1 Introduction↩︎

Multipartite entanglement provides a structural resource in quantum information theory. In this work, we study \(k\)-uniform states in \((\mathbb{C}^q)^{\otimes n}\), namely pure states whose \(k\)-body reduced density matrices are maximally mixed [1], [2]. These states also admit a coding-theoretic interpretation: a \(k\)-uniform state is equivalently a one-dimensional pure quantum error-correcting code with distance \(k+1\) [2], [3]. Thus, the existence or nonexistence of \(k\)-uniform states can be viewed as the existence or nonexistence of pure quantum codes of dimension one. Uniform states also appear in quantum information tasks such as secret sharing [4] and masking [5].

The existence problem for \(k\)-uniform states with prescribed parameters is central to this topic [1], [6]–[14]. A particularly important case is the extremal regime in which \(k\) reaches half of the number of parties; such states are called absolutely maximally entangled (AME) states. Orthogonal arrays and classical error-correcting codes give systematic families of \(k\)-uniform states [1], [6], [11]. AME states have also been connected with combinatorial designs and multiunitary matrices [7], while quantum combinatorial designs provide further constructions of \(k\)-uniform states [8]. More recent constructions go beyond the maximum-distance-separable-code framework and produce additional examples of \(k\)-uniform and AME states [13]. On the other hand, nonexistence results have been developed using quantum weight enumerators, shadow enumerators, and related linear-programming constraints [9], [12], [14]; for instance, the seven-qubit AME state was ruled out in [10]. In this line, deriving necessary conditions, or equivalently upper bounds on \(k\) in terms of the number of systems \(n\), is particularly important [9], [14]. A basic necessary condition for the existence of \(k\)-uniform states in \((\mathbb{C}^q)^{\otimes n}\) is \(k\le \lfloor n/2\rfloor,\) which follows either from the quantum Singleton bound [15], [16] or directly from the Schmidt-rank argument. The extremal case \(k=\lfloor n/2\rfloor\) corresponds to AME states, which represent the strongest form of multipartite entanglement among all pure states. However, AME states do not exist in general. For example, AME states in \((\mathbb{C}^2)^{\otimes n}\) exist if and only if \(n = 2,3,5,6\) [2], [9], [10], [17]. For larger dimension \(q\), many cases remain open, and we refer to the online table of AME states maintained by Huber and Wyderka [18]. Scott [2] proved the nonexistence of AME states in the following celebrated parity-dependent bound [2] in 2004, which rules out all AME states for large \(n\) compared to the dimension \(q\).

Beyond the AME regime, it is natural to study states that are close to AME states, that is, when \(k\) is close to \(\lfloor n/2\rfloor\). Following the notation in [19], for an integer \(l\ge 0\), we call an \((\lfloor n/2\rfloor-l)\)-uniform state an AME state of defect \(l\). The defect parameter quantifies the distance to the AME threshold and provides a convenient scale to formulate refined Scott-type bounds. For qubit systems, Rains’ shadow bounds [12], [17] and their improvements [20], [21] showed the nonexistence of AME states of defect approximately \(n/6\). For qutrit systems, Shi et al. [14] extended the shadow bounds and showed the nonexistence of AME states of defect approximately \(n/14\). There are no such systematic results for general dimension \(q\). Recently, Ning et al. [19] obtained explicit Scott-type bounds for defects \(l=1,2\) for general \(q\) and proposed a general asymptotic picture for fixed \(l\).

Based on Theorem 2, Ning et al.[19] further conjectured that for each fixed defect \(l\) there should exist a Scott-type nonexistence threshold with leading term \((2l+2)q^{2}\) (up to lower-order corrections) [19]. Our first result confirms the predicted leading behavior and, moreover, provides a closed-form bound with explicit lower-order terms and parity dependence. The result is stated below, which settles the Scott-type bound problem for arbitrary defect.

When \(l=0\), Theorem 3 recovers Scott’s bound in Theorem 1; when \(l=1,2\), it recovers the bounds of Ning et al.in Theorem 2. For \(q=2,3\), Theorem 3 does not reach the shadow-bound regimes for defects of order approximately \(n/6\) and \(n/14\), respectively. However, it gives the first explicit Scott-type bound for arbitrary defect \(l\) and arbitrary local dimension \(q\).

By the standard correspondence between \(k\)-uniform states and one-dimensional pure quantum error-correcting codes, the defect parameter also measures the deficiency from the quantum Singleton extremal regime. Then Theorem 3 can be read as a length bound for pure quantum codes \(((n,1,d))_q\) near the quantum Singleton regime. Indeed, the associated code distance is \(d=k+1\), and an AME state of defect \(l\) corresponds to \[d=\left\lfloor\frac{n}{2}\right\rfloor-l+1.\] Thus Theorem 3 rules out one-dimensional pure quantum codes with this distance whenever \(n\) exceeds the same parity-dependent threshold.

Further, as a consequence of Theorem 3, we derive an explicit asymptotic upper bound on \(k/n\) for \(k\)-uniform states at fixed local dimension \(q\).

