1 Introduction↩︎

Non-Hermitian Hamiltonians occur in several quantum settings [1]–[4]. In open dynamics, they are often used as effective generators [5]–[7].

When these Hamiltonians are pseudo-Hermitian, exceptional points [8], [9] and dynamical phase transitions take place [10]–[13].

The isomorphism between two Hilbert spaces \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\), endowed with the metric operators \(\eta_1\) and \(\eta_2\), respectively, means the existence of an invertible operator \(T:\mathcal{H}_{1}\to \mathcal{H}_{2}\) such that \[\langle Tx,\eta_2 T y\rangle = \langle x,\eta_1 y\rangle, \qquad \forall x,y\in\mathcal{H}_{1}.\] Equivalently, \(T^\dagger \eta_2 T=\eta_1\). Changing the metric without changing the representatives of states and operators mixes different Hilbert-space structures. This is the source of many apparent inconsistencies in the treatment of non-Hermitian models [14]. The simultaneous transport ensures the equivalence between Hilbert spaces defined by inner products induced by \(\eta_{1}\) and \(\eta_{2}\).

Several prescriptions are used in practice. Some authors use the original inner product and treat the non-Hermitian Hamiltonian as an effective generator of nonunitary evolution, as in open-system or postselected dynamics [7]. Other authors use the biorthogonal description. If the Hamiltonian is diagonalizable, right and left eigenvectors provide a biorthogonal resolution of the identity. When the spectrum is real, this structure can be used to construct a positive metric with respect to which the Hamiltonian is self-adjoint [15], [16]. In that case, the biorthogonal and quasi-Hermitian formulations are closely related.

Other prescriptions are based on the explicit construction of a physical metric operator. This includes formulations in which an additional operator, usually denoted by \(\mathcal{C}\), is used to define a positive inner product [16]. The situation becomes more subtle when the metric is time dependent, since in that case the metric, states, observables, and dynamical generator must be treated consistently within the same time-dependent Hilbert-space representation [17], [18].

For a pseudo-Hermitian Hamiltonian, there exists a similarity operator \(S=S^{\dagger}\) such that \(SH= H^{\dagger}S\). Depending on the model parameter space associated with a given Hamiltonian, the spectrum consists of real eigenvalues or the appearance of complex conjugate pairs. They correspond to the so-called exact symmetry phase and spontaneously broken symmetry phase. The boundary between the two phases consists of exceptional points.

In the exact-symmetry phase, one can construct a positive-definite metric operator \(S\) obeying \(SH=H^{\dagger} S\), which can be taken as a metric inducing a new Hilbert space. This is not the case for the spontaneously broken-symmetry regime and exceptional points. Starting from the similarity operator \(S\), the metric can be defined by using the Krein formalism.

If \[S=VDV^\dagger ,\] with \(D\) real and nonsingular, we define \[K=V \tilde{D}V^\dagger .\] where \(\tilde{D}\) is the absolute value of the diagonal matrix \(D\). This Krein-positivized operator \(K\) defines a Hilbert-space metric. However, \[KH \neq H^\dagger K .\]

In finite dimension, any two metrics are related by a congruence transformation: \[\eta_1=T^\dagger \eta_2 T.\] To be consistent, under this change states and operators must transform as \[x\mapsto Tx, \qquad A\mapsto TAT^{-1}.\]

We shall use uncertainty relations to test this consistency requirement. The Heisenberg–Robertson relation bounds from below the product of variances by the modulus of the expectation value of the commutator [19], [20]. Schrödinger’s refinement strengthens this bound by incorporating the covariance contribution [21]. Product-form relations can become trivial when one variance vanishes, even if the observables are not jointly sharp on the state. This limitation motivates sum-variance relations [22]. These stronger relations have also been investigated in experimental and operational contexts, showing their relevance beyond purely formal considerations [23]–[25].

In this direction, the authors of Ref.[14] introduced the notion of “good observables” in finite-dimensional non-Hermitian systems endowed with a \(G\)-metric inner product [14]. In their formulation, an operator \(O\) is physically admissible when it satisfies \(O^\dagger G=GO\), and the corresponding uncertainty relation is valid for pairs of such metric-compatible observables. From the point of view developed here, this condition can be understood geometrically as the self-adjointness condition in the Hilbert space defined by the metric \(G\).

Although the isomorphism between finite-dimensional Hilbert spaces is a well-known concept in linear algebra, the scope of this work is to provide a way to compare the abundant number of metrics adopted in the literature around the treatment of non-Hermitian Hamiltonians and the consistency of physical results.

The paper is organized as follows. Section 2 fixes the notation for metric representations and recalls the construction of intertwiners in different spectral regimes. Section 3 proves the transport rule for metrics related by congruence. In particular, this section studies the relation between an indefinite intertwiner \(S\), its positive transformed operator \(K\), and the condition for \(K\) to be a quasi-Hermitian metric for the Hamiltonian. Operator \(K\) allows the construction of a new Hilbert space \(\mathcal{H}_{K}\). We show that physical results computed in this new Hilbert space are equivalent to the ones computed in the original Hilbert space \(\mathcal{H}\).

Section 4 applies the formalism to a two-level non-Hermitian spin model and distinguishes the unbroken, broken, and exceptional-point regimes. Section 5 analyzes Robertson and sum-variance uncertainty relations under metric transport. Conclusions are given in Section 6.

