\lvert M_{2}^{0}\rangle_{q})=F(\lvert D_{2} ^{0}\rangle_{q=1},\lvert D_{2}^{0}\rangle_{q})=\frac{1}{2}+\frac{\sqrt{q}}{1+q} , \label{eq:fid}\tag{19}\] and is shown in Fig. 5 (left). For representation convenience, we consider \(q=e^{\eta}\). As expected, the fidelity equals one at \(q=1\) (\(\eta=0\)), where both states coincide with their undeformed counterparts. As \(\eta\) increases, the fidelity decreases monotonically, reflecting the gradual deviation from the original state caused by the standard deformation. Notice that the state fidelity between the deformed and undeformed configurations exhibits an asymptotic saturation rather than decaying to zero in the limit \(q\rightarrow \infty\). This behavior is deeply rooted in the Kashiwara crystal limit¹ [59]. As \(q\) approaches this critical limit, the continuous algebraic structure of the representation transitions into a rigid, combinatorially well-defined framework known as a crystal graph. Within a finite-dimensional Hilbert space—such as the spin-1/2 representation—the deformed states do not escape into orthogonal regions of the state space; instead, they smoothly converge toward the rigid basis vectors of the crystal lattice. This topological preservation bounds the geometric distance between the states, preventing total orthogonality and explaining the non-vanishing saturation of the quantum fidelity.

The fidelities between the \(h\)-deformed states and their corresponding undeformed states for two qubits are plotted in Fig. 5 (right) and can be written as: \[\begin{align} F( \ket{D_2^{-2}}_{h=0}, \ket{D_2^{-2}}_h)= \frac{16}{\left(h^{2}+4\right)^{2}}, \qquad F(\ket{M_2^{0}}_{h=0},\ket{M_2^{0}}_h)= \frac{2}{2+h^{2}}. \end{align}\]

As also observed in the \(q\)-deformation case, the fidelity \(F\) for both \(\lvert M_{2}^{0}\rangle_{h}\) and \(\lvert D_{2}^{-2}\rangle_{h}\) starts at \(1\) when \(h = 0\), where the states coincide exactly in the absence of deformation. As \(h\) increases, the fidelity decreases steadily toward zero, indicating that larger deformations push each deformed state farther away from its corresponding undeformed configuration. Notably, \(\lvert D_{2}^{-2}\rangle_{h}\) exhibits a higher sensitivity to the deformation than \(\lvert M_{2}^{0}\rangle_{h}\). By the time \(h \approx 6\), both fidelities drop below \(0.1\), rendering the deformed states nearly orthogonal to their original versions. The fact that \(\lvert D_{2}^{-2}\rangle_{h}\) loses fidelity faster suggests that its underlying algebraic or tensor product structure is more drastically reshaped by the \(h\)-deformation.

The rapid collapse stands in sharp contrast to the standard \(q\)-deformed case (Fig. 5, left), where the fidelity preserves a residual fraction and saturates at a non-zero value. This fundamental discrepancy is deeply rooted in the algebraic nature of each mechanism. The standard \(q\)-deformation smoothly guides the states toward the rigid, combinatorially well-defined framework of the Kashiwara crystal limit as the parameter approaches infinity. Within a finite-dimensional Hilbert space, this topological preservation prevents total orthogonality. Conversely, this saturation effect is entirely absent in the case of the Jordanian deformation. While the standard \(q\)-deformation scales the weights of the representation systematically, the \(h\)-deformation introduces non-symmetric terms involving nilpotent generators. As the deformation parameter increases, these nilpotent contributions dominate the transformation of the states, rapidly suppressing their overlap with the original configuration until complete orthogonality is reached. Consequently, while the \(q\)-deformation acts as a smoother mechanism that maintains a stable fidelity fraction, the Jordanian deformation produces a highly disruptive modification, allowing one to effectively distinguish the deformed state from its original ancestor with near certainty.

5.2 Deformed states with three qubits↩︎

When comparing the \(q\)-deformed states with \(3\) qubits with their corresponding undeformed counterparts, we find the following fidelities, shown in Fig. 6 (left) as a function of \(\eta\): \[\begin{align} & F(|D^{-1/2}_{3}\rangle_{q=1},|D^{-1/2}_{3}\rangle_{q})=F(|D^{1/2}_{3}\rangle_{q=1},|D^{1/2}_{3}\rangle_{q})=\frac{1+\sqrt{q}+q}{3(1-\sqrt{q}+q)}, \notag \\ & F(|M^{-1/2}_{3}\rangle_{q=1},|M^{-1/2}_{3}\rangle_{q})=\frac{(q^{3/2}+ 3q + 2)^{2}}{6(q+1)(q^{2}+q+1)}, \notag \\ & F(|M^{1/2}_{3}\rangle_{q=1},|M^{1/2}_{3}\rangle_{q})=\frac{\left((2 q+3) \sqrt{q}+1\right)^{2}}{6 (q+1) \left(q^{2}+q+1\right)}, \notag \\ & F(|V^{-1/2}_{3}\rangle_{q=1},|V^{-1/2}_{3}\rangle_{q})=F(|V^{1/2}_{3}\rangle_{q=1},|V^{1/2}_{3}\rangle_{q})=\frac{q+2 \sqrt{q}+1}{2 q+2}. \end{align}\]

Once again, the fidelity starts at \(1\) in the undeformed limit \(\eta=0\) and subsequently decreases as the deformation parameter increases. The impact of deformation varies slightly among different states. Regarding the specific configurations, the fidelities of \(\ket{D_3^{\pm1/2}}_{q}\) experience a steady decline before leveling off near \(0.35\) at large \(\eta\), indicating that a significant residual overlap survives the deformation. A comparable trend is found for \(\ket{V_3^{\pm1/2}}_{q}\), which stabilize around \(0.50\) and confirm a similar vulnerability to the \(q\)-mechanism. The remaining cases, however, deviate noticeably from this average behavior: while \(\ket{M_3^{-1/2}}_{q}\) proves highly sensitive—dropping rapidly toward an asymptote of approximately \(0.20\)—the \(\ket{M_3^{1/2}}_{q}\) state emerges as the most resilient configuration, decaying slowly to maintain a strong structural similarity to its original form at \(0.67\). As synthesized in Fig. 6 (left), the standard \(q\)-deformation induces distinct, parameter-dependent footprints based on the algebraic structure of each state, yet all share a common saturation profile. As discussed previously, this asymptotic freezing as \(\eta \to \infty\) is directly linked to the Kashiwara crystal limit, where the quantum state simplifies into a single dominant basis vector, forcing the fidelity to lock into a constant, non-vanishing value.

For the \(3\)-qubit states in the \(h\)-deformation case with \(3\) qubits, the following fidelities are found \[\begin{align} &&F( \ket{D_3^{-3}}_{h=0}, \ket{D_3^{-3}}_h)=\frac{16}{19 h^{4}+32 h^{2}+16}, \qquad F( \ket{D_3^{-1}}_{h=0}, \ket{D_3^{-1}}_h)=\frac{48}{9 h^{4}+32 h^{2}+48},\nonumber\\ &&F( \ket{M_3^{-1}}_{h=0}, \ket{M_3^{-1}}_h)=\frac{6}{5 h^{2}+6}, \qquad \qquad \qquad \, F( \ket{M_3^{1}}_{h=0}, \ket{M_3^{1}}_h)=\frac{2}{3 h^{2}+2} , \nonumber\\ &&F( \ket{V_3^{-1}}_{h=0}, \ket{V_3^{-1}}_h)= F( \ket{V_3^{1}}_{h=0}, \ket{V_3^{1}}_h)=\frac{2}{h^{2}+2}. \end{align}\] The results are shown in Fig. 6 (right). The behavior is totally analogous to that of two-qubit states: the fidelity decreases with \(h\) until it reaches zero for all states. The rate at which fidelity decreases depends on the state: \(|D_{3} ^{-3}\rangle_{h}\) and \(|D_{3}^{-1}\rangle_{h}\) lose fidelity more rapidly for increasing \(h\) compared to \(|V_{3}^{-1}\rangle_{h}\) and \(|M_{3}^{-1}\rangle_{h}\), while \(|M_{3}^{1}\rangle_{h}\) shows an intermediate behavior.

5.3 Deformed states with four qubits↩︎

A comparison between the \(q\)-deformed \(4-\)qubit states and their ideal undeformed counterparts yields the following fidelities: \[\begin{align} F(|D^{-1}_{4}\rangle_{q=1},|D^{-1}_{4}\rangle_{q}) & =F(|D^{1}_{4}\rangle_{q=1},|D^{1}_{4}\rangle_{q})=\frac{\left(\sqrt{q}+1\right)^{2}(q+1)}{4 \left(q^{2}+1\right)}, \notag \\ F(|D^{0}_{4}\rangle_{q=1},|D^{0}_{4}\rangle_{q}) & =F(|M^{0}_{4}\rangle_{q=1},|M^{0}_{4}\rangle_{q})=\frac{(q+1)^{2}\left(q+\sqrt{q}+1\right)}{6 \left(q-\sqrt{q}+1\right) \left(q^{2}+1\right)}, \notag \\ F(|M^{-1}_{4}\rangle_{q=1},|M^{-1}_{4}\rangle_{q}) & =\frac{\left(q+\sqrt{q}+1\right) \left(q^{3/2}+3 q-3 \sqrt{q}+3\right)^{2}}{12 \left(q-\sqrt{q}+1\right) \left(q^{3}+q^{2}+q+1\right)}, \notag \\ F(|M^{1}_{4}\rangle_{q=1},|M^{1}_{4}\rangle_{q}) & = \frac{\left(q+\sqrt{q}+1\right) \left(3 q^{3/2}-3 q+3 \sqrt{q}+1\right)^{2}}{12 (q+1) \left(q-\sqrt{q}+1\right) \left(q^{2}+1\right)}, \notag \\ F(|V^{-1}_{4}\rangle_{q=1},|V^{-1}_{4}\rangle_{q}) & = \frac{\left(q^{3/2}+3 q+2\right)^{2}}{6 (q+1) \left(q^{2}+q+1\right)}, \notag \\ F(|V^{0}_{4}\rangle_{q=1},|V^{0}_{4}\rangle_{q}) & = \frac{\left(5 q^{3/2}+q^{2}+5 \sqrt{q}+1\right)^{2}}{12 (q+1)^{2}\left(q^{2}+q+1\right)}, \notag \\ F(|V^{1}_{4}\rangle_{q=1},|V^{1}_{4}\rangle_{q}) & = \frac{\left(2 q^{3/2}+3 \sqrt{q}+1\right)^{2}}{6 (q+1) \left(q^{2}+q+1\right)}, \end{align}\] \[\begin{align} F(|T^{0}_{4}\rangle_{q=1},|T^{0}_{4}\rangle_{q}) & =\frac{\left(q^{3/2}+2 q^{2}+6 q+\sqrt{q}+2\right)^{2}}{12 (q+1)^{2}\left(q^{2}+q+1\right)}, \notag \\ F(|R^{-1}_{4}\rangle_{q=1},|R^{-1}_{4}\rangle_{q}) & =F(|R^{1}_{4}\rangle_{q=1},|R^{1}_{4}\rangle_{q})=\frac{(\sqrt{q}+1)^{2}}{2 (q+1)}, \notag \\ F(|R^{0}_{4}\rangle_{q=1},|R^{0}_{4}\rangle_{q}) & =F(|Q^{0}_{4}\rangle_{q=1},|Q^{0}_{4}\rangle_{q})=\frac{\left(\sqrt{q}+1\right)^{4}}{4 (q+1)^{2}}. \end{align}\] These fidelities are represented as functions of the deformation parameter in Figs. 7 and 8.