Ning et al.[19] also derived asymptotic upper bounds on \(k/n\) for fixed \(q\) through an implicit logarithmic inequality, reporting numerical constants for \(4\le q\le 9\). We will show that for every \(q\ge4\), the constants in Theorem 4 are strictly smaller than the bound obtainable from that asymptotic criterion in [19], thereby providing an analytic improvement. This result will be given in Proposition 8. For better clarity, in Table 1, we compare our explicit constants with the numerical constants for \(q\in\{4,5,6,7,8,9\}\) reported by Ning et al., see Table IV in [19].

At a high level, our proof is an enumerator-based linear-programming argument. We impose the standard constraints satisfied by quantum weight and unitary enumerators, linked by the MacWilliams-type identities [9], [19], [22], [23]. For an AME state of defect \(l\), these constraints contain a finite truncated MacWilliams subsystem involving \(2l\) non-negative coefficients \(A_{k+1},A_{k+2},\dots,A_{k+2l}\). The key contribution is an explicit infeasibility certificate for this truncated system once \(n\) exceeds the claimed threshold. This certificate applies uniformly to arbitrary defect \(l\), extends the previous analyses for \(l=1,2\) in [19], and yields the closed-form Scott-type bound in Theorem 3.

Organization. In Section 2 we recall the enumerator constraints needed in the proof. Section 3 establishes Theorem 3. Section 4 derives the asymptotic bound on \(k/n\) and compares it with the bounds of Ning et al. [19].

2 Preliminaries↩︎

2.1 Uniform entangled states↩︎

Throughout the paper, let \(q\ge 2\) and let \(\mathcal{H}=(\mathbb{C}^q)^{\otimes n}\) be the Hilbert space of an \(n\)-partite quantum system with local dimension \(q\). For integers \(a\le b\), write \([a,b]=\{a,a+1,\dots,b\}\) and \([n]=[1,n]\). A pure state in \(\mathcal{H}\) will be denoted by \(|\psi\rangle\), and its density matrix is \(\rho=|\psi\rangle\langle\psi|\). For a subset \(S\subseteq[n]\), we write \(S^c=[n]\setminus S\) and define the reduced density matrix on the parties in \(S\) by \(\rho_S=\operatorname{Tr}_{S^c}(\rho)\), where \(\operatorname{Tr}_{S^c}\) denotes the partial trace over the tensor factors indexed by \(S^c\). Thus \(\rho_S\) is a positive semidefinite matrix of trace one acting on \((\mathbb{C}^q)^{\otimes |S|}\). For standard background on density operators, partial traces, and reduced states, we refer to [24], [25].

A state \(|\psi\rangle\) is called \(k\)-uniform if every \(k\)-party reduced density matrix is maximally mixed, namely \[\rho_S=\frac{I_{q^k}}{q^k},\qquad \forall\, S\subseteq[n],\;|S|=k.\] Equivalently, every reduced density matrix on at most \(k\) parties is maximally mixed. Indeed, if \(T\subseteq[n]\) and \(|T|\le k\), then one can choose \(S\supseteq T\) with \(|S|=k\), and tracing \(\rho_S=I_{q^k}/q^k\) over \(S\setminus T\) gives \(\rho_T=I_{q^{|T|}}/q^{|T|}\). In this sense, \(k\)-uniformity says that no subsystem of size at most \(k\) contains any local information about the global pure state.

The largest possible value of \(k\) is \(\lfloor n/2\rfloor\). To see this, consider a bipartition \(S\sqcup S^c=[n]\). Since \(\rho\) is pure, the two reduced density matrices \(\rho_S\) and \(\rho_{S^c}\) have the same nonzero eigenvalues, counted with multiplicity. If \(\rho_S\) is maximally mixed, then \(\rho_S\) has rank \(q^{|S|}\). Hence \(q^{|S|}\le q^{n-|S|}\), and therefore \(|S|\le n/2\). The extremal case \(k=\lfloor n/2\rfloor\) is called an absolutely maximally entangled state, or an AME state.

We will also use the standard coding-theoretic interpretation. A \(k\)-uniform state in \((\mathbb{C}^q)^{\otimes n}\) is equivalently a one-dimensional pure quantum error-correcting code with parameters \(((n,1,k+1))_q\); in particular, its code distance is \(d=k+1\) [2], [3]. More generally, following Ning et al. [19], if \(k=\lfloor n/2\rfloor-l\) for some integer \(l\ge0\), we call a \(k\)-uniform state an AME state of defect \(l\). In code language, this corresponds to \[d=\left\lfloor\frac{n}{2}\right\rfloor-l+1.\] Thus the defect parameter measures the distance from the AME threshold, or equivalently the deficiency from the near-Singleton regime for one-dimensional pure quantum codes.