2 Formalism↩︎

Consider a finite dimensional Hilbert space \(\mathcal{H}\simeq\mathbb{C}^n\) equipped with the canonical inner product \(\langle .,.\rangle\). Let \(A\) be an operator for which there exists a self-adjoint operator \(\eta\) such that \[\eta A = A^{\dagger}\eta\] This condition implies that \(A\) is a pseudo-Hermitian operator. If \(\eta\) is positive definite, i.e \(\eta= \rho^{\dagger}\rho\), it can be used as a metric to define a new inner product \[\label{eta} \langle .,. \rangle_{\eta}=\langle .,\eta . \rangle\tag{1}\] In this case, operator \(A\) is self-adjoint in \(\mathcal{H}_{\eta}= (\mathcal{H}, \langle .,. \rangle_{\eta})\) where

\[\langle ., A. \rangle_{\eta} =\langle ., \eta A. \rangle = \langle ., A^{\dagger } \eta. \rangle =\langle A ., . \rangle_{\eta}.\] It is straightforward to show that there exists an isomorphism between this Hilbert space and the original one, provided that the operator \(A\) is related to a Hermitian operator \[a=\rho A \rho^{-1}\] in the original Hilbert space [26]–[28].

This observation also clarifies the meaning of the good-observable condition \(O^\dagger G=GO\) used in Ref. [14]. If \(G=\rho^\dagger \rho\), then an operator \(O\) satisfying this condition is self-adjoint in the metric Hilbert space \(\mathcal{H}_G\) and is mapped by the isomorphism to the Hermitian operator \(\rho O\rho^{-1}\) in the reference Hilbert space. Conversely, every Hermitian observable in the reference representation defines a good observable after the inverse transport.

This is consistent with the standard transformation rule under a change of basis. When passing from one basis to another, states and operators must be represented in the same basis. Hence, any operator defined in the initial representation has to be transformed to the new representation before its adjointness properties, expectation values or observable character can be evaluated.

For a pseudo-Hermitian Hamiltonian in the parameter region corresponding to the exact-symmetry phase, it is possible to construct a positive-definite metric operator.

In the spontaneously broken-symmetry regime and at exceptional points it is not possible to define a positive \(\eta\) that serves as a similarity operator between \(A\) and \(A^{\dagger}\).

In such cases, there is no unique canonical prescription in the literature. Instead, different prescriptions have been adopted.

One possible prescription is to keep the initial Hilbert-space structure and study the no-jump effective non-Hermitian Hamiltonian associated with a Lindblad dynamics [5], [7]. In this scheme, the trace of the density matrix is no longer preserved and a renormalization is needed at each instant of time.

Another possibility is to consider the so-called biorthonormal framework [15], [29]. In this approach, a diagonalizable non-Hermitian operator \(A\) is described by right and left eigenvectors, \[A |R_n\rangle = E_n |R_n\rangle, \qquad A^\dagger |L_n\rangle = E_n^* |L_n\rangle ,\]

normalized according to \[\label{ort} \langle L_m|R_n\rangle=\delta_{mn}.\tag{2}\] with the resolution of the identity \[\sum_n |R_n\rangle\langle L_n|=I,\]

Thus, any state in \(\mathcal{H}\) can be expanded as \[|\psi\rangle=\sum_n \langle L_n|\psi\rangle |R_n\rangle .\] The inner product calculated in Eq.(2 ), can be obtained by introducing the operator \[\label{S} S = \sum |L_n\rangle \langle L_n |\tag{3}\] such that \(\langle L_m|R_n\rangle=\langle R_m|R_n\rangle_{S}=\delta_{mn}\).

The operator \(S\) defines a suitable metric for the treatment of pseudo-Hermitian Hamiltonians with real spectra, and can also be considered for operators that do not belong to the pseudo-Hermitian class. It therefore defines the metric Hilbert space \[\mathcal{H}_{S} = \bigl(\mathcal{H},\langle\cdot,\cdot\rangle_{S}\bigr), \qquad \langle f,g\rangle_{S} = \langle f,S g\rangle .\]

The construction of such operator \(S\) was studied in [30], [31], where a finite-dimensional framework was developed for non-Hermitian dynamics. The central idea is that the construction of the metric, and hence of the inner product used to evaluate expectation values, depends on the spectral regime of the Hamiltonian.

In the parameter regime corresponding to the spontaneously broken phase, i.e. when the spectrum contains non-degenerate complex-conjugate pairs, the previous positive construction is no longer sufficient. One can instead construct a self-adjoint intertwining operator by pairing eigenvectors associated with conjugate eigenvalues. Schematically, such an operator has the form \[S= \sum_{j\leq i} \delta(\overline{E}_j-\overline{E}_i^{\,*}) \left( \alpha_j |L_j\rangle \langle L_i| + \alpha_j^* |L_i\rangle \langle L_j| \right),\] with suitable complex coefficients. This operator restores the intertwining relation \[SA=A^\dagger S,\] but it is not positive definite. Hence it defines an indefinite inner product rather than a Hilbert-space metric.

This leads naturally to a Krein-space interpretation. Since \(S\) is self-adjoint, its spectral subspaces can be separated into positive and negative parts. Thus one may write \[\mathcal{H}=\mathcal{H}_+\oplus\mathcal{H}_-,\] and decompose \[S=S_+ + S_-,\] where \(S_+\) is positive on \(\mathcal{H}_+\) and \(S_-\) is negative on \(\mathcal{H}_-\). A positive operator can then be obtained by reversing the sign on the negative subspace: \[K=S_+-S_- .\] Equivalently, if \[S=V D V^\dagger,\] with \(D\) real and nonsingular, one may define \[K=V|D|V^\dagger ,\] where \(|D|\) is obtained by replacing each eigenvalue of \(D\) by its absolute value. The operator \(K\) is self-adjoint and positive definite, and therefore defines the Hilbert space \[\mathcal{H}_{K} = \bigl(\mathcal{H},\langle\cdot,\cdot\rangle_{K}\bigr), \qquad \langle x,y\rangle_{K} = \langle x,K y\rangle .\] However, the positivity of \(K\) should not be confused with pseudo-Hermiticity of \(A\) with respect to \(K\). In general, \[K A\neq A^\dagger K .\] Thus the Krein-positivized metric defines a positive Hilbert-space geometry, but it does not necessarily make the original Hamiltonian self-adjoint in that metric. This distinction is crucial when comparing expectation values, observables and uncertainty relations across metric representations.