The same general trend in fidelity as it increases is observed as in the cases of \(N=2\) and 3 qubits. The robustness of the fidelity also depends on the class of entanglement being considered. Particular configurations such as \(\ket{D_4^{\pm1}}_{q}\) and \(\ket{D_4^{0}}_{q}\), and especially \(\ket{M_4^{-1}}_{q}\) and \(\ket{V_4^{0}}_{q}\), exhibit the most pronounced reduction in fidelity as \(\eta\) increases.

Crucially, these different rates of fidelity decay cannot be explained solely in terms of the number of computational-basis components appearing in each state. Rather, the relevant point is how the deformation redistributes the weight among those components, and in particular whether the component selected in the strong-deformation regime was already relevant in the undeformed state.

There are two distinct mechanisms that can lead to a slower decay of the fidelity. The first occurs for states involving a smaller number of computational-basis components, such as the states \(\ket{R_4^{\pm1}}_{q}\). Since these states are supported on fewer configurations, the selection of one dominant component by the deformation still leaves a relatively large overlap with the undeformed state. In this case, robustness is mainly a consequence of the reduced number of components among which the initial probability weight is distributed.

The second mechanism is different. A state may contain several components, but the component favored by the deformation may already carry a large weight in the undeformed state. This is the case of \(|M_4^{1}\rangle_q\). Although this state contains four computational-basis components, the deformation drives the state towards the component that already has the largest amplitude at \(q=1\). As a result, the fidelity remains large even in the strong-deformation regime. Therefore, the robustness of \(|M_4^{1}\rangle_q\) does not come from having fewer components, but from the fact that the deformation selects the component that was already dominant in the undeformed state.

On the other hand, fidelities arising when comparing the \(h\)-deformed states with \(4\) qubits with their corresponding undeformed counterparts can be written as \[\begin{align} F(|D^{-4}_{4}\rangle_{h=0},|D^{-4}_{4}\rangle_{h}) & =\frac{256}{81 h^{8}+848 h^{6}+2144 h^{4}+1280 h^{2}+256}, \notag \\ F(|D^{-2}_{4}\rangle_{h=0},|D^{-2}_{4}\rangle_{h}) & =\frac{16}{41 h^{4}+40 h^{2}+16}, \notag \\ F(|D^{0}_{4}\rangle_{h=0},|D^{0}_{4}\rangle_{h}) & =\frac{24}{9 h^{4}+20 h^{2}+24}, \notag \\ F(|M^{-2}_{4}\rangle_{h=0},|M^{-2}_{4}\rangle_{h}) & =\frac{48}{9 h^{6}+59 h^{4}+104 h^{2}+48}, \notag \\ F(|M^{0}_{4}\rangle_{h=0},|M^{0}_{4}\rangle_{h}) & =\frac{6}{11 h^{2}+6}, \notag \\ F(|M^{2}_{4}\rangle_{h=0},|M^{2}_{4}\rangle_{h}) & =\frac{1}{3 h^{2}+1},\notag\\ F(|V^{-2}_{4}\rangle_{h=0},|V^{-2}_{4}\rangle_{h}) & =\frac{96}{9 h^{6}+62 h^{4}+128 h^{2}+96}, \notag \\ F(|V^{0}_{4}\rangle_{h=0},|V^{0}_{4}\rangle_{h}) & =\frac{6}{7 h^{2}+6}, \notag \\ F(|V^{2}_{4}\rangle_{h=0},|V^{2}_{4}\rangle_{h}) & =\frac{2}{3 h^{2}+2}, \notag \\ F(|T^{0}_{4}\rangle_{h=0},|T^{0}_{4}\rangle_{h}) & =\frac{12}{9 h^{4}+20 h^{2}+12}, \notag \\ F(|R^{-2}_{4}\rangle_{h=0},|R^{-2}_{4}\rangle_{h}) & =\frac{32}{h^{6}+10 h^{4}+32 h^{2}+32}, \notag \\ F(|R^{0}_{4}\rangle_{h=0},|R^{0}_{4}\rangle_{h}) & =F(|R^{2}_{4}\rangle_{h=0},|R^{2}_{4}\rangle_{h})=\frac{2}{h^{2}+2}, \notag \\ F(|Q^{0}_{4}\rangle_{h=0},|Q^{0}_{4}\rangle_{h}) & =\frac{4}{h^{4}+4 h^{2}+4}. \end{align}\] These fidelities are represented as functions of the deformation parameter in Figs. 9 and 10. Again, the fidelity decreases monotonically as \(h\) increases from the undeformed limit (\(F=1\)) to zero. As the deformation increases, the overlap with the reference state gradually decreases.

As in the 3-qubit case, the rate of fidelity decay strongly depends on the internal structure of the state. Dicke states, such as \(\ket{D^{-4}_{4}}_{h}\) and \(\ket{D^{-2}_{4}}_{h}\), exhibit a markedly faster decay than other multipartite configurations like \(\ket{V^{0}_{4}}_{h}\) and \(\ket{R^{2}_{4}}_{h}\). This difference can be traced back to the highly symmetric nature of Dicke states, which are constructed as balanced superpositions of all permutations of the computational-basis configurations with a fixed number of excitations. The Jordanian \(h\)-deformation, characterized by its asymmetric coproduct and nilpotent contributions, breaks this symmetry and induces a rapid redistribution of amplitudes, thereby suppressing the overlap with the undeformed state. In contrast, states with less symmetric or more heterogeneous structures are less sensitive to this mechanism, as their correlations are not critically tied to a fine-tuned permutation symmetry.

5.4 Fidelities of \(q\)-Dicke states for any \(N\)↩︎

While previous sections focused on small system sizes, the algebraic structure of Dicke states allows for a deeper analytical treatment. Unlike non-Dicke states, which require the step-by-step application of the coproduct (or through \(q\)-Clebsch–Gordan coefficients) without a general closed expression, Dicke states can be constructed via explicit closed-form formulas. This enables us to derive exact analytical expressions for the fidelity under both deformation frameworks that hold for any \(N\), including the asymptotic limit \(N \to \infty\). In this subsection, we will focus on the standard deformation, whereas the non-standard one will be addressed in the following subsection.

Consider a one-dimensional spin chain of length \(N\) containing \(k\) excitations. The standard Dicke state in the undeformed limit \(q \to 1\) is a totally symmetric and uniformly weighted superposition of all possible basis states with exactly \(k\) excitations, as defined in Eq. 2 . So far, we have labelled the Dicke states for the standard deformation as \(|D_N^m\rangle_q\), where \(m\) is the eigenvalue of the \(L_z\) operator in the corresponding \(j = N/2\) irreducible representation. In this Section, for the sake of simplicity in the notation, we will parameterize the Dicke states by their number of excitations \(k\), instead of by the \(L_z\) eigenvalue \(m\). Thus, we define \(\ket{D_N^{(k)}} \equiv \ket{D_N^{m=k-N/2}}\), where \(k\) represents the number of up-spins.

The \(q\)-deformed analogue of the aforementioned Dicke state, arising for instance from the coproduct action of \(U_{q}(\mathfrak{sl}(2, \mathbb{R}))\), introduces a parameter-dependent weight for each basis configuration. It generates coefficients that depend on a weight function \(p(S)\) [60]: \[\label{eq31} \vert D^{(k)}_{N}\rangle_{q}= \frac{1}{\sqrt{\binom{N}{k}_{q}}}\sum_{S \in \mathcal{B}_{N,k}}q^{\frac{1}{2} p(S)}|S\rangle,\tag{20}\] where \(\binom{N}{k}_{q}\) is the \(q\)-binomial coefficient defined as \[\begin{align} \binom{N}{k}_{q}=\frac{[N]_{q}!}{[N-k]_{q}!\,\, [k]_{q}!},\qquad \textrm{with} \,\,\, [N]_q! = [N]_q [N-1]_q \cdots [1]_q, \end{align}\] and \(p(S)\) is a configuration-dependent integer exponent that characterizes the deformation \[\begin{align} p(S)=-k\left( \frac{N+1}{2}\right) + \sum_{i=1}^{k}n_{i}, \end{align}\] with \(n_{i}\) being the position of the \(i\)-th excitation in the chain.

Since the \(q\)-deformed Dicke state is normalized (\({}_{q}\langle D^{(k)}_{N}| D^{(k)}_{N} \rangle_{q}= 1\)) and the basis \(\{\vert S \rangle\}\) is orthonormal, the following identity strictly holds: \[\sum_{S \in \mathcal{B}_{N,k}}\left( q^{\frac{1}{2} p(S)}\right)^{2}= \sum_{S \in \mathcal{B}_{N,k}}q^{p(S)}= \binom{N}{k}_{q}.\] This establishes a fundamental polynomial identity: the sum of the variable raised to the weight function over all permutations generates the deformed binomial coefficient in that same variable. This means that for any scalar parameter \(A\), we have \(\sum_{S}A^{p(S)}= \binom{N}{k}_{A}\).

Now, we evaluate the overlap (probability amplitude) between the standard undeformed state and the \(q\)-deformed state using their inner product: \[\label{eq:overlap95exact} _{q=1}\langle D^{(k)}_{N}\vert D^{(k)}_{N}\rangle _{q}= \frac{1}{\sqrt{\binom{N}{k}}}\frac{1}{\sqrt{\binom{N}{k}_{q}}} \sum_{S \in \mathcal{B}_{N,k}}q^{\frac{1}{2} p(S)}.\tag{21}\] Using the polynomial identity from above and evaluating it at the variable \(A = q^{1/2}\), the summation reduces exactly to: \[\sum_{S \in \mathcal{B}_{N,k}}(q^{1/2})^{p(S)}= \binom{N}{k}_{q^{1/2}}.\] Therefore, the overlap becomes: \[_{q=0} \langle D^{(k)}_{N}\vert D^{(k)}_{N}\rangle _{q}= \frac{\binom{N}{k}_{q^{1/2}}}{\sqrt{\binom{N}{k}\binom{N}{k}_{q}}} .\] The fidelity \(F\), defined for pure states (as Dicke ones) as the squared modulus of the overlap \(F = |_{q=0}\langle D^{(k)}_{N}\vert D^{(k)}_{N}\rangle _{q}|^{2}\), can thus be expressed in a compact form explicitly independent of the exact basis configuration as: \[F(|D^{(k)}_{N}\rangle_{q=1},|D^{(k)}_{N}\rangle_{q}) = \frac{\left[ \binom{N}{k}_{q^{1/2}}\right]^{2}}{\binom{N}{k}\binom{N}{k}_{q}}, \label{eq:fidelity95q95finite}\tag{22}\] assuming² \(q \in \mathbb{R}^{+}\). While this formalism gives the exact evaluation for any finite \(q\), its underlying mechanics dictate the system’s behavior in the extreme deformation limit, \(q \to \infty\).