2.2 Weight and unitary enumerators↩︎

We now recall the enumerator formalism used in the proof, following the quantum weight-enumerator framework of Shor–Laflamme and Rains [22], [23]. Choose an orthogonal unitary error basis \(\{e_a\}_{a=0}^{q^2-1}\subseteq \mathbb{C}^{q\times q}\) satisfying \(e_0=I_q\) and \(\operatorname{Tr}(e_a^\dagger e_b)=q\delta_{ab}\). For an \(n\)-fold tensor product \(E=e_{a_1}\otimes e_{a_2}\otimes\cdots\otimes e_{a_n}\), define its weight by \(\mathop{\mathrm{wt}}(E)=|\{r\in[n]:a_r\ne0\}|\). Thus the weight counts the number of tensor factors on which the error operator is nontrivial.

The weight enumerator of the pure state \(\rho=|\psi\rangle\langle\psi|\) is the polynomial \(A(x,y)=\sum_{i=0}^n A_i x^{n-i}y^i\), where \[A_i=\sum_{\mathop{\mathrm{wt}}(E)=i}|\operatorname{Tr}(E\rho)|^2.\] The coefficients \(A_i\) are nonnegative by definition, and \(A_0=|\operatorname{Tr}(\rho)|^2=1\). One should think of \(A_i\) as measuring the total strength of the \(i\)-body correlations of the state with respect to the chosen error basis. If \(|\psi\rangle\) is \(k\)-uniform, then all nontrivial observables supported on at most \(k\) parties have zero expectation [2], [19], so \[A_1=A_2=\cdots=A_k=0,\quad A_i\geq 0\text{ for }i\in [k+1,n].\] This is the first source of linear constraints in the proof.

The second enumerator is expressed in terms of purities of reduced density matrices. For \(S\subseteq[n]\), define \(A'_S=\operatorname{Tr}(\rho_S^2)\). This quantity is the purity of the reduced state on \(S\). For each \(i\in[0,n]\), set \[A'_i=\sum_{|S|=i}A'_S =\sum_{|S|=i} \operatorname{Tr}(\rho_S^2)\] and define the unitary enumerator \(A'(x,y)=\sum_{i=0}^n A'_i x^{n-i}y^i\). Since \(\rho\) is pure, the reductions \(\rho_S\) and \(\rho_{S^c}\) have the same nonzero eigenvalues. Hence \(\operatorname{Tr}(\rho_S^2)=\operatorname{Tr}(\rho_{S^c}^2)\), and after summing over all subsets of size \(i\) we get the symmetry \[A'_i=A'_{n-i}, ~i\in [0,n].\] For any density matrix \(\sigma\) on a \(d\)-dimensional Hilbert space, one has \(\operatorname{Tr}(\sigma^2)\ge 1/d\), with equality precisely when \(\sigma=I_d/d\); this follows from Cauchy–Schwarz applied to the eigenvalues of \(\sigma\) [24], [25]. Hence for every \(S\) with \(|S|=i\), one has \(\operatorname{Tr}(\rho_S^2)\ge 1/q^i\). So for any \(i\in [0,n]\), \[A'_i\ge q^{-i}\binom{n}{i}.\] If \(|\psi\rangle\) is \(k\)-uniform and \(i\le k\), then every \(i\)-party reduced density matrix is maximally mixed, so \(\operatorname{Tr}(\rho_S^2)=1/q^i\) for \(|S|=i\). Therefore \[A'_i=q^{-i}\binom{n}{i} \text{ for 0\le i\le k.}\]

2.3 Enumerator constraints↩︎

The weight and unitary enumerators are linked by a quantum MacWilliams-type identity, originating from the enumerator theory of quantum error-correcting codes [9], [22], [23]. In the present rank-one setting of a pure state, the form needed in this paper is \[\label{eq:mcwlm} q^iA'_i=\sum_{j=0}^i A_j\binom{n-j}{i-j},\qquad 0\le i\le n.\tag{1}\]

For completeness, we indicate why this relation holds. The reduced density matrix \(\rho_S\) can be expanded in the tensor-product error basis supported inside \(S\). Orthogonality of the basis gives \[q^{|S|}\operatorname{Tr}(\rho_S^2) = \sum_{\mathop{\mathrm{supp}}(E)\subseteq S}|\operatorname{Tr}(E\rho)|^2.\] Now sum this identity over all subsets \(S\subseteq[n]\) with \(|S|=i\). An error operator of weight \(j\) is supported in exactly \(\binom{n-j}{i-j}\) subsets \(S\) of size \(i\). This gives Eq. (1 ).

We collect all the constraints on the two kinds of enumerators in the following lemma, which will be used to deduce the contradiction in the rest of the paper.