At exceptional points the situation is even more singular. In finite dimension, an exceptional point is a set in the parameter space at which eigenvalues and eigenvectors coalesce and the Hamiltonian is no longer diagonalizable. The operator must then be represented by its Jordan decomposition, \[A=\widetilde{P} B\widetilde{P}^{-1},\] where the columns of \(\widetilde{P}\) are generalized eigenvectors \(\{ |\overline{v}_{i} \rangle \}\) and \(B\) a Jordan block matrix. In this case, the metric construction must be formulated in terms of Jordan chains and dual generalized eigenvectors. One introduces a generalized symmetry operator \(S_J\), schematically written as \[\label{SB} S_B= \sum_{j\leq i} \delta(\overline{E}_j-\overline{E}_i^{\,*}) \left( \alpha_j |L_j\rangle \langle\overline{v}_i| + \alpha_j^* |L_i\rangle \langle\overline{v}_j| \right),\tag{4}\] which satisfies \[S_B A=A^\dagger S_B .\] As in the broken diagonalizable case, \(S_B\) is generally indefinite. After separating its positive and negative spectral parts, one obtains a positive operator \[S_{K}=S_{B,+}-S_{B,-},\] which defines an inner product adapted to the Jordan representation.

Notice that \[\label{eq:S-KJ} S=KJ=JK,\tag{5}\] where \(J\) is a fundamental symmetry satisfying \(J^2=I\), \(J^\dagger=J\), and \([J,K]=0\). The indefinite form associated with \(S\) can therefore be written as \[\label{eq:indefinite-form-KJ} [x,y]_S= \langle x,Jy\rangle_K.\tag{6}\]

Finally, if \(A\) is a general non-Hermitian Hamiltonian that is not pseudo-Hermitian, its spectrum is not necessarily real or arranged in complex-conjugate pairs, and \(A\) and \(A^\dagger\) are not isospectral in the pseudo-Hermitian sense. In this case one may still introduce a positive operator \[S_g= \sum_{j=1}^{N_{\max}} |L_j\rangle \langle L_j| ,\] and define \[\langle f,g\rangle_{S_g} = \langle f|S_g g\rangle .\] However, \[S_g A\neq A^\dagger S_g .\] Therefore \(S_g\) provides a positive inner product for evaluating expectation values, but it does not turn \(A\) into a pseudo-Hermitian operator.

Whatever Hilbert-space structure is chosen, and whatever metric is used to define the corresponding inner product, finite-dimensionality ensures that these metric descriptions can be related. More precisely, all positive definite metrics on a finite-dimensional vector space are connected by congruence. Hence, different metric representations may be identified, but only after specifying how the associated states and operators are transformed. We shall make this identification explicit by describing the consistent transport of metrics, states and operators between different representations. This is essential for the analysis of physical quantities, such as expectation values and uncertainty relations

3 Relation between \(\mathcal{H}\) and \(\mathcal{H}_{K}\)↩︎

Let \(\eta_1>0\) and \(\eta_2>0\) be the matrix representation of a positive definite sesquilinear form. It is known that there exists \(T\in GL(n)\) such that \[\eta_1=T^\dagger\eta_2T .\] In fact, we can take \[T=\eta_2^{-1/2}\eta_1^{1/2}.\] or a self-adjoint choice \[\label{eq:T-general} T=\eta_2^{-1/2}\left(\eta_2^{1/2}\eta_{1}\eta_2^{1/2}\right)^{1/2}\eta_2^{-1/2},\tag{7}\]

The congruence relation between metrics, together with the corresponding transformation of states and operators described before, can be applied to the case of the metric \(K\) with respect to the initial metric structure of the space, defined by the inner product \(\langle \cdot,\cdot\rangle\).

The metric \(K\) may also be represented in initial Hilbert-space coordinates. With the spectral notation above, choose \[\label{eq:W-def} T:=V \tilde{D}^{-1/2}.\tag{8}\] Then \(T^\dagger K T=I\). Accordingly, a \(K\)-state \(\tilde{x}\) is represented by \(x_{\rm }=T^{-1}\tilde{x}\), and a \(K\)-observable \(A\) by \(A=T^{-1} \tilde{A}T\). The matrix elements are preserved: \[\label{eq:K-canonical} \langle \tilde{x},\tilde{A} \tilde{y}\rangle_K = \langle T^{-1}x,(T^{-1}AT)T^{-1}y\rangle .\tag{9}\] Thus the passage to initial coordinates is not only a change of coordinates for the metric, but a simultaneous transport of the full representation. By Proposition 1, this transport gives the isomorphic relation between these two Hilbert-spaces.

Disregarding these linear-algebraic notions, or overlooking the full mathematical framework underlying them, may give rise to conceptual and formal inconsistencies in the analysis of systems governed by non-Hermitian Hamiltonians.

4 A two-level non-Hermitian spin model↩︎

We now apply the metric-transport constructed in the previous section to the two-dimensional reduction of the collective-spin Hamiltonian studied in Ref. [31]. This model arises as a finite-dimensional non-Hermitian spin Hamiltonian describing a system of \(N\) collective spins interacting through a dissipative one-axis twisting mechanism [32], [33]. In its general form, the Hamiltonian can be written as \[H=-\frac{\omega}{2}+H_{\mathrm{OAT}}+H_d,\] where \[H_{\mathrm{OAT}}=-\frac{\lambda}{2N}S_z^2, \qquad H_d=2i\kappa S_x .\] Here \(S=(S_x,S_y,S_z)\) denotes the collective spin operator of the system, whose components satisfy the usual angular momentum commutation relations \([S_i,S_j]=i\epsilon_{ijk}S_k\). The corresponding Hilbert space has dimension \(2S+1\), so that the model provides a natural finite-dimensional setting in which the role of metric structures can be analyzed explicitly. The term \(H_{\mathrm{OAT}}\) represents the one-axis twisting interaction, which is commonly associated with spin squeezing effects. The term \(H_d\), being non-Hermitian, introduces an effective dissipative contribution. Depending on the values of the parameters \(\lambda\) and \(\kappa\), the spectrum may be real, may contain complex-conjugate pairs, or may develop exceptional points. For this reason, the model is particularly useful for illustrating the difference between quasi-Hermitian regimes, broken-symmetry regimes and singular exceptional-point limits.