In the computational basis representation 20 , the amplitudes scale as \(q^{\frac{1}{2}p(S)}\). When approaching the \(q \to \infty\) threshold, the maximally-weighted state is the one maximizing the function \(p(S)\). The maximum possible value of \(p(S)\) for a system of size \(N\) with \(k\) excitations is exactly \(k(N-k)/2\). This upper bound is reached by a unique basis state in which all the excitations are maximally clustered at one end of the chain, \(\vert \downarrow \dots\downarrow\uparrow\dots\uparrow\rangle\).

Because the amplitude of this unique maximal-weight state \(|S_{\text{max}}\rangle\) grows exponentially faster than all other states, the normalization naturally drives all competing amplitudes to zero. Physically, this induces a spatial “collapse” of the \(q\)-Dicke state. While for \(q=1\) the system exists as a completely delocalized and highly entangled superposition, at \(q \to \infty\) it collapses into a completely localized separable product state: \[\lim_{q \to \infty}\vert D^{(k)}_{N}\rangle_{q}= \vert S_{\text{max}}\rangle .\]

We can independently verify the analytical limit of the fidelity via our exact closed formula, by taking the asymptotic expansion of the \(q\)-binomial for \(q \gg 1\): \[\binom{N}{k}_{q}\sim q^{\frac{k(N-k)}{2}}.\] This exponent can be rigorously derived by taking the limit of the symmetric \(q\)-number for \(q \gg 1\), which behaves as \[[y]_{q}= \frac{q^{y/2}- q^{-y/2}}{q^{1/2}- q^{-1/2}}\sim q^{\frac{y-1}{2}} .\] Consequently, the asymptotic scaling of the \(q\)-factorial is given by \[[n]_{q}! \sim \prod_{j=1}^{n}q^{\frac{j-1}{2}}= q^{\frac{n(n-1)}{4}}.\] Substituting this expression back into the definition of the \(q\)-binomial coefficient yields \[\binom{N}{k}_{q}= \frac{[N]_{q}!}{[N-k]_{q}! [k]_{q}!}\sim q^{E},\] where the exponent \(E\) is precisely \[E = \frac{N(N-1)}{4}- \frac{(N-k)(N-k-1)}{4}- \frac{k(k-1)}{4} = \frac{k(N-k)}{2}.\]

Applying this limiting behavior to the exact analytical expression derived previously: \[\lim_{q \to \infty} F(|D^{(k)}_{N}\rangle_{q=0},|D^{(k)}_{N}\rangle_{q}) = \frac{\left[ (q^{1/2})^{\frac{k(N-k)}{2}}\right]^{2}}{\binom{N}{k}\,\, q^{\frac{k(N-k)}{2}}} = \frac{q^{\frac{k(N-k)}{2}}}{\binom{N}{ k }\,\, q^{\frac{k(N-k)}{2}}}= \frac{1}{\binom{N}{k}} . \label{eq:fidelity95limit95q95inf}\tag{23}\] The \(q\)-dependent factors cancel out, leading to a purely combinatorial result.

From a logical consistency viewpoint, this analytical result perfectly aligns with the physical interpretation of the state collapse (Kashiwara crystal limit). If the deformed state in the \(q \to \infty\) limit approaches a single fully localized basis state \(\vert S_{\text{max}}\rangle\), and the non-deformed reference state (\(q =1\)) is a uniform maximal superposition over all \(\binom{N}{k}\) possible configurations, the transition overlap is simply the geometric projection of the uniform state onto the fully localized state. The squared magnitude of this trivial projection yields \(1/\binom{N}{k}\) directly. These results explain the numerical findings for small \(N\).

5.4.1 Fidelity per site in the thermodynamic limit↩︎

To analyze the thermodynamic limit of the fidelity between the \(q\)-deformed Dicke state and its undeformed counterpart, we define the corresponding scaling variables. Let \(N\) be the system size and \(k\) the number of excitations. We introduce the macroscopic excitation density \(x = k/N\), where \(x \in [0,1]\), and parametrize the quantum deformation as \(q = e^\eta\).

The fidelity per site, \(f\), is defined through the thermodynamic rate function of the total fidelity \(F\) as[61], [62] \[\ln f(x,\eta) = \lim_{N \to \infty} \frac{1}{N} \ln F(x,\eta).\] We study this quantity in two distinct scaling regimes of the deformation parameter \(\eta\).

5.4.1.1 Weak deformation regime (\(\eta \ll 1\))

The thermodynamic limit \(N\to\infty\) and the weak deformation limit \(\eta\to 0\) do not commute, as we will see below from the small-\(\eta\) expansion of the exact fidelity of Eq. 22 . Making an expansion in \(\eta\), one finds \[[n]_q \approx n\left[ 1+\frac{\eta^2}{24}(n^2-1) +\mathcal{O}(\eta^4) \right],\] so \[[n]_{q^{1/2}} \approx n\left[ 1+\frac{\eta^2}{96}(n^2-1) +\mathcal{O}(\eta^4) \right].\] Expanding the logarithm of the symmetric \(q\)-numbers for \(\eta\ll1\), one finds: \[\ln [n]_q \approx \ln n + \frac{\eta^2}{24}(n^2 - 1) + \mathcal{O}(\eta^4), \quad \ln [n]_{q^{1/2}} \approx \ln n + \frac{\eta^2}{96}(n^2 - 1) + \mathcal{O}(\eta^4).\] Transforming the operator product of the quantum binomial coefficient \(\binom{N}{k}_q\) into a sum of logarithms yields: \[\ln \binom{N}{k}_q = \ln \binom{N}{k} + \frac{\eta^2}{24} \left[ \sum_{n=1}^N (n^2 - 1) - \sum_{n=1}^k (n^2 - 1) - \sum_{n=1}^{N-k} (n^2 - 1) \right].\] Evaluating the sums using Faulhaber’s formula \[S(y) = \sum_{n=1}^y (n^2-1) = \frac{y(y-1)(2y+5)}{6},\] the difference algebraically collapses to \(S(N) - S(k) - S(N-k) = k(N - k)(N + 1)\). Introducing the macroscopic filling fraction \(x = k/N\), the deformed coefficients are approximated as: \[\begin{align} \ln \binom{N}{k}_q &= \ln \binom{N}{k} + \frac{\eta^2}{24}N^2(N + 1)x(1 - x) + \mathcal{O}(\eta^4), \\ \ln \binom{N}{k}_{q^{1/2}} &= \ln \binom{N}{k} + \frac{\eta^2}{96}N^2(N + 1)x(1 - x) + \mathcal{O}(\eta^4). \end{align}\] Therefore, the logarithm of the fidelity behaves as \[\frac{1}{N}\ln F(x,\eta) = -\frac{\eta^2}{48}N(N+1)x(1-x) +\mathcal{O}(\eta^4).\]

This expression shows that keeping \(\eta\) fixed while taking \(N\to\infty\) leads to an immediate orthogonality catastrophe, since the intensive logarithmic fidelity diverges as \(N^2\eta^2\). Hence, in order to obtain a finite thermodynamic rate function, the deformation parameter must scale inversely with the system size. We therefore introduce the double scaling limit \[\eta=\frac{\tilde{\eta}}{N},\] where \(\tilde{\eta}\) is a finite intensive control parameter. This scaling in the deformation parameter \(\eta\) was also derived in [17] in order to keep the extensibility of the Kittel–Shore model (maintaining a finite quotient between energy and volume in the thermodynamic limit).

Substituting this scaling into the previous expression gives \[\frac{1}{N}\ln F (x,\tilde{\eta}) = -\frac{\tilde{\eta}^2}{48} \left(1+\frac{1}{N}\right) x(1-x) +\mathcal{O}\!\left(\frac{\tilde{\eta}^4}{N^2}\right).\] Consequently, in the thermodynamic limit, \[\ln f(x,\tilde{\eta}) \equiv \lim_{N\to\infty}\frac{1}{N}\ln F (x,\tilde{\eta}) = -\frac{\tilde{\eta}^2}{48}x(1-x)+ \mathcal{O}\!\left(\tilde{\eta}^4\right).\]

This result identifies the weak-deformation regime as a Gaussian decay regime for the fidelity per site. The decay rate is maximal at half filling, \(x=1/2\), where \[\ln f\left(\frac{1}{2},\tilde{\eta}\right) = -\frac{\tilde{\eta}^2}{192}.\] Thus the Dicke state with \(k=N/2\) excitations presents the lowest fidelity per site in the weak \(q\)-deformation regime. This state is also the one which exhibits maximal permutation entropy. Notice that a Dicke state \(\lvert D_N^{(k)} \rangle\) is a uniform superposition (with identical coefficients) of all computational basis states that contain exactly \(k\) up-spins (or excitations) and \(N-k\) down-spins. The number of states satisfying this condition is given by the binomial coefficient \(\binom{N}{k}\). One can associate with an “entropy” that measures how dispersed or distributed the state is across the computational basis. At the point \(x=1/2\), the Dicke state is composed of the largest possible number of computational basis vectors. By possessing the maximum number of entangled and interchangeable components (permutations), it exhibits maximal permutation entropy.

5.4.1.2 Strong deformation regime (\(\eta\to \infty\))

In the regime where \(\eta\) is held fixed as a large constant while \(N \to \infty\) (which formally includes the extreme limit \(\eta \to \infty\)), the asymptotic behavior is thus entirely dictated by the asymptotic expansion of the \(q\)-binomial coefficients in Eq. 22 . Remarkably, for any finite \(\eta\) independent of \(N\), the leading order terms induced by the quantum deformation exactly cancel out between the numerator and the denominator in Eq. 22 . Consequently, the asymptotic behavior is given by the classical binomial coefficient present in the denominator. Introducing the density \(x = k/N\) and applying Stirling’s approximation to the binomial coefficient, the logarithm of the total fidelity reads: \[\ln F(x) \simeq -\ln \binom{N}{xN} \simeq N \left[ x \ln x + (1-x) \ln(1-x) \right].\] Dividing by the system size \(N\), we find that the fidelity per site \(\ln f\) collapses to a constant independent of the perturbation strength \(\eta\) and entirely determined by the classical Shannon entropy of the system \(\mathcal{H}(x)\): \[\ln f(x) = x \ln x + (1-x) \ln(1-x) = - \mathcal{H}(x).\] Then, the fidelity decreases as the filling fraction moves away from the extremal values \(x=0\) and \(x=1\), since \(\mathcal{H}(0)=\mathcal{H}(1)=0\) and \(f=1\). The minimum value is reached at half filling, \(x=1/2\) (i.e. \(k=N/2\)), where \(\mathcal{H}(1/2)=\ln 2\) and therefore \[\ln f(1/2)=-\ln 2, \qquad F(1/2)\sim 2^{-N}.\] Thus, for any fixed filling fraction \(0<x<1\), the total fidelity approaches zero in the thermodynamic limit, while the fidelity per site remains finite. Consequently, the weak- and strong-deformation regimes are proven to be different: for small deformations, the fidelity shows a Gaussian dependence on the deformation parameter, whereas the strong-deformation regime is controlled by the binary Shannon entropy \(\mathcal{H}(x)\), independently of the value of the deformation parameter.