2.4 Main ideas↩︎

Lemma 1 translates the existence of a \(k\)-uniform state into a linear-programming feasibility problem over quantum enumerators. In the positive-defect part of the proof of Theorem 3, we fix \(l\ge1\), equivalently \(k=\lfloor n/2\rfloor-l\). Since \(A_1,\dots,A_k\) vanish, the first potentially nonzero weight-enumerator coefficients are \(A_{k+1},A_{k+2},\dots\). We extract from the MacWilliams-type relation ?? the truncated subsystem with \[i=k+1,k+2,\dots,k+2l.\] After the known terms \(A_0=1\) and \(A_1=\cdots=A_k=0\) are separated, this subsystem gives a lower-triangular linear system for \[A_{k+1},A_{k+2},\dots,A_{k+2l}.\]

The symmetry and lower bounds for the unitary enumerator in Eq. ?? provide the remaining constraints. Thus the nonexistence problem reduces to proving infeasibility of a finite truncated MacWilliams linear program. The main point of the proof is to construct an explicit certificate which annihilates the free parameters in this truncated system and forces a negative value for a nonnegative enumerator expression. This gives a contradiction to the enumerator constraints and yields the claimed Scott-type bound.

3 Proof of Theorem 3↩︎

Suppose that a \(k\)-uniform state \(\ket{\psi}\) exists with \(k=\lfloor n/2\rfloor-l\); we will derive a contradiction. The case \(l=0\) is Scott’s bound [2], while \(l=1,2\) is covered by Ning et al. [19]. Hence we assume \(l\ge 3\) from now on. Throughout the proof, we use the convention that \(\binom{a}{b}=0\) whenever \(b<0\) or \(b>a\). Following the enumerator constraints framework recalled in Lemma 1, we first isolate a finite block of Eq. ?? . Put \[M\triangleq\begin{pmatrix} \binom{n-k-1}{0} & & & \\[2pt] \binom{n-k-1}{1} & \binom{n-k-2}{0} & & \\[2pt] \vdots & \vdots & \ddots & \\[2pt] \binom{n-k-1}{2l-1} & \binom{n-k-2}{2l-2} & \cdots & \binom{n-k-2l}{0} \end{pmatrix}_{2l\times 2l}, \qquad M_{i,j}\triangleq\begin{cases} \binom{n-k-j}{i-j}, &i\ge j,\\ 0, &i<j. \end{cases}\] This lower triangular matrix will be used to collect the unknown coefficients. For \(i=k+1,\dots,k+2l\), Eq. ?? gives the equivalent system \[\label{eqM} M\begin{pmatrix} A_{k+1} \\[2pt] A_{k+2} \\[2pt] \vdots \\[2pt] A_{k+2l} \end{pmatrix}=\begin{pmatrix} q^{k+1}A_{k+1}' \\[2pt] q^{k+2}A_{k+2}' \\[2pt] \vdots \\[2pt] q^{k+2l}A_{k+2l}' \end{pmatrix}-\begin{pmatrix} \binom{n}{k+1} \\[2pt] \binom{n}{k+2} \\[2pt] \vdots \\[2pt] \binom{n}{k+2l} \end{pmatrix}.\tag{2}\] A direct binomial inversion gives \[M^{-1}=\begin{pmatrix} \binom{n-k-1}{0} & & & \\[2pt] -\binom{n-k-1}{1} & \binom{n-k-2}{0} & & \\[2pt] \vdots & \vdots & \ddots & \\[2pt] -\binom{n-k-1}{2l-1} & \binom{n-k-2}{2l-2} & \cdots & \binom{n-k-2l}{0} \end{pmatrix}_{2l\times 2l}, \qquad (M^{-1})_{i,j}=\begin{cases} (-1)^{i+j}\binom{n-k-j}{i-j}, &i\ge j,\\ 0, &i<j. \end{cases}\] Write \(\mathbf{v}_i\) for the \(i\)-th column of \(M^{-1}\), where \(1\le i\le 2l\).

3.1 The Case of \(n\) Even↩︎

Assume first that \(n=2k+2l\). By the symmetry in Eq. ?? , the last primed coefficient in Eq. 2 is known: \[A_{k+2l}'=A_k'=\frac{1}{q^k}\binom{n}{k}.\] The remaining quantities \[A_{k+1},A_{k+2},\dots,A_{k+2l},A_{k+1}',A_{k+2}',\dots,A_{k+2l-1}'\] are treated as variables.

Step 1: We first give general forms of \(A_{k+1},A_{k+2},\dots,A_{k+2l},A_{k+1}',A_{k+2}',\dots,A_{k+2l-1}'\) satisfying Eq. 2 , which will be in Eq. 5 and 6 . Incorporating the symmetry relations among the primed variables, Eq. 2 becomes \[\label{eq95even} L_{(3l-1)\times(4l-1)}\begin{pmatrix} A_{k+1} \\ \vdots \\ A_{k+2l} \\ A_{k+1}' \\ \vdots \\ A_{k+2l-1}' \end{pmatrix}=\begin{pmatrix} -\binom{n}{k+1} \\ \vdots \\ -\binom{n}{k+2l-1} \\ (q^{2l}-1)\binom{n}{k+2l} \\ O_{(l-1)\times 1} \end{pmatrix},\tag{3}\] where \[L\triangleq\begin{pmatrix} M_{2l\times 2l} & P_{2l\times(2l-1)}\\ O_{(l-1)\times 2l} & Q_{(l-1)\times (2l-1)} \end{pmatrix}\] and \[P\triangleq\begin{pmatrix} -q^{k+1} & & & &\\ & -q^{k+2} & & &\\ & & \ddots & &\\ & & & -q^{k+2l-1} &\\ 0& 0 & \cdots & 0 \end{pmatrix},\qquad Q\triangleq\begin{pmatrix} 1 & \cdots & 0 & 0 & 0 & \cdots & -1 \\ & \ddots & 0 & 0 & 0 & \iddots & \\ & & 1 & 0 & -1 & & \end{pmatrix},\] and \(O\) is the all zero matrix.