In what follows, we focus on the two-dimensional representation of this Hamiltonian. This reduction is sufficient to display explicitly all the main mechanisms discussed in the previous section. The same formal procedure can be reproduced in higher-dimensional spin representations [34].

In a two-dimensional Hilbert space, the model is given by \[\label{eq:H2} H_2(\omega,\lambda,\kappa)= \begin{pmatrix} \dfrac{\lambda}{4}-\dfrac{\omega}{2} & -i\kappa\\[1mm] -i\kappa & -\dfrac{\lambda}{4}-\dfrac{\omega}{2} \end{pmatrix}.\tag{10}\] Its eigenvalues are \[\label{eq:eigs} E_\pm=-\frac{\omega}{2}\pm\frac{1}{4}\sqrt{\Delta}, \qquad \Delta=\lambda^2-16\kappa^2.\tag{11}\] Thus the model has three different spectral regimes: the exact or unbroken phase, where \(\Delta>0\); the spontaneously broken phase, where \(\Delta<0\); and the exceptional point, where \(\Delta=0\). The aim of this section is not only to construct metrics in these regimes, but to show how the interpretation of these metrics changes according to the spectral properties of \(H_2\).

A general Hermitian intertwining operator used in the two-level model can be written as \[\label{S-param} S= \begin{pmatrix} \beta & i\alpha\\[1mm] -i\alpha &\quad \dfrac{\alpha\lambda}{2\kappa}-\beta \end{pmatrix}, \qquad \alpha,\beta\in\mathbb{R},\tag{12}\] and it satisfies the pseudo-Hermiticity relation \[S H_2=H_2^\dagger S .\] The eigenvalues of \(S\) are \[\label{eq:S-eigs} \sigma_\pm=\frac{\alpha\lambda\pm\Theta}{4\kappa}, \qquad \Theta=\sqrt{ 16(\alpha^2+\beta^2)\kappa^2 -8\alpha\beta\kappa\lambda +\alpha^2\lambda^2}.\tag{13}\] Therefore the sign of \(S\) depends on the region of parameter space. This point is essential: when \(S\) is positive, it may define a Hilbert-space metric compatible with \(H_2\); when \(S\) is indefinite, its positive modulus defines a geometry, but not necessarily a quasi-Hermitian representation of the same Hamiltonian.

4.1 Exact/unbroken phase↩︎

In the exact phase, \(\Delta>0\), the spectrum of \(H_2\) is real. Let \(\delta=\sqrt{\Delta}\), and the left eigenvectors \(|L_{i} \rangle\) are given by \[|L_{1}\rangle=\{-\frac{i (\delta -\lambda )}{4 \kappa },1\} \quad |L_{2}\rangle= \{ \frac{i (\delta +\lambda )}{4 \kappa },1 \}\] Then the operator \(S\) given in Eq.(3 ), becomes \[S_{0}= \left( \begin{array}{cc} \frac{\lambda ^2}{4 \kappa ^2}-2 & \frac{i \lambda }{2 \kappa } \\ -\frac{i \lambda }{2 \kappa } & 2 \\ \end{array} \right)\] in coincidence with Eq.(12 ), for \(\alpha= \frac{\lambda }{2 \kappa }\) and \(\beta= \frac{\lambda ^2}{4 \kappa ^2}-2\).

This is the standard quasi-Hermitian situation. In this case, the intertwiner \(S\) in Eq. 12 can be chosen positive, then \(S\) itself defines another positive metric for the same real-spectrum Hamiltonian. In that case the two metric representations, \(I\) and \(S\), are related by congruence. Namely, there exists \(T\in GL(2)\) such that \[T. S.T^{\dagger} =I.\] where \[T=\frac{\sqrt{2}}{r \sqrt{\lambda ^4-r^2}}\left( \begin{array}{cc} \kappa \left(r r_{+}+\Delta ^2 r_{-}\right) & 4 i \kappa ^2 \lambda r_{-} \\ -4 i \kappa ^2 \lambda r_{-} & \kappa \left(r r_{+}-\Delta ^2 r_{-}\right) \\ \end{array} \right)\] being \[\begin{align} r&=&\sqrt{256 \kappa ^4-16 \kappa ^2 \lambda ^2+\lambda ^4}\\ r_{\pm}&=& \sqrt{\lambda ^2-r} \pm \sqrt{\lambda ^2+r} \end{align}\] According to Proposition 2, the two descriptions are equivalent only if states and operators are transported together: \[\tilde{x}=Tx, \qquad \tilde{A}=T AT^{-1}.\] Then, all matrix elements are preserved: \[\langle x,Ay\rangle_S = \left\langle T x, \left(T A T^{-1}\right)T y \right\rangle = \left\langle \tilde{x}, \tilde{A} \tilde{y} \right\rangle .\] Thus, in the unbroken phase, the freedom in the choice of a positive metric does not create an ambiguity if the complete metric representation is transported consistently. Expectation values, variances and Robertson-type uncertainty relations are then preserved under the change of metric representation.

4.2 Spontaneously broken phase↩︎

We now consider the spontaneously broken phase, \(\Delta<0\). Writing \[\delta_b=\sqrt{16\kappa^2-\lambda^2},\] the eigenvalues are \[\label{eq:eigs-broken} E_\pm=-\frac{\omega}{2}\pm\frac{i}{2}\delta_b .\tag{14}\] Thus the Hamiltonian has a complex-conjugate pair of eigenvalues. This already implies that \(H_2\) cannot be self-adjoint with respect to any positive definite metric. Indeed, if there existed \(\eta>0\) such that \[\eta H_2=H_2^\dagger\eta,\] then \[h=\eta^{1/2}H_2\eta^{-1/2}\] would be Hermitian in the reference Hilbert space. Consequently, the spectrum of \(H_2\) would have to be real, contradicting Eq. 14 .