5.5 Fidelities for \(h\)-Dicke states for any \(N\)↩︎

Quantum fidelities must always be evaluated using properly normalized states to ensure a well-defined physical interpretation. Since the \(h\)-deformed Clebsch–Gordan coefficients do not, in general, produce normalized states, we distinguish between the algebraic (non-normalized) states \(\ket{\tilde{\Phi}_h}\) and their normalized counterparts \(\ket{\Phi}_h\), which are related by means of: \[\ket{\Phi}_h = \frac{\ket{\tilde{\Phi}}_h}{\sqrt{_h\langle \tilde{\Phi}| \tilde{\Phi}\rangle_{h}}}.\]

The non-normalized \(h\)-deformed Dicke states \(|\tilde{D}^{-N}_{N}\rangle_{h}\) are expressed as a superposition of the computational basis states \(\ket{S}= \ket{s_1 s_2 \dots s_N}\) belonging to the basis \(\mathcal{B}_N\) defined in Eq. 18 as explained in [15], i.e.: \[\label{eq58}|\tilde{D}^{-N}_{N}\rangle_{h}= \sum_{S\in \mathcal{B}_N} \xi_{n_\uparrow}(P_{\uparrow}(S)) \left(\frac{h}{2}\right)^{n_\uparrow(S)} |S\rangle,\tag{24}\] where \(P_{\uparrow}(S) = \{j \mid s_{j}= \uparrow\}\) is the set of indices indicating the positions of the up-spins in \(\ket{S}\), and \(n_{\uparrow}(S) = |P_{\uparrow} (S)|\) is the total number of up-spins in \(S\). The structural factor \({\xi}(P_{\uparrow}(S))\) can be elegantly expressed in terms of elementary symmetric polynomials. Given the positional function \(g(j) = 2j - N - 1\), we let \(e_{m}(P_{\uparrow})\) be the \(m\)-th elementary symmetric polynomial of the values \(\{g(j) \mid j \in P_{\uparrow}\}\): \[\begin{align} e_{0}(P_{\uparrow}) & = 1,\nonumber \\ e_{1}(P_{\uparrow}) & = \sum_{j \in P_\uparrow}g(j), \nonumber \\ e_{2}(P_{\uparrow}) & = \sum_{j_1 < j_2 \in P_\uparrow}g(j_{1})g(j_{2}), \nonumber \\ & \dots \nonumber \\ e_{m}(P_{\uparrow}) & = \sum_{j_1 < j_2 < \dots < j_m \in P_\uparrow}g(j_{1})g(j_{2})\dots g(j_{m}). \end{align}\] By convention, \(e_{m}(P_{\uparrow}) = 0\) if \(m > n_{\uparrow}\). The coefficient \(\xi\) is completely defined as: \[{\xi}_{n_\uparrow}(P_{\uparrow}) = \sum_{m=0}^{\lfloor n_\uparrow/2 \rfloor}\mathcal{C}_{m}(N) e_{n_\uparrow-2m} (P_{\uparrow}), \label{eq:gamma95general}\tag{25}\] where \(\mathcal{C}_{m}(N)\) are particular polynomial coefficients in \(N\) (\(\mathcal{C}_{0}(N) = 1\), \(\mathcal{C}_{1}(N) = N\), \(\mathcal{C}_{2}(N)=N(3N-2)\)), whose leading terms precisely scale as \(( 2m-1)!! N^{m}\) [15].

In the undeformed limit (\(h \to 0\)), the only surviving term is \(|\downarrow\downarrow\dots\downarrow\rangle\), which corresponds to zero excitations (\(n_{\uparrow}= 0\)). The term with zero excitations in \(|\tilde{ D}^{-N}_{N}\rangle_{h}\) is the one with coefficient \(\xi_{0}= \mathcal{C}_{0}e_{0}= 1\). Consequently, there is only one nonvanishing contribution to the fidelity, leading to \[\label{eq:fidelity95D}F(|D^{-N}_{N}\rangle_{h=0},|D^{-N}_{N}\rangle_{h})= \frac{|_{h=0}\langle D^{-N}_{N}| \tilde{D}^{-N}_{N}\rangle_{h}|^{2}}{|_h\langle \tilde{D}^{-N}_{N}| \tilde{D}^{-N}_{N}\rangle_{h} |} = \frac{1}{\displaystyle \sum_{S \in \mathcal{B}_N}\left[ \xi_{n_\uparrow}(P_{\uparrow}(S)) \right]^{2}\left(\frac{h}{2}\right)^{2n_\uparrow(S)}} .\tag{26}\]

Another interesting state with an analytical closed formula [15] is the \(h\)-Dicke state for a single excitation, also called \(W\)-state (\(| D^{(1)}_{N}\rangle_{h} \equiv | W_{N}\rangle_{h}\)). Its non-normalized form is given by: \[\label{eq64}|\tilde{W}_{N}\rangle_{h}= \sum_{S\in \mathcal{B}_N}{\xi}^{\prime}_{n_\uparrow}(P_{\uparrow} (S))\left(\frac{h}{2}\right)^{n_\uparrow(S)-1}|S\rangle,\tag{27}\] with: \[\begin{align} \label{eq65}{\xi}^{\prime}_{n_\uparrow}(P_{\uparrow}) & = \sum_{m=0}^{\lfloor (n_\uparrow-1)/2 \rfloor}(2m+1) \mathcal{C}_{m}(N) e_{n_\uparrow-1-2m}(P_{\uparrow}) \nonumber \\ & - 2 \sum_{m=0}^{\lfloor (n_\uparrow-3)/2 \rfloor}\binom{2m+3}{3}\mathcal{C}_{m}(N) e_{n_\uparrow-3-2m}(P_{\uparrow}). \end{align}\tag{28}\] In the limit \(h\to 0\), the only surviving contributions of \(|\tilde{W}_{N}\rangle_{h}\) are those with \(n_\uparrow=1\). Hence, \(\ket{ W_N}_{h=0}\) is the uniform superposition of the \(N\) single-excitation computational basis states: \[\ket{W_N}_{h=0} = \frac{1}{\sqrt{N}} \sum_{\substack{S \in \mathcal{B}_N \\ n_\uparrow(S)=1}} \ket S.\] Since there are \(N\) terms with one excitation in the deformed state \(\ket{\tilde{W}_N}_h\), the fidelity is \[\label{eq:fidelity95W}F(|W_{N}\rangle_{h=0},|W_{N}\rangle_{h})= \frac{|_{h=0}\langle W_{N}| \tilde{W}_{N}\rangle_{h}|^{2}}{|_h\langle \tilde{W}_{N}| \tilde{W}_{N}\rangle_{h} |} = \frac{N}{\displaystyle \sum_{S \in \mathcal{B}_N} \left[\xi'_{n_\uparrow}(P_{\uparrow}(S))\right]^2 \left(\frac{h}{2}\right)^{2n_\uparrow(S)-2}} .\tag{29}\]

It is possible to obtain the rest of \(h\)-deformed Dicke states for any number of excitations by applying the coproduct operator \(\Delta_{h}^{(N)}(Z_{+})\) (Eq. 10 ) iteratively starting from the deformed ground state \(|\tilde{D}^{-N}_{N}\rangle_{h}\). The action of the coproduct is highly non-trivial, but when applied to spin-1/2 representation states \(\ket \Psi\), since \(Z_{+}^{2}\ket{\downarrow}=0\), it reduces to [63]: \[\begin{align} \Delta_{h}^{(N)}(Z_{+})\ket{\Psi} & = \sum_{l=1}^{N}\left( 1^{\otimes (l-1)}\otimes Z_{+}\otimes 1^{\otimes (N-l)}\right)\ket{\Psi}\nonumber \\ & \quad - \frac{h^{2}}{2}\sum_{1 \le l < j <t \le N}\left( 1^{\otimes (l-1)}\otimes Z_{+}\otimes 1^{\otimes (j-l-1)}\otimes Z_{+}\right.\nonumber \\ & \quad \quad \left.\otimes 1^{\otimes (t-j-1)}\otimes Z_{+}\otimes 1^{\otimes (N-t)}\right)\ket{\Psi}. \label{eq:z43simp} \end{align}\tag{30}\]

This explicit action becomes particularly transparent in the macroscopic limit (\(N\to\infty\)), where the structure of the deformed ground state simplifies. In this regime, the states \(|\tilde{D}^{-N}_{N}\rangle_h\) can be approximated as [63] \[|\tilde{D}^{-N}_{N}\rangle_{h}= \sum_{S\in \mathcal{B}_N}{\xi}_{n_\uparrow}(S) \left(\frac{h}{2}\right )^{n_\uparrow(S)}|S\rangle\approx \sum_{S \in \mathcal{B}_N}\sum_{m=0}^{\lfloor n_\uparrow/2 \rfloor} (2m-1)!! N^{m}e_{n_\uparrow-2m}(P_{\uparrow})\left(\frac{h}{2}\right)^{n_\uparrow(S)} |S\rangle. \label{dicke95fundamental95N}\tag{31}\] Then, a clear separation of scales emerges in the coproduct expansion. While the structural coefficients scale as \(\mathcal{O}(N^{m})\), the elementary symmetric polynomials scale as \(e_{n_\uparrow-2m}\sim \mathcal{O}(N^{n_\uparrow-2m})\). So the overall scaling behaves as \(\mathcal{O}(N^{m}) \times \mathcal{O} (N^{n_\uparrow-2m}) = \mathcal{O}(N^{n_\uparrow-m})\). This strict monotonic decrease in the power of \(N\) with respect to \(m\) implies that the \(m=0\) contribution dominates the macroscopic structure. Thus, at leading order, the dominant state is governed by \(e_{n_\uparrow}(P_\uparrow)\). In the same large-\(N\) regime, the term involving the \(h\)-parameter in the coproduct 30 produces contributions involving elementary symmetric polynomials of lower degree and is therefore subleading with respect to the undeformed term of \(\Delta_{h}^{(N)}(Z_{+})\). This means that if \(N \to \infty\), the coproduct 30 can be well approximated by its undeformed version (with \(h=0\)). When applying this coproduct operator to the state 31 , the operation consists of sequentially traversing all positions and adding an up-spin. This leads to the following closed-form asymptotic expression for the macroscopic Dicke state, strictly valid if the number of times \(i\) the coproduct is applied is such that \(i \ll N\): \[\label{uwzygvxo} | D_N^{-N+2i}\rangle_h \propto \sum_{S\in \mathcal{B}_N} e_{n_\uparrow(S)-i}(P_\uparrow(S)) \left(\frac{h}{2}\right)^{n_\uparrow(S)-i} |S\rangle .\tag{32}\]

To systematically quantify the impact of the quantum deformation on the macroscopic properties of the system, we derive below the general expression for the fidelity \(F(_{h=0}\langle D_{N}^{-N+2i}| D_{N}^{-N+2i}\rangle_{h})\). We operate in the asymptotic regime (\(N \to \infty\)), where the \(h\)-Dicke states can be approximated by Eq. [eq:asymptotic95dicke95general].