The invertibility of \(M\), together with the displayed block form, gives \(\mathrm{rank}\,L=3l-1\). Thus the solution set of Eq. 3 is an \(l\)-dimensional affine subspace. We parametrize its homogeneous part as follows. Define vectors in \(\mathbb{R}^{4l-1}\) by \[\mathbf{y}_1\triangleq\begin{pmatrix} q^{k+1}\mathbf{v}_1 \\ 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix},\qquad \mathbf{y}_2\triangleq\begin{pmatrix} q^{k+2}\mathbf{v}_2 \\ 0 \\ 1 \\ \vdots \\ 0 \end{pmatrix},\qquad\cdots,\qquad \mathbf{y}_{2l-1}\triangleq\begin{pmatrix} q^{k+2l-1}\mathbf{v}_{2l-1} \\ 0 \\ 0 \\ \vdots \\ 1 \end{pmatrix}.\] A direct check gives \[L(\mathbf{y}_1+\mathbf{y}_{2l-1})=\cdots=L(\mathbf{y}_{l-1}+\mathbf{y}_{l+1})=L\mathbf{y}_l=0.\] Set \[\mathbf{w}_1\triangleq\mathbf{y}_1+\mathbf{y}_{2l-1},\qquad\cdots,\qquad \mathbf{w}_{l-1}\triangleq\mathbf{y}_{l-1}+\mathbf{y}_{l+1}, \qquad \mathbf{w}_l\triangleq\mathbf{y}_l.\] The last \(2l-1\) coordinates show that the vectors in \(W:=\{\mathbf{w}_i\mid1\le i\le l\}\) are linearly independent. Since the nullity is \(l\), this set forms a basis of \(\mathrm{Ker}\,L\).

We next choose a particular solution of Eq. 3 . Write \[\mathbf{y}_{\mathrm p}=(a_{k+1},\dots,a_{k+2l},a_{k+1}',\dots,a_{k+2l-1}')^{\mathsf T}.\] The primed coordinates are assigned using Eq. ?? and the symmetry \(A_i'=A_{n-i}'\): set \[\begin{align} a_{k+1}'=a_{k+2l-1}'&=\frac{1}{q^{k+1}}\binom{n}{k+1},\\ &\vdots\\ a_{k+l-1}'=a_{k+l+1}'&=\frac{1}{q^{k+l-1}}\binom{n}{k+l-1},\\ a_{k+l}'&=\frac{1}{q^{k+l}}\binom{n}{k+l}. \end{align}\] Inserting these values and \(A_{k+2l}'=\frac{1}{q^k}\binom{n}{k}\) into Eq. 2 gives \(a_{k+1}=\dots=a_{k+l}=0\) and \[\label{value95even} \begin{align} a_{k+l+1}&=(q^2-1)\binom{n}{k+l+1},\\ a_{k+l+2}&=(q^4-1)\binom{n}{k+l+2}-(n-(k+l+1))(q^2-1)\binom{n}{k+l+1}. \end{align}\tag{4}\] The other entries \(a_{k+l+i}\) are determined uniquely for \(1\le i\le l\), but only \(a_{k+l+1}\) and \(a_{k+l+2}\) will be used below. Since \(\mathbf{y}_{\mathrm p}\) satisfies Eq. 3 , every solution of Eq. 3 has the form \[\label{sol95even} \mathbf{y}=t_1\mathbf{w}_1+\cdots+t_{l}\mathbf{w}_{l}+\mathbf{y}_{\mathrm p}, \qquad \text{ for some } t_1,\dots,t_l\in\mathbb{R}.\tag{5}\] Project \(\mathbf{y}\) and \(\mathbf{y}_{\mathrm{p}}\) onto their first \(2l\) coordinates, which correspond to \(A_{k+1},A_{k+2},\dots,A_{k+2l}\). Denote the projections by \(\tilde{\mathbf{y}}\) and \(\tilde{\mathbf{y}}_{\mathrm{p}}\). Then \[\label{eq:ytilde95even} \begin{align} \tilde{\mathbf{y}}&=t_1(q^{k+1}\mathbf{v}_1+q^{k+2l-1}\mathbf{v}_{2l-1})+\cdots+t_{l-1}(q^{k+l-1}\mathbf{v}_{l-1}+q^{k+l+1}\mathbf{v}_{l+1})+t_lq^{k+l}\mathbf{v}_l+\tilde{\mathbf{y}}_{\mathrm p}\\ &=(\mathbf{v}_1,\cdots,\mathbf{v}_{2l})(t_1q^{k+1},\cdots,t_lq^{k+l},t_{l-1}q^{k+l+1},\cdots,t_1q^{k+2l-1},0)^{\mathsf T}+\tilde{\mathbf{y}}_{\mathrm p}\\ &=M^{-1}\boldsymbol{\mu}+\tilde{\mathbf{y}}_{\mathrm p}, \end{align}\tag{6}\] where \[\boldsymbol{\mu}=(t_1q^{k+1},\cdots,t_lq^{k+l},t_{l-1}q^{k+l+1},\cdots,t_1q^{k+2l-1},0)^{\mathsf T}.\]

