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arXiv:2608.13907

Robust Quantum Extremal Numbers

Wanchen Zhang, Zicheng Han, Xiande Zhang

quant-phSubmitted Aug 14, 2026• Announced Aug 17, 2026• 0 citations
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Abstract

Absolutely maximally entangled states require every reduction of at most half of the parties to be maximally mixed, a condition that is both rigid and often impossible for qubit systems. Previous work introduced the quantum extremal number, which maximizes the number of exactly maximally mixed half-body marginals, and determined the exact value Qex(8,4)=56. The present work develops a robust extension of this extremal problem. For a subsystem $A$, the marginal maximal-mixing defect is defined by \[ D_A=2^{|A|}\operatorname{Tr}(ρ_A^2)-1 =2^{|A|}\left\|ρ_A-\frac{I_A}{2^{|A|}}\right\|_2^2, \] and $Q_{\mathrm{ex},\varepsilon}^{D}(n,k)$ is defined as the maximum number of $k$-body marginals satisfying $D_A\leq\varepsilon$ in an $n$-qubit pure state. This counting problem differs from approximate $k$-uniformity, which requires all $k$-body marginals to obey a common error bound. For pure states on $4m$ qubits, the following local stability inequality is established: \[ \sum_{i\in T}D_{T\setminus\{i\}}\geq1 \qquad (|T|=2m+1). \] It follows that, whenever $\varepsilon<1/(2m+1)$, the hypergraph of $\varepsilon$-good $2m$-subsets is $K_{2m+1}^{(2m)}$-free. Combined with the known exact eight-qubit construction, this yields the stability plateau \[ Q_{\mathrm{ex},\varepsilon}^{D}(8,4)=56, \qquad 0\leq\varepsilon<\frac15. \] For odd systems of $2k+1$ qubits, the exact forbidden hypergraph $H_k$ is used to derive explicit finite-error stability radii. In particular, $Q_{\mathrm{ex},\varepsilon}^{D}(9,4)\leq120$ for $0\leq\varepsilon<1/17$. These results turn exact quantum Turán obstructions into quantitative robustness statements and identify intervals on which quantum extremal numbers are stable under imperfect marginal mixedness.

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Absolutely Maximally Entangled States of $2q$ Parties in Every Odd Prime-Power Dimension $q$

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Absolutely maximally entangled (AME) states represent an extreme form of multipartite entanglement: every reduced system containing at most half of the parties is maximally mixed. These states provide perfect tensors and optimal quantum error-correcting codes, yet their existence is known only in restricted parameter regimes. For every odd prime power $q=p^e\ge3$, we construct a stabilizer $\mathrm{AME}(2q,q)$ state whose normalized one-party projection yields a stabilizer $\mathrm{AME}(2q-1,q)$ state. A closed-form $q\times q$ bordered-circulant matrix $A_q$ over $\mathbb{F}_{q^2}$ generates a Hermitian self-dual maximum distance separable (MDS) code $[2q,q,q+1]_{q^2}$, which lies outside the extended Reed-Solomon classes. In suitable bases, the amplitude tensors define normalized $q$-unitary complex Hadamard matrices of order $q^q$ with $p$th-root phases. Additional constructions yield $\mathrm{AME}(q+3,q)$ states and families at intermediate particle numbers through explicit rescalings of selected submatrices. We also provide nine explicit parent matrices and the corresponding one-party projections.

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Pure multiparticle quantum states are called absolutely maximally entangled if all reduced states obtained by tracing out at least half of the particles are maximally mixed. We provide a method to characterize these states for a general multiparticle system. With that, we prove that a seven-qubit state whose three-body marginals are all maximally mixed, or equivalently, a pure ((7,1,4))_{2} quantum error correcting code, does not exist. Furthermore, we obtain an upper limit on the possible number of maximally mixed three-body marginals and identify the state saturating the bound. This solves the seven-particle problem as the last open case concerning maximally entangled states of qubits.