5.5.1 Fidelity per site in the thermodynamic limit↩︎

Within this leading thermodynamic approximation, the finite-size fidelity is defined as the squared overlap between the normalized deformed state and the undeformed Dicke state. Since the latter is a uniform symmetric superposition of \(\binom{N}{i}\) computational basis configurations, and the deformed state assigns unit weight to each of these configurations in the limit \(h\to 0\), the overlap is entirely determined by their number. Therefore, one obtains \[F( |D_{N}^{-N+2i}\rangle_{h=0}, \,\big|\, D_{N}^{-N+2i}\rangle_{h}) = \frac{\binom{N}{i}}{\mathcal{Z}_i(h,N)},\] where \(\mathcal{Z}_i(h,N)\) plays the role of a structural partition function and comes from the squared norm of the corresponding non-normalized deformed state \[\label{eq:partition95function} \mathcal{Z}_{i}(h, N) = \sum_{\substack{S\in\mathcal{B}_N}} \left( \frac{h}{2} \right)^{2(n_\uparrow(S)-i)} \left[e_{n_\uparrow-i}(P_\uparrow(S))\right]^2.\tag{33}\]

Taking the logarithm and dividing by the system size, one finds the following fidelity per site: \[\ln f(_{h=0}\langle D_{N}^{-N+2i}| D_{N}^{-N+2i}\rangle_{h}) = \lim_{N \to \infty} \frac{1}{N} \ln F(_{h=0}\langle D_{N}^{-N+2i}| D_{N}^{-N+2i}\rangle_{h})= \lim_{N \to \infty} \left( - \frac{1}{N} \ln \mathcal{Z}_i(h, N) \right),\] where we have used the fact that the initial configurational entropy \(\frac{1}{N} \ln \binom{N}{i}\) vanishes in the macroscopic scaling \(N\to \infty\) for any finite number of initial excitations \(i \ll N\). For fixed \(i\) (or, more generally, for any number of excitations such that \(i/N\to 0\) as \(N\to\infty\)), the dependence of \(\mathcal{Z}_i(h,N)\) on \(i\) only contributes through terms that are negligible after division by \(N\). Therefore, if \(i \ll N\) the fidelity per site \(f(_{h=0}\langle D_{N}^{-N+2i}| D_{N}^{-N+2i}\rangle_{h})\) coincides with \(f(_{h=0}\langle D_{N}^{-N}| D_{N}^{-N}\rangle_{h})\). It is thus sufficient to evaluate the leading contribution coming from \(i=0\).

For \(i=0\), the relevant term of \(\mathcal{Z}_i(h, N)\) is \([e_{n_\uparrow}(P_\uparrow)]^2\). Since \(P_\uparrow\) is a subset of exactly \({n_\uparrow}\) sites, the \({n_\uparrow}\)-th elementary symmetric polynomial reduces to the product of all values in the subset, \[e_{n_\uparrow}(P_\uparrow)=\prod_{j\in P_\uparrow} g(j),\] so the corresponding partition function becomes \[\mathcal{Z}_0(h,N) = \sum_{S\in\mathcal{B}_N} \left(\frac{h}{2}\right)^{2n_\uparrow(S)} \prod_{j\in P_\uparrow(S)} g(j)^2 .\] By the generating function of elementary symmetric polynomials, this sum factorizes as \[\mathcal{Z}_0(h,N) = \prod_{j=1}^{N} \left[ 1+\left(\frac{h}{2}\right)^2 g(j)^2 \right].\] This factorization shows that, at leading order, each lattice site contributes through a local weight that depends explicitly on its position.

Using \(g(j)=2j-N-1\), the fidelity per site is therefore determined by \[\ln f(h) = -\lim_{N\to\infty} \frac{1}{N} \sum_{j=1}^{N} \ln\!\left[ 1+\frac{h^2}{4}(2j-N-1)^2 \right].\] Introducing the continuous variable \(x=j/N\), one has \(2j-N-1\approx N(2x-1)\) in the thermodynamic limit, and the sum becomes \[\ln f(h) \approx -\int_0^1 \ln\!\left[ 1+\frac{h^2N^2}{4}(2x-1)^2 \right]dx .\]

This exact integral immediately reveals the conditions for the orthogonality catastrophe. If the deformation \(h\) remains a finite constant as \(N \to \infty\), the argument of the logarithm grows as \(\mathcal{O}(N^2)\), causing the integral to diverge logarithmically and forcing \(f(h) \to 0\) (i.e., \(\ln f(h) \to -\infty\)). The deformed macroscopic state becomes strictly orthogonal to the undeformed state, independently of the fixed value of \(h\). To prevent this catastrophic loss of fidelity and preserve a macroscopic overlap, the deformation parameter must be dynamically rescaled with the system size. The exact mathematical form of the integral precisely dictates the critical scaling: \[h = \frac{\tilde{h}}{N},\] where \(\tilde{h}\) is a system-size independent constant (analogous to the double scaling limit \(\eta = \tilde{\eta}/N\) discussed for the standard deformation). Under this critical scaling, the macroscopic \(N\) dependence strictly cancels out. By substituting \(y = x - 1/2\), the integral is solved analytically without any truncations or approximations. Thus, the continuum limit of the leading thermodynamic contribution for the thermodynamic rate function of any finitely-excited initial state (\(i \ll N\)) is given by the formula: \[\ln f(\tilde{h}) \approx - 2 \int_{0}^{1/2} \ln \left[ 1 + \tilde{h}^2 y^2 \right] dy = 2 - \ln\left(1 + \frac{\tilde{h}^2}{4}\right) - \frac{4}{\tilde{h}} \arctan\left(\frac{\tilde{h}}{2}\right). \label{eq:exact95fidelity95closed}\tag{34}\] We now analyze the closed-form expression for the fidelity per site of Eq. 34 in the two extremal regimes of the rescaled deformation parameter.

5.5.1.1 Weak Deformation Regime (\(\tilde{h} \ll 1\)):

performing a Taylor expansion of the logarithmic and arctangent functions around \(\tilde{h} = 0\) yields the leading-order behavior of the thermodynamic rate function: \[\label{eq:weak95h95limit} \ln f(\tilde{h}) \approx 2 - \left(\frac{\tilde{h}^2}{4}\right) - \frac{4}{\tilde{h}} \left( \frac{\tilde{h}}{2} - \frac{\tilde{h}^3}{24} \right) + \mathcal{O}\!\left(\tilde{h}^4\right)= - \frac{\tilde{h}^2}{12}+ \mathcal{O}\!\left(\tilde{h}^4\right),\tag{35}\] showing a Gaussian dependence on the rescaled deformation parameter. The absence of a linear term means that the first deviation from the undeformed case is quadratic in \(\tilde{h}\). Therefore, for small deformations, the fidelity per site remains close to one and decreases smoothly as \(\tilde{h}\) increases.

5.5.1.2 Strong Deformation Regime (\(\tilde{h} \gg 1\)):

Conversely, when the rescaled deformation is extremely large, \(\arctan(\tilde{h}/2) \to \pi/2\) and \(\ln(1 + \tilde{h}^2/4) \approx 2 \ln(\tilde{h}) - \ln(4)\). The thermodynamic rate function simplifies asymptotically to: \[\label{eq:strong95h95limit} \ln f(\tilde{h}) \approx 2 - 2 \ln(\tilde{h}) + \ln(4) - \frac{2\pi}{\tilde{h}},\tag{36}\] which implies that, for \(\tilde{h}\gg1\), \[f(\tilde{h})\approx\frac{4e^2}{\tilde{h}^2}.\] Thus, in the strong-deformation regime, the fidelity per site decays algebraically with the rescaled deformation parameter, in contrast with the Gaussian behaviour found for weak deformation.

5.6 Comparison of Thermodynamic Scaling in \(q\)- and \(h\)-Deformations↩︎

From the fidelity viewpoint, both the standard \(q\)-deformation and the Jordanian \(h\)-deformation require a critical rescaling of the deformation parameter, of order \(N^{-1}\), in order to obtain a finite and non-trivial fidelity per site in the thermodynamic limit. However, only in the standard \(q\)-deformation this rescaling is also required for spectral reasons.

In the standard Drinfeld-Jimbo \(q\)-deformation, such as in the \(q\)-deformed Kittel–Shore model [17], the deformation modifies the macroscopic eigenvalues themselves. Setting \(q = e^{\eta}\), the modified spectrum transforms proportionally to \(\sinh(\eta X/2)/\sinh(\eta/2)\). To prevent the energy from diverging exponentially—which would violate thermodynamic extensivity, where energy must scale proportionally to the system volume \(N\)—the argument of the hyperbolic sine must remain finite. Consequently, the deformation parameter naturally follows the standard intensive mean-field scaling law: \(\eta =\tilde{\eta}/N\). Under this explicit spectral constraint, the system remains thermodynamically well-behaved [17]. The same scaling has been found here by imposing that the fidelity per site goes to a finite (non-vanishing) value.

By contrast, the Jordanian \(h\)-deformation does not change the eigenvalues of the \(h\)-Kittel–Shore model but it acts purely on the internal combinatorics of the spin lattice. The algebraic operators generate position-dependent factors that break permutation symmetry. As demonstrated by the asymptotic fidelity per site, in the weak-deformation regime one finds \[\ln f(\tilde{h})=-\frac{\tilde{h}^2}{12},\] so that the total fidelity scales as \[F\sim \exp\left(-\frac{N\tilde{h}^2}{12}\right).\] The role of the critical scaling is not to keep the total fidelity finite in the thermodynamic limit, but rather to obtain a finite and non-trivial fidelity per site. Without the scaling \(h=\tilde{h}/N\), the rate function itself diverges and the fidelity per site collapses. Crucially, this scaling behavior cannot be inferred from the spectrum of the \(h\)-Kittel–Shore model, as the \(h\)-deformation leaves the macroscopic energy eigenvalues unmodified.

6 Conclusions↩︎

In this work, we study the impact of standard (\(q\)-) and non-standard (\(h\)-, Jordanian) deformations of \(\mathcal{U}(\mathfrak{sl}(2, \mathbb{R}))\) on quantum systems of different number of qubits. We present a comparison between the experimental data and the theoretical predictions for both deformations. When unexpected states are created with negligible probability, the \(q\)-deformation provides an accurate description by differently reweighting the basis components according to the value of \(q\). Conversely, when the probability of these new states becomes non-negligible, the \(h\)-deformation is required for proper modeling. In particular, for the latter scenario, a combination of both deformations may yield a superior description, as it simultaneously allows for the reweighting of the expected components and the emergence of new ones. A natural candidate is the hybrid \((q,h)\)-deformation, obtained by combining both \(q\)- and \(h\)-deformations. Despite this mixed origin, the hybrid deformation is still of the standard type, since it can be derived from the standard \(q\)-deformation through a nonlinear change of basis, either at the level of the quantum group coordinates [64] or of the quantum algebra generators [65]. This possibility could lead to a more accurate description of the experimental states and will be explored in future work.

To quantify the impact of these algebraic deformations, we compute the theoretical fidelity between the deformed states and their undeformed counterparts (\(q=1\) or \(h=0\)). This quantum overlap serves as a direct measure of how the state space deviates from the original scenario as the deformation parameters and the number of qubits \(N\) vary. In all cases studied, the fidelity decreases with increasing deformation, though displaying distinct behaviors. The \(q\)-deformation causes a smooth, monotonic decay that stabilizes to a non-zero constant, leading to configurations that still retain a residual algebraic connection to the initial symmetric structures. Conversely, the \(h\)-deformation induces a faster and stronger loss of fidelity, rapidly orthogonalizing the states with respect to the original ones and drastically altering the overall correlation architecture. Furthermore, these behavioral differences become profound in the macroscopic limit for arbitrary \(N\)-qubit Dicke states. As \(N\) grows, the \(q\)-deformation geometrically drives the symmetric superposition towards a fully localized, separable product state in the strong-deformation regime (\(q \to \infty\)). Assuring a stable macroscopic limit (\(N \to \infty\)) thus requires a standard intensive deformation parameter scaling of \(N^{-1}\), which agrees with a previous result when imposing the thermodynamic extensivity of the \(q\)-Kittel–Shore spectrum. Similarly, the \(h\)-deformation mathematically alters the spatial combinatorics of the spin lattice, inducing an orthogonality catastrophe that also demands a critical scaling of \(N^{-1}\) to stabilize the macroscopic quantum states. In this case, this result cannot be obtained from the thermodynamic extensivity of the \(h\)-Kittel–Shore spectrum, since its eigenvalues are \(h\) independent.