Step 2: Now assume \(n>2(l+1)q^2+2(l-1)\) and we deduce a contradiction. To rule out a defect-\(l\) absolutely maximally entangled state in this range, it is enough to prove that for every \(t_1,\dots,t_l\in\mathbb{R}\), the vector \(\mathbf{y}\) in Eq. 5 has a negative component. That is, some \(A_i\) or \(A_j'\) must be negative, which contradicts Lemma 1. Suppose instead that some choice of \(t_1,\dots,t_l\) makes all components of \[\mathbf{y}=t_1\mathbf{w}_1+\cdots+t_{l}\mathbf{w}_{l}+\mathbf{y}_{\mathrm p}\] nonnegative. Then \(\tilde{\mathbf{y}}=M^{-1}\boldsymbol{\mu}+\tilde{\mathbf{y}}_{\mathrm p}\) is componentwise nonnegative as well. Assume that we have a vector \[\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_{2l})^{\mathsf T}\in\mathbb{R}_{\ge 0}^{2l}\] satisfying \[\label{aim95even} \boldsymbol{\lambda}^{\mathsf T}M^{-1}\boldsymbol{\mu}=0.\tag{7}\] Then \[\boldsymbol{\lambda}^{\mathsf T}\tilde{\mathbf{y}}=\boldsymbol{\lambda}^{\mathsf T}(M^{-1}\boldsymbol{\mu}+\tilde{\mathbf{y}}_{\mathrm p})=\boldsymbol{\lambda}^{\mathsf T}\tilde{\mathbf{y}}_{\mathrm p}\ge 0.\] In this way, we transfer the necessary condition \(\mathbf{y}\ge 0\) for some \(t_1,\dots,t_l\in\mathbb{R}\) to \(\boldsymbol{\lambda}^{\mathsf T}\tilde{\mathbf{y}}_{\mathrm p}\geq 0\) for a specific \(\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_{2l})^{\mathsf T}\in\mathbb{R}_{\ge 0}^{2l}\). If there exists such a \(\boldsymbol{\lambda}\) satisfying Eq. 7 but \(\boldsymbol{\lambda}^{\mathsf T}\tilde{\mathbf{y}}_{\mathrm p}<0\), then we have a contradiction. Next, we show such a vector \(\boldsymbol{\lambda}\) exists.

First, we show a vector \(\boldsymbol{\lambda}\) satisfying Eq. 7 exists by presetting some of the entries.

Assuming Claim 5, the preceding construction gives \[\boldsymbol{\lambda}^{\mathsf T}\tilde{\mathbf{y}}=\boldsymbol{\lambda}^{\mathsf T}\tilde{\mathbf{y}}_{\mathrm p}=\sum_{i=1}^{2l}\lambda_{i}a_{k+i}=la_{k+l+1}+\frac{l+1}{n-(k+l+1)}a_{k+l+2}\ge 0.\] Using Eq. 4 , this becomes \[l(q^2-1)\binom{n}{k+l+1}+\frac{l+1}{n-(k+l+1)}\left((q^4-1)\binom{n}{k+l+2}-(n-(k+l+1))(q^2-1)\binom{n}{k+l+1}\right)\ge 0.\] Cancelling the common terms gives \[\frac{l+1}{n-(k+l+1)}(q^4-1)\binom{n}{k+l+2}\ge(q^2-1)\binom{n}{k+l+1}.\] Using the identity \(\binom{n}{k+l+2}=\frac{n-(k+l+1)}{k+l+2}\binom{n}{k+l+1}\), we obtain \[\frac{l+1}{k+l+2}(q^2+1)\ge1, \qquad\text{i.e.}\qquad k\le(l+1)q^2-1.\] Consequently \(n=2k+2l\le2(l+1)q^2+2(l-1)\), contradicting the assumed lower bound. It remains to prove Claim 5.

We first record a binomial identity used in the verification of the claim, whose proof is given in the Appendix.

We now verify Claim 5.

3.2 The Case of \(n\) Odd↩︎

Now let \(n=2k+2l+1\). In this case, all the \(4l\) variables in Eq. 2 \[A_{k+1},A_{k+2},\dots,A_{k+2l},A_{k+1}',A_{k+2}',\dots,A_{k+2l}'\] are unknown. The proof follows the same strategy as the even case, but the parity changes the symmetry constraints. To make the comparison transparent, we keep the notation of the even case, with the necessary modifications.