This spectral condition is also visible at the level of the intertwiner \(S\). Since \[\Theta^2=(\alpha\lambda-4\beta\kappa)^2+16\alpha^2\kappa^2,\] one has, in the broken region \(|\lambda|<4|\kappa|\) and for \(\alpha\neq 0\), \[\Theta>|\alpha\lambda|.\] Therefore, the eigenvalues of \(S\) in Eq.(13 ) have opposite signs \[\kappa>0\Rightarrow \sigma_-<0< \sigma_+, \qquad \kappa<0\Rightarrow \sigma_+ < 0 < \sigma_-.\] and \(S\) is not positive definite. It defines instead an indefinite metric structure.

Following the Krein-type construction discussed in the previous sections, we diagonalize \(S\) as \(S=VDV^\dagger\), and define its positive modulus by replacing the eigenvalues of \(S\) with their absolute values: \[\label{eq:SK-model} K=V \tilde{D}V^\dagger.\tag{15}\] Explicitly, the diagonalizing unitary matrix may be written as \[V= \left( \begin{array}{cc} -\dfrac{i (\alpha \lambda -4 \beta \kappa +\Theta )} {\sqrt{2 \Theta ^2-\Theta (8 \beta \kappa -2 \alpha \lambda )}} & \dfrac{i (-\alpha \lambda +4 \beta \kappa +\Theta )} {\sqrt{\Theta (8 \beta \kappa -2 \alpha \lambda )+2 \Theta ^2}} \\[3mm] \dfrac{4 \alpha \kappa } {\sqrt{2 \Theta ^2-\Theta (8 \beta \kappa -2 \alpha \lambda )}} & \dfrac{4 \alpha \kappa } {\sqrt{\Theta (8 \beta \kappa -2 \alpha \lambda )+2 \Theta ^2}} \end{array} \right).\] The Krein-positivized metric is then \[\label{eq:SK-explicit} K= \frac{1}{\Theta} \left( \begin{array}{cc} s & i \alpha ^2 \lambda \\[1mm] -i \alpha ^2 \lambda & s+\tfrac{\alpha ^2 \lambda ^2}{2 \kappa }-2 \alpha \beta \lambda \end{array} \right),\tag{16}\] where \(s=4\kappa(\alpha^2+\beta^2)-\alpha\beta\lambda\).

The main point is that \(K\) is positive, but positivity alone does not imply quasi-Hermiticity of \(H_2\). Indeed, \(K H_2\neq H_2^\dagger K\). This is exactly the fact identified in Proposition 1. Equivalently, if \[S=KJ, \qquad J=\operatorname{sgn}(S)=\left( \begin{array}{cc} -1 & 0 \\ 0 & 1 \\ \end{array} \right),\] then \[H_2^{\sharp K}=JH_2J .\] Thus \(K\) is a quasi-Hermitian metric for \(H_2\) if and only if \[[J,H_2]=0.\]

The two positive geometries \(I\) and \(K\) can be related by the transport rule of Proposition 2. If \(T_{\rm br}\in GL(2)\) is chosen so that \[T_{br}K. T_{br}^{\dagger}=I,\] then the corresponding dictionary is \[x_{\rm br}=T_{br}x, \qquad A_{\rm br}=T_{br}AT_{br}^{-1}.\] where \[T_{br}=\frac{\sqrt{\kappa }}{\Theta \sqrt{\Theta ^2-\alpha ^2 \lambda ^2}} \left( \begin{array}{cc} \Theta_{-} (4 \beta \kappa -\alpha \lambda )+\Theta \Theta_{+} & 4 i \alpha \kappa \Theta_{-} \\ -4 i \alpha \kappa \Theta_{-} & \Theta_{-} (\alpha \lambda -4 \beta \kappa )+\Theta \Theta_{+} \\ \end{array} \right)\] being \(\Theta_{\pm}= \sqrt{\Theta -\alpha \lambda } \pm \sqrt{\Theta +\alpha \lambda}\) Thus \[\langle x,Ay\rangle_{K} = \left\langle T_{br}x, \left(T_{br}AT_{br}^{-1}\right)T_{br}y \right\rangle = \left\langle x_{br}, A_{br}y_{br} \right\rangle.\] This equation is the concrete form, for the broken two-level model, of the general metric-transport principle. It shows that different metrics can be compared only if the states and the operators are transported together. If one changes the metric but keeps the same coordinate representatives of states and observables, one is mixing different Hilbert-space geometries. Apparent anomalies in expectation values, variances or Robertson-type uncertainty relations may then be artifacts of this mismatch rather than genuine physical effects [14].

4.3 Exceptional point↩︎

At the exceptional point the Hamiltonian is not diagonalizable and its eigenvectors coalesce [35]. We consider the boundary \[\Delta=\lambda^2-16\kappa^2=0,\] and, for simplicity, take \(\lambda=4\kappa\) and \(\kappa\neq 0\). The Hamiltonian and its adjoint are then \[H_{\rm EP} = \begin{pmatrix} \kappa-\dfrac{\omega}{2} & -i\kappa\\[1mm] -i\kappa & -\kappa-\dfrac{\omega}{2} \end{pmatrix}, \qquad H_{\rm EP}^{\dagger} = \begin{pmatrix} \kappa-\dfrac{\omega}{2} & i\kappa\\[1mm] i\kappa & -\kappa-\dfrac{\omega}{2} \end{pmatrix}.\] Both eigenvalues collapse into \[E_{\rm EP}=-\frac{\omega}{2},\] and \(H_{\rm EP}\) has a second-order exceptional point.