Moreover, in future articles, we plan to extend this analysis by investigating in detail the behavior of various entanglement measures, such as von Neumann and Rényi entropies, tangles, and logarithmic negativities under these deformations. This will allow for a more complete characterization of the generation and suppression of quantum correlations.

Acknowledgments↩︎

The authors acknowledge partial support from the grants PID2023-148373NB-I00 funded by MCIN/ AEI / 10.13039/501100011033/FEDER – UE, as well as BU011P25 funded by the Regional Government of Castilla y León (Junta de Castilla y León). V. Mariscal acknowledges support from the Universidad de Burgos through its PhD grant program.

7 \(q\)-deformed states of up to 4 qubits↩︎

In this section, we collect the normalized states associated with the irreducible representations of the standard deformation \(\mathcal{U}_q \left(\mathfrak{sl}(2, \mathbb{R})\right)\). We will use the notation \(|X_N^m\rangle_q\), where \(N\) is the number of qubits (which corresponds mathematically to the \(N\)-fold tensor product of the fundamental spin-\(1/2\) representation), and \(m\) the eigenvalue of the \(L_z\) operator in the corresponding \(j = N/2\) irreducible representation. We also use the computational basis \[\begin{align} && \mathcal{B}_N= \left\{ \ket{s_1s_2\cdots s_N}\;:\; s_\ell\in\{\uparrow,\downarrow\},\;\ell=1,\ldots,N \right\},\nonumber\\ &&\textrm{where}\quad \ket{s_1s_2\cdots s_N} = \ket{s_1}\otimes\ket{s_2}\otimes\cdots\otimes\ket{s_N}. \end{align}\] This basis constitutes an orthonormal one of the composite Hilbert space \(\mathcal{H}_N = (\mathbb{C}^2)^{\otimes N}\). We also define \[\left\{ |\uparrow\rangle := \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad |\downarrow\rangle := \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right\}.\] For \(2\) qubits, we use the notation present in Fig. 11 for labelling the states.

The triplet of \(q\)-Dicke states (\(j=1\)) is given by: \[\begin{align} && \ket{D^{-1}_2}_q = \ket{\downarrow\downarrow}, \nonumber\\ &&\ket{D^{0}_2}_q = \frac{1}{\sqrt{[2]_{q}}} (q^{1/4}\ket{\downarrow\uparrow} + q^{-1/4} \ket{\uparrow\downarrow}), \nonumber\\ &&\ket{D^1_2}_q =\ket{\uparrow\uparrow}, \label{eq:2qqubits} \end{align}\tag{37}\] while the singlet (\(j=0\)) can be written as: \[\begin{align} \ket{M^0_2}_q =\frac{1}{\sqrt{[2]_{q}}} \left(-q^{-1/4}\left|\downarrow\uparrow\right\rangle + q^{1/4}\left|\uparrow \downarrow\right\rangle\right). \end{align}\] For \(3\) qubits, we use the notation present in Fig. 12 to label the states.

The quadruplet of \(q\)-Dicke states (with \(j=3/2\)) is: \[\begin{align} &&\ket{D^{-3/2}_3}_q =|\downarrow\downarrow\downarrow\rangle, \nonumber\\ &&\ket{D^{-1/2}_3}_q = \frac{1}{\sqrt{[3]_{q}}}\left( q^{1/2}|\downarrow\downarrow\uparrow\rangle +|\downarrow\uparrow\downarrow\rangle+ q^{-1/2}|\uparrow\downarrow\downarrow\rangle\right), \nonumber\\ &&\ket{D^{1/2}_3}_q =\frac{1}{\sqrt{[3]_{q}}}\left(q^{1/2} |\downarrow\uparrow\uparrow\rangle +|\uparrow\downarrow\uparrow\rangle + q^{-1/2}|\uparrow\uparrow\downarrow\rangle \right),\nonumber\\ &&\ket{D^{3/2}_3}_q =\left|\uparrow\uparrow\uparrow\right\rangle, \label{eq:3qubitsq} \end{align}\tag{38}\] and the two doublets of \(q\)-states (with \(j=1/2\)) can be expressed as \[\begin{align} &&|M^{-1/2}_{3}\rangle_{q}=\frac{1}{\sqrt{[2]_{q}}\sqrt{[3]_{q}}}\left(q^{1/4}\left|\uparrow\downarrow\downarrow\right\rangle+q^{3/4}\left|\downarrow\uparrow\downarrow\right\rangle-[2]_{q}\,\, q^{-1/4}\left|\downarrow\downarrow\uparrow\right\rangle\right), \nonumber\\ &&|M^{1/2}_{3}\rangle_{q}=\frac{1}{\sqrt{[2]_{q}}\sqrt{[3]_{q}}}\left([2]_{q}\,\, q^{1/4}\left|\uparrow\uparrow\downarrow\right\rangle-q^{-3/4}\left|\uparrow\downarrow\uparrow\right\rangle-q^{-1/4}\left|\downarrow\uparrow\uparrow\right\rangle\right), \nonumber\\ &&|V^{-1/2}_{3}\rangle_q=\frac{1}{\sqrt{[2]_{q}}}\left(q^{1/4}\left|\uparrow\downarrow\downarrow\right\rangle-q^{-1/4}\left|\downarrow\uparrow\downarrow\right\rangle\right), \nonumber\\ &&|V^{1/2}_{3}\rangle_q=\frac{1}{\sqrt{[2]_{q}}}\left(q^{1/4}\left|\uparrow\downarrow\uparrow\right\rangle-q^{-1/4}\left|\downarrow\uparrow\uparrow\right\rangle\right). \end{align}\] The corresponding deformed states for a \(4\)-qubit system are presented here for the first time, labeled according to Fig. 13.

Using the \(q\)-Clebsch–Gordan coefficients 9 , we obtain the quintuplet of \(q\)-Dicke states (with \(j=2\)) \[\begin{align} &&|D^{-2}_{4}\rangle_{q}=\left|\downarrow\downarrow\downarrow\downarrow\right\rangle, \nonumber\\ &&|D_{4}^{-1}\rangle_{q}=\frac{1}{\sqrt{[4]_{q}}}\left(q^{-3/4}\left|\uparrow\downarrow\downarrow\downarrow\right\rangle+q^{-1/4}\left|\downarrow\uparrow\downarrow\downarrow\right\rangle+q^{1/4}\left|\downarrow\downarrow\uparrow\downarrow\right\rangle+q^{3/4}\left|\downarrow\downarrow\downarrow\uparrow\right\rangle\right), \nonumber\\ &&|D^{0}_{4}\rangle_{q}=\frac{\sqrt{[2]_{q}}}{\sqrt{[3]_{q}}\sqrt{[4]_{q}}}\left(q^{-1}\left|\uparrow\uparrow\downarrow\downarrow\right\rangle+q^{-1/2}\left|\uparrow\downarrow\uparrow\downarrow\right\rangle+\left|\uparrow\downarrow\downarrow\uparrow\right\rangle+\left|\downarrow\uparrow\uparrow\downarrow\right\rangle+q^{1/2}\left|\downarrow\uparrow\downarrow\uparrow\right\rangle+q\left|\downarrow\downarrow\uparrow\uparrow\right\rangle\right), \nonumber\\ &&|D^{1}_{4}\rangle_{q}=\frac{1}{\sqrt{[4]_{q}}}\left(q^{3/4}\left|\downarrow\uparrow\uparrow\uparrow\right\rangle+q^{1/4}\left|\uparrow\downarrow\uparrow\uparrow\right\rangle+q^{-1/4}\left|\uparrow\uparrow\downarrow\uparrow\right\rangle+q^{-3/4}\left|\uparrow\uparrow\uparrow\downarrow\right\rangle\right), \nonumber\\ &&|D_{4}^{2}\rangle_{q}=\left|\uparrow\uparrow\uparrow\uparrow\right\rangle, \end{align}\] together with three triplets (with \(j=1\)) \[\begin{align} &&|M_{4}^{-1}\rangle_{q}=\frac{1}{\sqrt{[3]_{q}}\sqrt{[4]_{q}}}\left(q^{1/4}\left|\uparrow\downarrow\downarrow\downarrow\right\rangle+q^{3/4}\left|\downarrow\uparrow\downarrow\downarrow\right\rangle+q^{5/4}\left|\downarrow\downarrow\uparrow\downarrow\right\rangle-q^{-1/4}[3]_{q}\left|\downarrow\downarrow\downarrow\uparrow\right\rangle\right), \nonumber\\ &&|M_{4}^{0}\rangle_{q}=\frac{\sqrt{[2]_{q}}}{\sqrt{[3]_{q}}\sqrt{[4]_{q}}}(\left|\uparrow\uparrow\downarrow\downarrow\right\rangle+q^{1/2}\left|\uparrow\downarrow\uparrow\downarrow\right\rangle-q^{-1}\left|\uparrow\downarrow\downarrow\uparrow\right\rangle+q\left|\downarrow\uparrow\uparrow\downarrow\right\rangle-q^{-1/2}\left|\downarrow\uparrow\downarrow\uparrow\right\rangle-\left|\downarrow\downarrow\uparrow\uparrow\right\rangle), \nonumber\\ &&|M_{4}^{1}\rangle_{q}=\frac{1}{\sqrt{[3]_{q}}\sqrt{[4]_{q}}}\left(q^{1/4}[3]_{q}\left|\uparrow\uparrow\uparrow\downarrow\right\rangle-q^{-5/4}\left|\uparrow\uparrow\downarrow\uparrow\right\rangle-q^{-3/4}\left|\uparrow\downarrow\uparrow\uparrow\right\rangle-q^{-1/4}\left|\downarrow\uparrow\uparrow\uparrow\right\rangle\right),\nonumber\\ &&|V_{4}^{-1}\rangle_{q}=\frac{1}{\sqrt{[2]_{q}}\sqrt{[3]_{q}}}\left(q^{1/4}\left|\uparrow\downarrow\downarrow\downarrow\right\rangle+q^{3/4}\left|\downarrow\uparrow\downarrow\downarrow\right\rangle-q^{-1/4}[2]_{q}\left|\downarrow\downarrow\uparrow\downarrow\right\rangle\right), \nonumber\\ &&|V_{4}^{0}\rangle_{q}=\frac{1}{[2]_{q}\sqrt{[3]_{q}}}\left([2]_{q}\left|\uparrow\uparrow\downarrow\downarrow\right\rangle-q^{-1}\left|\uparrow\downarrow\uparrow\downarrow\right\rangle+q^{1/2}\left|\uparrow\downarrow\downarrow\uparrow\right\rangle-q^{-1/2}\left|\downarrow\uparrow\uparrow\downarrow\right\rangle+q\left|\downarrow\uparrow\downarrow\uparrow\right\rangle-[2]_{q}\left|\downarrow\downarrow\uparrow\uparrow\right\rangle\right), \nonumber\\ &&|V_{4}^{1}\rangle_{q}=\frac{1}{\sqrt{[2]_{q}}\sqrt{[3]_{q}}}\left(q^{1/4}[2]_{q}\left|\uparrow\uparrow\downarrow\uparrow\right\rangle-q^{-3/4}\left|\uparrow\downarrow\uparrow\uparrow\right\rangle-q^{-1/4}\left|\downarrow\uparrow\uparrow\uparrow\right\rangle\right),\nonumber\\ &&|R_{4}^{-1}\rangle_{q}=\frac{1}{\sqrt{[2]_{q}}}\left(q^{1/4}\left|\uparrow\downarrow\downarrow\downarrow\right\rangle-q^{-1/4}\left|\downarrow\uparrow\downarrow\downarrow\right\rangle\right), \nonumber\\ &&|R_{4}^{0}\rangle_{q}=\frac{1}{[2]_{q}}\left(\left|\uparrow\downarrow\uparrow\downarrow\right\rangle+q^{1/2}\left|\uparrow\downarrow\downarrow\uparrow\right\rangle-q^{-1/2}\left|\downarrow\uparrow\uparrow\downarrow\right\rangle-\left|\downarrow\uparrow\downarrow\uparrow\right\rangle\right), \nonumber\\ &&|R_{4}^{1}\rangle_{q}=\frac{1}{\sqrt{[2]_{q}}}\left(q^{1/4}\left|\uparrow\downarrow\uparrow\uparrow\right\rangle-q^{-1/4}\left|\downarrow\uparrow\uparrow\uparrow\right\rangle\right), \end{align}\] and two singlets (with \(j=0\)) \[\begin{align} &&|T_{4}^{0}\rangle_{q}=\frac{1}{[2]_{q}\sqrt{[3]_{q}}}\left(q^{1/2}[2]_{q}\left|\uparrow\uparrow\downarrow\downarrow\right\rangle-q^{-1/2}\left|\uparrow\downarrow\uparrow\downarrow\right\rangle-\left|\uparrow\downarrow\downarrow\uparrow\right\rangle-\left|\downarrow\uparrow\uparrow\downarrow\right\rangle-q^{1/2}\left|\downarrow\uparrow\downarrow\uparrow\right\rangle+q^{-1/2}[2]_{q}\left|\downarrow\downarrow\uparrow\uparrow\right\rangle\right),\nonumber\\ &&|Q_{4}^{0}\rangle_{q}=\frac{1}{[2]_{q}}\left(q^{1/2}\left|\uparrow\downarrow\uparrow\downarrow\right\rangle-\left|\uparrow\downarrow\downarrow\uparrow\right\rangle-\left|\downarrow\uparrow\uparrow\downarrow\right\rangle+q^{-1/2}\left|\downarrow\uparrow\downarrow\uparrow\right\rangle\right). \end{align}\]