Step 1: First, Eq. 2 becomes \[\label{eq95odd} L_{3l\times4l}\begin{pmatrix} A_{k+1} \\ \vdots \\ A_{k+2l} \\ A_{k+1}' \\ \vdots \\ A_{k+2l}' \end{pmatrix}=\begin{pmatrix} -\binom{n}{k+1} \\ \vdots \\ -\binom{n}{k+2l} \\ O_{l\times 1} \end{pmatrix},\tag{18}\] where \[L=\begin{pmatrix} M_{2l\times 2l} & P_{2l\times2l}\\ O_{l\times 2l} & Q_{l\times 2l} \end{pmatrix}\] with \[P=\begin{pmatrix} -q^{k+1} & & & &\\ & -q^{k+2} & & &\\ & & \ddots & &\\ & & & -q^{k+2l} &\\ \end{pmatrix},\qquad Q=\begin{pmatrix} 1 & \cdots & 0 & 0 & \cdots & -1 \\ & \ddots & 0 & 0 & \iddots & \\ & & 1 & -1 & & \end{pmatrix}.\]

In this case, \(\mathrm{rank}\,L=3l\). Thus the solution set of Eq. 18 is still an \(l\)-dimensional affine subspace in \(\mathbb{R}^{4l}\). We define \(2l\) similar vectors in \(\mathbb{R}^{4l}\): \[\mathbf{y}_1=\begin{pmatrix} q^{k+1}\mathbf{v}_1 \\ 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix},\qquad \mathbf{y}_2=\begin{pmatrix} q^{k+2}\mathbf{v}_2 \\ 0 \\ 1 \\ \vdots \\ 0 \end{pmatrix},\qquad\cdots,\qquad \mathbf{y}_{2l}=\begin{pmatrix} q^{k+2l}\mathbf{v}_{2l} \\ 0 \\ 0 \\ \vdots \\ 1 \end{pmatrix},\] which satisfy \(L(\mathbf{y}_1+\mathbf{y}_{2l})=L(\mathbf{y}_2+\mathbf{y}_{2l-1})=\cdots=L(\mathbf{y}_{l}+\mathbf{y}_{l+1})=0.\) This gives \(l\) solutions to the homogeneous part of Eq. 18 : \[\mathbf{w}_1=\mathbf{y}_1+\mathbf{y}_{2l},\qquad\mathbf{w}_2=\mathbf{y}_2+\mathbf{y}_{2l-1},\qquad\cdots,\qquad \mathbf{w}_{l}=\mathbf{y}_{l}+\mathbf{y}_{l+1},\] which are also linearly independent, and form a basis of \(\mathrm{Ker}\,L\).

The particular solution \(\mathbf{y}_{\mathrm p}=(a_{k+1},\dots,a_{k+2l},a_{k+1}',\dots,a_{k+2l}')^{\mathsf T}\) to Eq. 18 is chosen as before, which gives \[\begin{align} a_{k+1}'=a_{k+2l}'&=\frac{1}{q^{k+1}}\binom{n}{k+1},\\ &\vdots\\ a_{k+l}'=a_{k+l+1}'&=\frac{1}{q^{k+l}}\binom{n}{k+l}, \end{align}\] and \[\label{value95odd} \begin{align} a_{k+l+1}&=(q-1)\binom{n}{k+l+1},\\ a_{k+l+2}&=(q^3-1)\binom{n}{k+l+2}-(n-(k+l+1))(q-1)\binom{n}{k+l+1}. \end{align}\tag{19}\] Finally, each solution to Eq. 18 has the form \[\label{sol95odd} \mathbf{y}=t_1\mathbf{w}_1+\cdots+t_{l}\mathbf{w}_{l}+\mathbf{y}_{\mathrm p},\qquad t_1,\dots,t_l\in\mathbb{R},\tag{20}\] and its projection to the first \(2l\) coordinates satisfies \[\tilde{\mathbf{y}}=M^{-1}\boldsymbol{\mu}+\tilde{\mathbf{y}}_{\mathrm p}\] with \[\boldsymbol{\mu}=(t_1q^{k+1},\cdots,t_lq^{k+l},t_{l}q^{k+l+1},\cdots,t_1q^{k+2l})^{\mathsf T}.\]

Step 2: Now assume \[n>2(l+1)q^2+2(l+1)q+(2l-1).\] As in the even case, suppose for contradiction that some choice of \(t_1,\dots,t_l\) makes all components of \(\mathbf{y}\) in Eq. 20 nonnegative. Then its projection \[\tilde{\mathbf{y}}=M^{-1}\boldsymbol{\mu}+\tilde{\mathbf{y}}_{\mathrm p}\] is also componentwise nonnegative. We will construct a vector \[\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_{2l})^{\mathsf T}\in\mathbb{R}_{\ge0}^{2l}\] satisfying \[\label{aim95odd} \boldsymbol{\lambda}^{\mathsf T}M^{-1}\boldsymbol{\mu}=0.\tag{21}\] For such a vector, the assumed nonnegativity of \(\tilde{\mathbf{y}}\) implies \[\boldsymbol{\lambda}^{\mathsf T}\tilde{\mathbf{y}} = \boldsymbol{\lambda}^{\mathsf T}(M^{-1}\boldsymbol{\mu}+\tilde{\mathbf{y}}_{\mathrm p}) = \boldsymbol{\lambda}^{\mathsf T}\tilde{\mathbf{y}}_{\mathrm p} \ge0.\] We will show that this necessary inequality contradicts the assumed lower bound on \(n\).