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An Explicit Scott-Type Bound for Absolutely Maximally Entangled States with Arbitrary Defect

Shixuan Zeng, Xiande Zhang

Absolutely maximally entangled (AME) states and, more generally, $k$-uniform states in $(\C^q)^{\otimes n}$ are central objects in multipartite entanglement theory, with applications to quantum secret sharing, quantum masking, and quantum error correction. In the extremal case $k=\lfloor n/2\rfloor$, Scott (2004) proved a sharp nonexistence bound showing that AME states cannot exist once the number of parties $n$ exceeds a threshold of order $2q^{2}$ (with a parity dependence on $n$), where $q$ is the local dimension. Recently, Ning et al.\ studied \emph{defective} AME states (i.e., $k=\lfloor n/2\rfloor-l$ with $l>0$), gave explicit Scott-type bounds for defects $l=1,2$ and conjectured a general $(2l+2)q^{2}+o(q^{2})$ behavior. In this paper, we solve this conjecture and establish a fully explicit Scott-type upper bound for AME states with arbitrary defect $l\ge 0$, yielding Scott's bound for $l=0$ and Ning et al.'s bounds for $l=1,2$ as special cases. Equivalently, this gives nonexistence bounds for one-dimensional pure quantum error-correcting codes near the quantum Singleton regime. The proof uses a truncated MacWilliams linear-programming system and an explicit infeasibility certificate. As a direct application, we derive explicit asymptotic upper bounds on $k/n$ for fixed local dimension $q$, improving the implicit upper bounds given by Ning et al.

quant-pharXiv:2606.01943•IEEE Transactions on Information Theory•Jun 2, 2026•0 citations

A maximum concurrence criterion to investigate absolutely maximally entangled states

Subhasish Bag, Sarkar, Ramita, Prasanta Kumar Panigrahi

We propose a straightforward method to determine the maximal entanglement of pure states using the criterion of maximal I-concurrence, a measure of entanglement. The square of concurrence for a bipartition $X|X^\prime$ of a pure state is defined as $E^2_{X| X ^\prime}=2[1-tr({ρ_X}^2)]$. From this, we can infer that the concurrence $E_{X| X ^\prime}$ reaches its maximum when $tr({ρ_X}^2)$ is minimized. Using this approach, we identify numerous Absolutely Maximally Entangled (AME) pure states that exhibit maximal entanglement across all possible bipartitions. Conditions are derived for pure states to achieve maximal mixedness in all bipartitions, revealing that any pure state with an odd number of subsystem coefficients does not meet the AME criterion. Furthermore, we obtain equal maximal multipartite entangled pure states across all bipartitions using our maximal concurrence criterion.