8 \(h\)-deformed states of up to 4 qubits↩︎

In this section, we collect the normalized states associated with the irreducible representations of the non-standard deformation \(\mathcal{U}_h(\mathfrak{sl}(2,\mathbb{R}))\). We will use the notation \(\ket{X_N^{\tilde{m}}}_h\), where \(N\) is the number of qubits, corresponding mathematically to the \(N\)-fold tensor product of the fundamental spin-\(1/2\) representation, and \(\tilde{m}\) is the eigenvalue of the \(H=2J_z\) operator in the corresponding \(j=N/2\) irreducible representation. The computational basis used here is the same as the one introduced in Appendix 7.

For 2 qubits, the triplet of \(h\)-Dicke states (with \(j=1\)) is given by: \[\begin{align} &&\ket{D^{-2}_2}_h =\frac{4}{h^{2}+4} \left(\frac{h^{2}}{4}\ket{\uparrow\uparrow} -\frac{h}{2}\ket{\uparrow\downarrow}+\frac{h}{2}\ket{ \downarrow\uparrow} + \ket{\downarrow\downarrow}\right), \nonumber\\ &&\ket{D^{0}_2}_h = \frac{1}{\sqrt{2}} (\ket{\uparrow\downarrow}+\ket{ \downarrow\uparrow}), \nonumber\\ &&\ket{D^2_2}_h =\ket{\uparrow\uparrow}, \label{eq:2qubits} \end{align}\tag{39}\] while the singlet state (with \(j=0\)) is: \[\ket{M^0_2}_{h}=\frac{1}{\sqrt{h^{2}+2}}\left(-h |\uparrow\uparrow\rangle + \ket{\uparrow\downarrow} -\ket{ \downarrow\uparrow}\right). \label{Mh}\tag{40}\] For \(3\) qubits, we obtain the following quadruplet of \(h\)-Dicke states (with \(j=3/2\)): \[\begin{align} &&\ket{D^{-3}_3}_h =\frac{4}{\sqrt{19h^{4}+32h^{2}+16}}\left(\frac{3h^{2}}{4}\ket{\uparrow\uparrow\downarrow} -\frac{h^{2}}{4}\ket{\uparrow\downarrow\uparrow}-h\ket{ \uparrow\downarrow\downarrow} +\frac{3h^{2}}{4}\ket{\downarrow\uparrow\uparrow} +h\ket{ \downarrow\downarrow\uparrow}+ \ket{\downarrow\downarrow\downarrow}\right), \nonumber\\ &&\ket{D^{-1}_3}_h = \frac{4 \sqrt{3}}{\sqrt{9h^{4}+32h^{2}+48}}\left(\frac{\sqrt{3}h^{2}}{4}\ket{\uparrow\uparrow\uparrow} -\frac{h}{\sqrt{3}}\ket{\uparrow\uparrow\downarrow} +\frac{h}{\sqrt{3}} \ket{\downarrow\uparrow\uparrow}+ \frac{1}{\sqrt{3}} (\ket{ \uparrow\downarrow\downarrow} + \ket{ \downarrow\uparrow\downarrow} +\ket{ \downarrow\downarrow\uparrow} )\right), \nonumber\\ &&\ket{D^1_3}_h =\frac{1}{\sqrt{3}} (\ket{ \uparrow\uparrow\downarrow} + \ket{ \uparrow\downarrow\uparrow} +\ket{ \downarrow\uparrow\uparrow} ),\nonumber\\ &&\ket{D^3_3}_h =|\uparrow\uparrow\uparrow\rangle, \label{eq:3qubitsh} \end{align}\tag{41}\] together with two doublets of \(h\)-states (with \(j=1/2\)): \[\begin{align} &&\ket{M^{-1}_3}_h = \frac{\sqrt{6}}{\sqrt{5h^{2}+6}}\left(\frac{1}{\sqrt{6}} (-h\ket{ \uparrow\uparrow\downarrow}-2h\ket{ \downarrow\uparrow\uparrow}) +\frac{1}{\sqrt{6}}( \ket{ \uparrow\downarrow\downarrow} + \ket{ \downarrow\uparrow\downarrow}-2 \ket{ \downarrow\downarrow\uparrow})\right),\nonumber\\ &&\ket{M^{1}_3}_h= \frac{\sqrt{2}}{\sqrt{3h^{2}+2}}\left(-\frac{3h}{\sqrt{6}} \ket{\uparrow\uparrow\uparrow} +\frac{1}{\sqrt{6}} (2\ket{\uparrow\uparrow\downarrow}-\ket{\uparrow\downarrow\uparrow}-\ket{\downarrow\uparrow\uparrow})\right),\nonumber\\[2.5ex] &&\ket{V^{-1}_3}_h=\frac{\sqrt{2}}{\sqrt{h^{2}+2}}\left(-\frac{h}{\sqrt{2}} \ket{ \uparrow\uparrow\downarrow}+\frac{1}{\sqrt{2}} ( -\ket{ \downarrow\uparrow\downarrow}+\ket{ \uparrow\downarrow\downarrow}) \right),\nonumber\\ &&\ket{V^{1}_3}_h = \frac{\sqrt{2}}{\sqrt{h^{2}+2}}\left(-\frac{h}{\sqrt{2}} \ket{ \uparrow\uparrow\uparrow}+\frac{1}{\sqrt{2}} (- \ket{ \downarrow\uparrow\uparrow}+\ket{ \uparrow\downarrow\uparrow})\right). \label{eq:3qubitsa} \end{align}\tag{42}\] For \(4\) qubits, the quintuplet of \(h\)-Dicke states (with \(j=2\)) can be written as: \[\begin{align} &&|D^{-4}_{4}\rangle_{h}=\frac{1}{\sqrt{81 h^{8}+848 h^{6}+2144 h^{4}+1280 h^{2}+256}}\left(9h^4\left|\uparrow\uparrow\uparrow\uparrow\right\rangle -18h^3\left|\uparrow\uparrow\uparrow\downarrow\right\rangle +10h^3\left|\uparrow\uparrow\downarrow\uparrow\right\rangle \right. \nonumber\\ &&\qquad \qquad \left.+28h^2\left|\uparrow\uparrow\downarrow\downarrow\right\rangle -10h^3\left|\uparrow\downarrow\uparrow\uparrow\right\rangle +4h^2\left|\uparrow\downarrow\uparrow\downarrow\right\rangle -20h^2\left|\uparrow\downarrow\downarrow\uparrow\right\rangle -24h\left|\uparrow\downarrow\downarrow\downarrow\right\rangle +18h^3\left|\downarrow\uparrow\uparrow\uparrow\right\rangle \right.\nonumber \\ &&\qquad \qquad\left.+12h^2\left|\downarrow\uparrow\uparrow\downarrow\right\rangle +4h^2\left|\downarrow\uparrow\downarrow\uparrow\right\rangle -8h\left|\downarrow\uparrow\downarrow\downarrow\right\rangle +28h^2\left|\downarrow\downarrow\uparrow\uparrow\right\rangle +8h\left|\downarrow\downarrow\uparrow\downarrow\right\rangle +24h\left|\downarrow\downarrow\downarrow\uparrow\right\rangle +16\left|\downarrow\downarrow\downarrow\downarrow\right\rangle \right), \nonumber\\ &&|D_{4}^{-2}\rangle_{h}=\frac{1}{2 \sqrt{41 h^{4}+40 h^{2}+16}}\left(9h^2\left|\uparrow\uparrow\uparrow\downarrow\right\rangle +h^2\left|\uparrow\uparrow\downarrow\uparrow\right\rangle -8h\left|\uparrow\uparrow\downarrow\downarrow\right\rangle +h^2\left|\uparrow\downarrow\uparrow\uparrow\right\rangle -4h\left|\uparrow\downarrow\uparrow\downarrow\right\rangle \right.\nonumber \\ &&\qquad \qquad\left.+4\left|\uparrow\downarrow\downarrow\downarrow\right\rangle +9h^2\left|\downarrow\uparrow\uparrow\uparrow\right\rangle +4h\left|\downarrow\uparrow\downarrow\uparrow\right\rangle +4\left|\downarrow\uparrow\downarrow\downarrow\right\rangle +8h\left|\downarrow\downarrow\uparrow\uparrow\right\rangle +4\left|\downarrow\downarrow\uparrow\downarrow\right\rangle +4\left|\downarrow\downarrow\downarrow\uparrow\right\rangle\right), \nonumber\\ && |D^{0}_{4}\rangle_{h}=\frac{1}{\sqrt{9 h^{4}+20 h^{2}+24}}\left(3h^2\left|\uparrow\uparrow\uparrow\uparrow\right\rangle -3h\left|\uparrow\uparrow\uparrow\downarrow\right\rangle -h\left|\uparrow\uparrow\downarrow\uparrow\right\rangle +2\left|\uparrow\uparrow\downarrow\downarrow\right\rangle +h\left|\uparrow\downarrow\uparrow\uparrow\right\rangle +2\left|\uparrow\downarrow\uparrow\downarrow\right\rangle\right. \nonumber \\ &&\qquad \qquad \left.+2\left|\uparrow\downarrow\downarrow\uparrow\right\rangle +3h\left|\downarrow\uparrow\uparrow\uparrow\right\rangle +2\left|\downarrow\uparrow\uparrow\downarrow\right\rangle +2\left|\downarrow\uparrow\downarrow\uparrow\right\rangle +2\left|\downarrow\downarrow\uparrow\uparrow\right\rangle\right), \nonumber\\ &&|D^{2}_{4}\rangle_{h}=\frac{1}{2}|\uparrow\uparrow\uparrow\downarrow\rangle +\frac{1}{2}|\uparrow\uparrow\downarrow\uparrow\rangle +\frac{1}{2}|\uparrow\downarrow\uparrow\uparrow\rangle +\frac{1}{2}|\downarrow\uparrow\uparrow\uparrow\rangle, \nonumber\\ &&|D_{4}^{4}\rangle_{h}=|\uparrow\uparrow\uparrow\uparrow\rangle, \end{align}\] while the three triplets (with \(j=1\)) are \[\begin{align} && |M_{4}^{-2}\rangle_{h}=\frac{1}{2 \sqrt{9 h^{6}+59 h^{4}+104 h^{2}+48}}\left(-6h^3\left|\uparrow\uparrow\uparrow\uparrow\right\rangle +9h^2\left|\uparrow\uparrow\uparrow\downarrow\right\rangle -3h^2\left|\uparrow\uparrow\downarrow\uparrow\right\rangle -8h\left|\uparrow\uparrow\downarrow\downarrow\right\rangle \right.