The proof of Claim 6 is similar to that of Claim 5, so we move it to the Appendix. Assuming Claim 6, the preceding construction gives \[\boldsymbol{\lambda}^{\mathsf T}\tilde{\mathbf{y}}=\boldsymbol{\lambda}^{\mathsf T}\tilde{\mathbf{y}}_{\mathrm p}=\sum_{i=1}^{2l}\lambda_{i}a_{k+i}=la_{k+l+1}+\frac{l+1}{n-(k+l+1)}a_{k+l+2}\ge 0.\] Using Eq. 19 , this becomes \[l(q-1)\binom{n}{k+l+1}+\frac{l+1}{n-(k+l+1)}((q^3-1)\binom{n}{k+l+2}-(n-(k+l+1))(q-1)\binom{n}{k+l+1})\ge 0.\] After cancelling the common terms, we get \[\frac{l+1}{n-(k+l+1)}(q^3-1)\binom{n}{k+l+2}\ge(q-1)\binom{n}{k+l+1}.\] The identity \(\binom{n}{k+l+2}=\frac{n-(k+l+1)}{k+l+2}\binom{n}{k+l+1}\) then gives \[\frac{l+1}{k+l+2}(q^2+q+1)\ge1, \qquad\text{i.e.}\qquad k\le(l+1)q^2+(l+1)q-1.\] Consequently \(n=2k+2l+1\le2(l+1)q^2+2(l+1)q+(2l-1)\), contradicting the assumed lower bound.

4 Asymptotic Upper Bound of \(k\)-Uniform States↩︎

In this section, we apply Theorem 3 to derive an explicit asymptotic upper bound on the ratio \(k/n\) for \(k\)-uniform states in \((\mathbb{C}^q)^{\otimes n}\) when the local dimension \(q\) is fixed.

We next compare Eq. ?? with the asymptotic criterion of Ning et al. [19]. Define, for \(0\le \theta<1/2\), \[\label{eq:ning95rate95function} \begin{align} \Phi_q(\theta)&\triangleq\frac{1}{2}\log(1-2\theta) +(1-\theta)\log\left(\frac{1-\theta}{1-2\theta}\right) +\theta\log 2-(1-2\theta)\log q. \end{align}\tag{22}\] Theorem IV.5 of Ning et al.states that if \(\Phi_q(\theta)>0\), then every \(k\)-uniform state in \((\mathbb{C}^q)^{\otimes n}\) satisfies \(k\le \theta n\) for all sufficiently large \(n\) [19]. Thus the strongest asymptotic constant obtainable from this criterion is \[\frac{k}{n}\leq \theta_{\mathrm N}(q):=\inf\{\theta\in[0,1/2):\Phi_q(\theta)>0\}.\]

Next, we show that the parity-free asymptotic bound in Eq. ?? is strictly stronger than the bound obtained from the asymptotic criterion bound \(\theta_{\mathrm N}(q)\) for every \(q\ge4\).

Proposition 8 shows that Theorem 7 gives a strictly stronger asymptotic upper bound on the ratio \(k/n\). A comparison for small \(q\in\{4,5,6,7,8,9\}\) has been listed in Table 1.

5 Conclusion↩︎

In this paper, we resolved the Scott-type nonexistence-bound conjecture posed by Ning et al.[19] for \(k\)-uniform states with arbitrary defect. Equivalently, the result gives explicit length bounds for one-dimensional pure quantum error-correcting codes near the quantum Singleton regime. The proof is based on a truncated MacWilliams linear-programming system for the quantum weight enumerator and the unitary enumerator, together with an explicit infeasibility certificate. This yields a closed-form Scott-type upper bound on the length \(n\) in terms of \((q,l)\) for all \(q\ge2\) and \(l\in\mathbb{Z}_{\ge0}\).

As a direct application, the explicit bound gives new asymptotic upper bounds on the ratio \(k/n\) for \(k\)-uniform states with fixed local dimension \(q\). Compared with the work in [19], our results are explicit and stronger for every \(q\ge4\). As shown in Table 1, the present bound does not improve the shadow-inequality bounds for \(q=2,3\) [14], [17]. It would be interesting to explore whether shadow-inequality methods can yield comparable Scott-type bounds for general local dimension \(q\).

6 sec:Appendix↩︎

References↩︎

---

1. School of the Gifted Young, University of Science and Technology of China. zsx0515@mail.ustc.edu.cn↩︎

2. School of Mathematical Sciences, University of Science and Technology of China; Hefei National Laboratory. drzhangx@ustc.edu.cn↩︎