The Hermitian intertwining operator obtained from Eq. 4 at the exceptional point is \[S_{\rm EP} = \begin{pmatrix} \beta & i\alpha\\[1mm] -i\alpha & -\beta+2\alpha \end{pmatrix}.\] Its eigenvalues are \[s_\pm = \alpha\pm\sqrt{\alpha^2+(\beta-\alpha)^2}.\] Hence, for \(\alpha\neq\beta\), \(S_{\rm EP}\) is indefinite. As in the broken phase, one may pass from this indefinite form to a positive operator by Krein positivization.

For \(\alpha\neq\beta\), a positive geometry can be obtained from the Krein-positivized form. It can be related to the standard Hilbert-space geometry by a congruence transformation. In the present two-dimensional case one may choose \[T= \frac{1}{2q(\alpha - \beta)} \begin{pmatrix} q q_{+}-(\alpha -\beta ) q_{-} & i \alpha q_{-} \\[1mm] -i \alpha q_{-} & \alpha q_{+}-\beta (q_{+}+q_{-}) \end{pmatrix},\] where \[q= \sqrt{2 \alpha ^2-2 \alpha \beta +\beta ^2}, \qquad q_{\pm}= \sqrt{q - \alpha} \pm \sqrt{q + \alpha}.\] This operator implements the metric transport between the positive geometry associated with the exceptional-point limit and the reference Hilbert space, provided the same transport is applied simultaneously to states and operators.

5 Uncertainty relation under metric transport↩︎

We now examine uncertainty relations from the viewpoint of metric transport [22], [23], [25], [34]. The aim is not to derive a new uncertainty principle for non-Hermitian systems, but to emphasize that uncertainty inequalities are representation-consistent only when expectation values, variances, states, adjoints and observables are evaluated within the same metric structure.

The standard Heisenberg–Robertson relation bounds the product of variances from below by the expectation value of the commutator. However, as was emphasized in Ref. [22], product-form uncertainty relations may become trivial when one of the variances vanishes, even if the observables are not jointly sharp on the state. This motivates stronger relations involving sums of variances, such as \(\Delta A^2+\Delta B^2\), whose lower bounds may remain nontrivial in these limiting situations.

Uncertainty relations for non-Hermitian operators have been discussed from several viewpoints, including formulations based on metric-compatible or “good” observables [14], [25], [36]. The point emphasized here is complementary. Even standard Robertson-type inequalities, or their stronger sum-variance versions, may appear to fail if the metric is changed while the same coordinate representatives of states and observables are kept fixed. Such an apparent failure does not indicate a violation of the uncertainty principle, but rather a mismatch between different metric representations.

Let \(\eta_1>0\), \(\eta_2>0\), and let \(T\in GL(n)\) be such that \[\eta_1=T^\dagger\eta_2T .\] Under the simultaneous transport \[\label{diccionario} \tilde{x}=Tx, \qquad \tilde{A}=TAT^{-1}, \qquad \tilde{B}=TBT^{-1},\tag{17}\] the identities derived in the previous section give \[\langle A\rangle_{\eta_1,x} = \langle \tilde{A}\rangle_{\eta_2,\tilde{x}}, \qquad \Delta^{2}_{\eta_1,x}A = \Delta^{2}_{\eta_2,\tilde{x}}\tilde{A},\] and similarly for \(B\). Here, expectation values are always understood as normalized metric expectation values: \[\langle A\rangle_{\eta,x} = \frac{\langle x,\eta A x\rangle}{\langle x,\eta x\rangle}.\] The commutator and covariance terms are transported in the same way: \[\langle[A,B]\rangle_{\eta_1,x} = \langle[\tilde{A},\tilde{B}]\rangle_{\eta_2,\tilde{x}}.\] Therefore, \[\Delta_{\eta_1,x} A\,\Delta_{\eta_1,x} B \geq \frac{1}{2} \left| \langle[A,B]\rangle_{\eta_1,x} \right|\] holds if and only if \[\Delta_{\eta_2,\tilde{x}} \tilde{A}\, \Delta_{\eta_2,\tilde{x}} \tilde{B} \geq \frac{1}{2} \left| \langle[\tilde{A},\tilde{B}]\rangle_{\eta_2,\tilde{x}} \right|\] is valid in the transported representation.

The same observation applies to sum uncertainty relations. The Maccone-Pati type inequality in the Hilbert space \(\mathcal{H}_{\eta}\) is given by [22] \[\Delta_{\eta,x}^2 A+\Delta_{\eta,x}^2 B \geq \pm i\langle[A,B]\rangle_{\eta,x} + \left| \langle x,(A\pm iB)x^\perp\rangle_{\eta} \right|^2.\] Here \(x^\perp\) satisfies \(\langle x,x^\perp\rangle_\eta=0\), and the sign is chosen so that the commutator contribution is non-negative. A second sum-type bound is \[\Delta_{\eta,x}^2 A+\Delta_{\eta,x}^2 B \geq \frac{1}{2} \left| \langle x^\perp_{A+B},(A+B)x\rangle_{\eta} \right|^2 ,\] where \[x^\perp_{A+B} = \frac{ (A+B-\langle A+B\rangle_{\eta,x})x }{ \Delta_{\eta,x}(A+B) }\] whenever \(\Delta_{\eta,x}(A+B)\neq0\). Equivalently, this last inequality can be written as \[\Delta_{\eta,x}^2 A+\Delta_{\eta,x}^2 B \geq \frac{1}{2} \Delta_{\eta,x}^2(A+B).\] As in the case of the Robertson inequality, the construction of the new Hilbert space endowed with \(\eta_{2}\) through Eq. 17 , together with the corresponding transport of states and operators, guarantees that analogous sum-uncertainty relations remain valid in the transported metric representation.