\nonumber \\ &&\qquad \qquad+5h^2\left|\uparrow\downarrow\uparrow\uparrow\right\rangle -4h\left|\uparrow\downarrow\uparrow\downarrow\right\rangle +8h\left|\uparrow\downarrow\downarrow\uparrow\right\rangle +4\left|\uparrow\downarrow\downarrow\downarrow\right\rangle -11h^2\left|\downarrow\uparrow\uparrow\uparrow\right\rangle -4h\left|\downarrow\uparrow\downarrow\uparrow\right\rangle +4\left|\downarrow\uparrow\downarrow\downarrow\right\rangle\nonumber \\ &&\qquad \qquad\left.-16h\left|\downarrow\downarrow\uparrow\uparrow\right\rangle +4\left|\downarrow\downarrow\uparrow\downarrow\right\rangle -12\left|\downarrow\downarrow\downarrow\uparrow\right\rangle \right), \nonumber\\ &&|M_{4}^{0}\rangle_{h}=\frac{1}{2\sqrt{11 h^{2}+6}} \left(-3h\left|\uparrow\uparrow\uparrow\downarrow\right\rangle -h\left|\uparrow\uparrow\downarrow\uparrow\right\rangle +2\left|\uparrow\uparrow\downarrow\downarrow\right\rangle -3h\left|\uparrow\downarrow\uparrow\uparrow\right\rangle +2\left|\uparrow\downarrow\uparrow\downarrow\right\rangle -2\left|\uparrow\downarrow\downarrow\uparrow\right\rangle\right. \nonumber \\ &&\qquad \qquad\left.-5h\left|\downarrow\uparrow\uparrow\uparrow\right\rangle +2\left|\downarrow\uparrow\uparrow\downarrow\right\rangle -2\left|\downarrow\uparrow\downarrow\uparrow\right\rangle -2\left|\downarrow\downarrow\uparrow\uparrow\right\rangle\right), \nonumber\\ &&|M_{4}^{2}\rangle_{h}=-\frac{1}{2 \sqrt{9 h^{2}+3}}\left(6h\left|\uparrow\uparrow\uparrow\uparrow\right\rangle -3\left|\uparrow\uparrow\uparrow\downarrow\right\rangle +\left|\uparrow\uparrow\downarrow\uparrow\right\rangle +\left|\uparrow\downarrow\uparrow\uparrow\right\rangle +\left|\downarrow\uparrow\uparrow\uparrow\right\rangle\right),\nonumber\\ && |V_{4}^{-2}\rangle_{h}=\frac{1}{\sqrt{9 h^{6}+62 h^{4}+128 h^{2}+96}}\left(-3h^3\left|\uparrow\uparrow\uparrow\uparrow\right\rangle +6h^2\left|\uparrow\uparrow\uparrow\downarrow\right\rangle -8h\left|\uparrow\uparrow\downarrow\downarrow\right\rangle -h^2\left|\uparrow\downarrow\uparrow\uparrow\right\rangle +2h\left|\uparrow\downarrow\uparrow\downarrow\right\rangle \right. \nonumber \\ &&\qquad \qquad\left.+2h\left|\uparrow\downarrow\downarrow\uparrow\right\rangle +4\left|\uparrow\downarrow\downarrow\downarrow\right\rangle -5h^2\left|\downarrow\uparrow\uparrow\uparrow\right\rangle -6h\left|\downarrow\uparrow\uparrow\downarrow\right\rangle +2h\left|\downarrow\uparrow\downarrow\uparrow\right\rangle +4\left|\downarrow\uparrow\downarrow\downarrow\right\rangle -4h\left|\downarrow\downarrow\uparrow\uparrow\right\rangle -8\left|\downarrow\downarrow\uparrow\downarrow\right\rangle \right), \nonumber\\ && |V_{4}^{0}\rangle_{h}=-\frac{1}{\sqrt{2} \sqrt{7 h^{2}+6}}\left(3h\left|\uparrow\uparrow\uparrow\downarrow\right\rangle +h\left|\uparrow\uparrow\downarrow\uparrow\right\rangle -2\left|\uparrow\uparrow\downarrow\downarrow\right\rangle +\left|\uparrow\downarrow\uparrow\downarrow\right\rangle -\left|\uparrow\downarrow\downarrow\uparrow\right\rangle +2h\left|\downarrow\uparrow\uparrow\uparrow\right\rangle +\left|\downarrow\uparrow\uparrow\downarrow\right\rangle \right. \nonumber \\ &&\qquad \qquad\left.-\left|\downarrow\uparrow\downarrow\uparrow\right\rangle +2\left|\downarrow\downarrow\uparrow\uparrow\right\rangle\right), \nonumber\\ &&|V_{4}^{2}\rangle_{h}=\frac{1}{\sqrt{9 h^{2}+6}}\left(-3h\left|\uparrow\uparrow\uparrow\uparrow\right\rangle +2\left|\uparrow\uparrow\downarrow\uparrow\right\rangle -\left|\uparrow\downarrow\uparrow\uparrow\right\rangle -\left|\downarrow\uparrow\uparrow\uparrow\right\rangle\right),\nonumber\\ &&|R_{4}^{-2}\rangle_{h}=\frac{1}{\left(h^{2}+4\right)\sqrt{h^2+2}}\left(-h^3\left|\uparrow\uparrow\uparrow\uparrow\right\rangle +2h^2\left|\uparrow\uparrow\uparrow\downarrow\right\rangle -2h^2\left|\uparrow\uparrow\downarrow\uparrow\right\rangle -4h\left|\uparrow\uparrow\downarrow\downarrow\right\rangle +h^2\left|\uparrow\downarrow\uparrow\uparrow\right\rangle \right. \nonumber \\ &&\qquad \qquad\left.-2h\left|\uparrow\downarrow\uparrow\downarrow\right\rangle +2h\left|\uparrow\downarrow\downarrow\uparrow\right\rangle +4\left|\uparrow\downarrow\downarrow\downarrow\right\rangle -h^2\left|\downarrow\uparrow\uparrow\uparrow\right\rangle +2h\left|\downarrow\uparrow\uparrow\downarrow\right\rangle -2h\left|\downarrow\uparrow\downarrow\uparrow\right\rangle -4\left|\downarrow\uparrow\downarrow\downarrow\right\rangle\right),\nonumber\\ &&|R_{4}^{0}\rangle_{h}=-\frac{1}{\sqrt{2} \sqrt{h^{2}+2}}\left(h\left|\uparrow\uparrow\uparrow\downarrow\right\rangle +h\left|\uparrow\uparrow\downarrow\uparrow\right\rangle -\left|\uparrow\downarrow\uparrow\downarrow\right\rangle -\left|\uparrow\downarrow\downarrow\uparrow\right\rangle +\left|\downarrow\uparrow\uparrow\downarrow\right\rangle +\left|\downarrow\uparrow\downarrow\uparrow\right\rangle\right), \nonumber\\ && |R_{4}^{2}\rangle_{h}=\frac{1}{\sqrt{h^{2}+2}}\left(-h\left|\uparrow\uparrow\uparrow\uparrow\right\rangle +\left|\uparrow\downarrow\uparrow\uparrow\right\rangle -\left|\downarrow\uparrow\uparrow\uparrow\right\rangle\right). \end{align}\] Finally, we also obtain two singlets (with \(j=0\)): \[\begin{align} &&|T_{4}^{0}\rangle_{h}=\frac{1}{\sqrt{9 h^{4}+20 h^{2}+12}}\left(3h^2\left|\uparrow\uparrow\uparrow\uparrow\right\rangle -3h\left|\uparrow\uparrow\uparrow\downarrow\right\rangle -h\left|\uparrow\uparrow\downarrow\uparrow\right\rangle +2\left|\uparrow\uparrow\downarrow\downarrow\right\rangle +h\left|\uparrow\downarrow\uparrow\uparrow\right\rangle -\left|\uparrow\downarrow\uparrow\downarrow\right\rangle \right. \nonumber \\ &&\qquad \qquad\left.-\left|\uparrow\downarrow\downarrow\uparrow\right\rangle +3h\left|\downarrow\uparrow\uparrow\uparrow\right\rangle -\left|\downarrow\uparrow\uparrow\downarrow\right\rangle -\left|\downarrow\uparrow\downarrow\uparrow\right\rangle +2\left|\downarrow\downarrow\uparrow\uparrow\right\rangle\right), \nonumber\\ &&|Q_{4}^{0}\rangle_{h}=\frac{1}{h^{2}+2}\left(h^2\left|\uparrow\uparrow\uparrow\uparrow\right\rangle -h\left|\uparrow\uparrow\uparrow\downarrow\right\rangle +h\left|\uparrow\uparrow\downarrow\uparrow\right\rangle -h\left|\uparrow\downarrow\uparrow\uparrow\right\rangle +\left|\uparrow\downarrow\uparrow\downarrow\right\rangle -\left|\uparrow\downarrow\downarrow\uparrow\right\rangle +h\left|\downarrow\uparrow\uparrow\uparrow\right\rangle -\left|\downarrow\uparrow\uparrow\downarrow\right\rangle \right. \nonumber \\ &&\qquad \qquad \left.+\left|\downarrow\uparrow\downarrow\uparrow\right\rangle\right). \end{align}\]

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