5.1 Example↩︎

The purpose of this example is not to construct a counterexample to the Robertson–Heisenberg uncertainty relation. Rather, it is to show explicitly how an apparent violation may arise when the metric is changed but the coordinate representatives of the state and of the operators are kept fixed. In such a situation, the quantities entering the uncertainty relation are no longer evaluated within a single metric representation.

We now specialize the example discussed in Section 4 to the broken-symmetry region by considering the particular parameter choice \(\lambda=12\kappa/5\). For this specific case, the matrix constructed in Eq. 16 is given by

\[K(\alpha,\beta)= \frac{1}{\sqrt{34\alpha^2-30\alpha\beta+25\beta^2}} \begin{pmatrix} 5\alpha^2-3\alpha\beta+5\beta^2 & 3i\alpha^2 \\[1mm] -3i\alpha^2 & \frac{43\alpha^2-45\alpha\beta+25\beta^2}{5} \end{pmatrix}.\]

For compactness, we introduce

\[u=\sqrt{25\alpha^2-30\alpha\beta+25\beta^2}, \qquad v=\sqrt{34\alpha^2-30\alpha\beta+25\beta^2},\] \[c=5\alpha^2-3\alpha\beta+5\beta^2, \qquad d=43\alpha^2-45\alpha\beta+25\beta^2.\]

We take

\[A=\sigma_x, \qquad B=\sigma_y, \qquad \psi_K=N_K(x,y)^T, \qquad N_K=\left(\frac{c_\psi}{5v}\right)^{-1/2} =\sqrt{\frac{5v}{c_\psi}}, \qquad x,y\in\mathbb{R},\]

with \(x^2+y^2\neq 0\). With these definitions, the \(K\)-norm of the fixed coordinate vector is

\[\langle \psi_K,\psi_K\rangle_K=1, \qquad c_\psi=5cx^2+dy^2.\]

Let us first perform the computation in the inconsistent way. Namely, we change the metric from the canonical one to \(K(\alpha,\beta)\), but keep fixed the same coordinate representatives \(A\), \(B\), and \(\psi\). Using the \(K\)-metric expectation values, one finds

\[\langle A\rangle_{K,\psi_K}= \frac{ 15i\alpha^2(x^2-y^2)+2v^2xy }{c_\psi}.\]

Similarly,

\[\langle B\rangle_{K,\psi_K}= \frac{ -15\alpha^2(x^2+y^2) + 6i\alpha(3\alpha-5\beta)xy }{c_\psi}.\]

These expectation values are not, in general, real. This already indicates that the fixed Pauli representatives \(A=\sigma_x\) and \(B=\sigma_y\) are not observables in the Hilbert space endowed with the metric \(K\).

The corresponding variances are

\[\Delta_{K,\psi_K}^2 A= \frac{ u^2v^2(x^2-y^2)^2 }{c_\psi^2}.\]

For the second observable one obtains

\[\Delta_{K,\psi_K}^2 B= \frac{ u^2v^2(x^2+y^2)^2 }{c_\psi^2}.\]

Therefore,

\[\Delta_{K,\psi_K}A\Delta_{K,\psi_K}B= \frac{ u^2v^2|x^4-y^4| }{c_\psi^2}.\]

On the other hand, the commutator term evaluated with the same fixed representatives gives

\[\frac{1}{2} \left| \langle[A,B]\rangle_{K,\psi_K} \right|= \frac{ \sqrt{ 900\alpha^4x^2y^2+ \left(5cx^2-dy^2\right)^2 } }{c_\psi}.\]

Equivalently, the difference between the squared left-hand side and the squared right-hand side of the Robertson product inequality is

\[\left( \Delta_{K,\psi_K}A\Delta_{K,\psi_K}B \right)^2 -\frac{1}{4} \left| \langle[A,B]\rangle_{K,\psi_K} \right|^2 = \frac{ u^4v^4(x^4-y^4)^2 }{c_\psi^4} -\frac{ 900\alpha^4x^2y^2+ \left(5cx^2-dy^2\right)^2 }{c_\psi^2}.\]

This quantity can be negative for suitable values of the parameters and of the state. Thus, in the non-transported computation one may obtain the apparent violation

\[\Delta_{K,\psi_K}A\Delta_{K,\psi_K}B < \frac{1}{2} \left| \langle[A,B]\rangle_{K,\psi_K} \right|.\]

This apparent violation occurs because the operators \(A\) and \(B\) have not been transported together with the metric. In particular, with respect to the \(K\)-metric adjoint

\[X^{\sharp_K}=K^{-1}X^\dagger K,\]

the fixed Pauli representatives are not, in general, \(K\)-self-adjoint:

\[A^{\sharp_K}\neq A, \qquad B^{\sharp_K}\neq B.\]

Thus the Robertson-Heisenberg inequality is being applied outside the metric representation in which its hypotheses are satisfied.

Let us now repeat the computation consistently. We use the metric transport operator \(T\) defined by

\[\eta_1=T^\dagger K T,\]

where \(\eta_1\) denotes the initial metric representation. In the present case, the initial representation is the canonical one, so one may take \(\eta_1=I\). The state and operators must be transported simultaneously according to

\[\psi_0=\frac{1}{\sqrt{x^2+y^2}}(x,y)^T, \qquad \widetilde{\psi}=T\psi_0, \qquad \widetilde{A}=TAT^{-1}, \qquad \widetilde{B}=TBT^{-1}.\]

In the \(K\)-metric representation, the corresponding adjoints are

\[\widetilde{A}^{\sharp_K}=K^{-1}\widetilde{A}^\dagger K, \qquad \widetilde{B}^{\sharp_K} =K^{-1}\widetilde{B}^\dagger K.\]

A direct calculation gives

\[\widetilde{A}^{\sharp_K}=\widetilde{A} =\frac{1}{uv} \begin{pmatrix} -15i\alpha^2 & d\\ 5c & 15i\alpha^2 \end{pmatrix}.\]

Likewise,

\[\widetilde{B}^{\sharp_K}=\widetilde{B} =\frac{1}{v^2(u^2+uv)} \begin{pmatrix} 45\alpha^3(5\beta-3\alpha) & \frac{i}{2} \left[ 225\alpha^4- \left(d+uv\right)^2 \right]\\ -\frac{i}{2} \left[ 225\alpha^4- \left(5c+uv\right)^2 \right] & 45\alpha^3(3\alpha-5\beta) \end{pmatrix}.\]

Therefore, the transported operators are genuine observables in the transported metric representation.

The corresponding expectation values are

\[\langle \widetilde{A}\rangle_{K,\widetilde{\psi}}= \frac{2xy}{x^2+y^2}, \qquad \langle \widetilde{B}\rangle_{K,\widetilde{\psi}}=0.\]

Moreover, the variances are

\[\Delta_{K,\widetilde{\psi}}^2\widetilde{A}= \frac{(x^2-y^2)^2}{(x^2+y^2)^2}, \qquad \Delta_{K,\widetilde{\psi}}^2\widetilde{B}=1.\]

Thus

\[\Delta_{K,\widetilde{\psi}}\widetilde{A} \Delta_{K,\widetilde{\psi}}\widetilde{B} = \frac{|x^2-y^2|}{x^2+y^2}.\]

The commutator term is

\[\frac{1}{2} \left| \langle[\widetilde{A},\widetilde{B}] \rangle_{K,\widetilde{\psi}} \right| = \frac{|x^2-y^2|}{x^2+y^2}.\]

Consequently,

\[\Delta_{K,\widetilde{\psi}}\widetilde{A} \Delta_{K,\widetilde{\psi}}\widetilde{B} = \frac{1}{2} \left| \langle[\widetilde{A},\widetilde{B}] \rangle_{K,\widetilde{\psi}} \right|.\]

With the transported state and operators, the Robertson–Heisenberg relation is restored. In this example, the transported computation saturates the Robertson product inequality.

This completes the comparison between the inconsistent and the transported computations. The apparent failure of the Robertson product form comes from mixing inequivalent metric representations.

6 Conclusions↩︎

We have analyzed the consistency of metric representations in finite-dimensional non-Hermitian quantum mechanics. The main point of the work is to stress that the physical description of the system can be given in different Hilbert spaces, through the construction of an isomorphism.

For pseudo-Hermitian operators, we considered theirKrein decomposition \[S=KJ,\] where \(K=|S|\) is positive and \(J=\operatorname{sgn}(S)\) is the associated fundamental symmetry. Although \(K\) defines a positive Hilbert-space geometry, it does not necessarily define a quasi-Hermitian metric for the original Hamiltonian. Indeed, the \(K\)-adjoint of \(H\) is given by \[H^{\sharp_K}=JHJ,\] and therefore \(K\) is compatible with \(H\) as a quasi-Hermitian metric if and only if \[[J,H]=0.\] Nevertheless, \(K\) is a metric and can be used to define a new Hilbert space \(\mathcal{H}_{K}\).

In Section 3 explicit isomorphism between the Hilbert space endowed with the Krein-positivized metric \(K=|S|\) and the usual Hilbert space has been constructed. That means that both Hilbert spaces can be used in defining the dynamics of non-Hermitian systems.

In Section 4 a two-level non-Hermitian spin model has been used to illustrate the main statements of our theoretical discussion. In the unbroken regime, where the spectrum is real, a metric may be used to obtain a quasi-Hermitian representation of the Hamiltonian. In the spontaneously broken regime and at the exceptional points, the Krein-positivized operator \(K\) still defines a positive geometry, but the Hamiltonian is not \(K\)-self-adjoint in the parameter space of the model.

We also used the Robertson uncertainty relation as consistency test. Once states, observables, and their adjoints are evaluated within a single metric Hilbert space, the uncertainty relations retain their physical meaning.

In this sense, our results provide a geometric interpretation of the good-observable prescription of Ref. [14]: good observables are precisely the operators that belong to a given metric representation, or equivalently those obtained by transporting ordinary Hermitian observables through the Hilbert-space isomorphism. This shows that the validity of uncertainty relations is not tied to a fixed coordinate representation, but to the consistent transport of the metric, the states, the adjoint operation and the observables.

The finite-dimensional setting considered here avoids domain subtleties and makes the transport mechanism explicit. In infinite-dimensional systems, additional conditions on domains, boundedness, invertibility, and metric dynamics are required.

7 ↩︎

The purpose of the following lemma is to separate two issues that are often identified too quickly. Although the Krein-positivized operator \(K=|S|\) is a positive metric, the original non-Hermitian Hamiltonian need not be \(K\)-self-adjoint. Hence the admissible observables in the Hilbert space \(\mathcal{H}_K\) must be characterized independently. The lemma below provides this characterization for the two-dimensional broken-symmetry metric used in the main text.

This lemma identifies the class of operators that are self-adjoint in the positive geometry defined by \(K\). In particular, inserting the entries of the original Hamiltonian \(H_2\) into Lemma 1 gives the condition \[\label{eq:condition-H2-SK} \kappa= \frac{\alpha\beta\lambda}{2(\alpha^2+\beta^2)}.\tag{18}\] Under this condition, \[\Delta = \lambda^2-16\kappa^2 = \lambda^2 \left[ 1-\frac{4\alpha^2\beta^2}{(\alpha^2+\beta^2)^2} \right] = \lambda^2 \frac{(\alpha^2-\beta^2)^2}{(\alpha^2+\beta^2)^2} \geq 0.\] Therefore \(H_2\) can be \(K\)-self-adjoint only outside the broken region. Except for the borderline case \(\alpha^2=\beta^2\), where \(\Delta=0\), condition 18 places the model in the unbroken regime \(\Delta>0\). In that regime, however, the original intertwining operator \(S\) may already be chosen positive, and the Krein positivization is not needed.

References↩︎