arXiv:2501.15477•Jan 26, 2025•0 citations
Absolutely maximally entangled states require every reduction of at most half of the parties to be maximally mixed, a condition that is both rigid and often impossible for qubit systems. Previous work introduced the quantum extremal number, which maximizes the number of exactly maximally mixed half-body marginals, and determined the exact value Qex(8,4)=56. The present work develops a robust extension of this extremal problem. For a subsystem AAA, the marginal maximal-mixing defect is defined by [ D_A=2^{|A|}\operatorname{Tr}(ρ_A^2)-1 =2^{|A|}\left|ρ_A-\frac{I_A}{2^{|A|}}\right|_2^2, ] and Qex,εD(n,k)Q_{\mathrm{ex},\varepsilon}^{D}(n,k)Qex,εD​(n,k) is defined as the maximum number of kkk-body marginals satisfying DA≤εD_A\leq\varepsilonDA​≤ε in an nnn-qubit pure state. This counting problem differs from approximate kkk-uniformity, which requires all kkk-body marginals to obey a common error bound.
For pure states on 4m4m4m qubits, the following local stability inequality is established: [ \sum_{i\in T}D_{T\setminus{i}}\geq1 \qquad (|T|=2m+1). ] It follows that, whenever ε<1/(2m+1)\varepsilon<1/(2m+1)ε<1/(2m+1), the hypergraph of ε\varepsilonε-good 2m2m2m-subsets is K2m+1(2m)K_{2m+1}^{(2m)}K2m+1(2m)​-free. Combined with the known exact eight-qubit construction, this yields the stability plateau [ Q_{\mathrm{ex},\varepsilon}^{D}(8,4)=56, \qquad 0\leq\varepsilon<\frac15. ] For odd systems of 2k+12k+12k+1 qubits, the exact forbidden hypergraph HkH_kHk​ is used to derive explicit finite-error stability radii. In particular, Qex,εD(9,4)≤120Q_{\mathrm{ex},\varepsilon}^{D}(9,4)\leq120Qex,εD​(9,4)≤120 for 0≤ε<1/170\leq\varepsilon<1/170≤ε<1/17. These results turn exact quantum Turán obstructions into quantitative robustness statements and identify intervals on which quantum extremal numbers are stable under imperfect marginal mixedness.
A pure multipartite quantum state is called absolutely maximally entangled if all reductions of no more than half of the parties are maximally mixed. However, an nnn-qubit absolutely maximally entangled state exists only when nnn equals 2, 3, 5, and 6. A natural question arises when it does not exist: Which nnn-qubit pure state has the largest number of maximally mixed $\ensuremath{\lfloor}n/2\ensuremath{\rfloor}$-party reductions? We denote this number by Qex(n)\mathrm{Qex}(n)Qex(n). It was shown that Qex(4)=4\mathrm{Qex}(4)=4Qex(4)=4 by Higuchi and Sudbery [Phys. Lett. A 273, 213 (2000)] and Qex(7)=32\mathrm{Qex}(7)=32Qex(7)=32 by Huber et al. [Phys. Rev. Lett. 118, 200502 (2017)]. In this paper, we give a general upper bound of Qex(n)\mathrm{Qex}(n)Qex(n) by linking the well-known Tur\'an problem in graph theory and provide lower bounds using constructive and probabilistic methods. In particular, we show that Qex(8)=56\mathrm{Qex}(8)=56Qex(8)=56, which is the third known value for this problem.
Absolutely Maximally Entangled States of 2q2q2q Parties in Every Odd Prime-Power Dimension qqq
Absolutely maximally entangled (AME) states represent an extreme form of multipartite entanglement: every reduced system containing at most half of the parties is maximally mixed. These states provide perfect tensors and optimal quantum error-correcting codes, yet their existence is known only in restricted parameter regimes. For every odd prime power q=pe≥3q=p^e\ge3q=pe≥3, we construct a stabilizer AME(2q,q)\mathrm{AME}(2q,q)AME(2q,q) state whose normalized one-party projection yields a stabilizer AME(2q−1,q)\mathrm{AME}(2q-1,q)AME(2q−1,q) state. A closed-form q×qq\times qq×q bordered-circulant matrix AqA_qAq​ over Fq2\mathbb{F}_{q^2}Fq2​ generates a Hermitian self-dual maximum distance separable (MDS) code [2q,q,q+1]q2[2q,q,q+1]_{q^2}[2q,q,q+1]q2​, which lies outside the extended Reed-Solomon classes. In suitable bases, the amplitude tensors define normalized qqq-unitary complex Hadamard matrices of order qqq^qqq with pppth-root phases. Additional constructions yield AME(q+3,q)\mathrm{AME}(q+3,q)AME(q+3,q) states and families at intermediate particle numbers through explicit rescalings of selected submatrices. We also provide nine explicit parent matrices and the corresponding one-party projections.
Pure multiparticle quantum states are called absolutely maximally entangled if all reduced states obtained by tracing out at least half of the particles are maximally mixed. We provide a method to characterize these states for a general multiparticle system. With that, we prove that a seven-qubit state whose three-body marginals are all maximally mixed, or equivalently, a pure ((7,1,4))_{2} quantum error correcting code, does not exist. Furthermore, we obtain an upper limit on the possible number of maximally mixed three-body marginals and identify the state saturating the bound. This solves the seven-particle problem as the last open case concerning maximally entangled states of qubits.
Absolutely maximally entangled (AME) states are pure multi-partite generalizations of the bipartite maximally entangled states with the property that all reduced states of at most half the system size are in the maximally mixed state. AME states are of interest for multipartite teleportation and quantum secret sharing and have recently found new applications in the context of high-energy physics in toy models realizing the AdS/CFT-correspondence. We work out in detail the connection between AME states of minimal support and classical maximum distance separable (MDS) error correcting codes and, in particular, provide explicit closed form expressions for AME states of nnn parties with local dimension qqq a power of a prime for all q≥n−1q \geq n-1q≥n−1. Building on this, we construct a generalization of the Bell-basis consisting of AME states and develop a stabilizer formalism for AME states. For every q≥n−1q \geq n-1q≥n−1 prime we show how to construct QECCs that encode a logical qudit into a subspace spanned by AME states. Under a conjecture for which we provide numerical evidence, this construction produces a family of quantum error correcting codes [ ⁣[n,1,n/2] ⁣]q[\![n,1,n/2]\!]_q[[n,1,n/2]]q​ for nnn even, saturating the quantum Singleton bound. We show that our conjecture is equivalent to the existence of an operator whose support cannot be decreased by multiplying it with stabilizer products and explicitly construct the codes up to n=8n = 8n=8.
Absolutely maximally entangled (AME) states and, more generally, kkk-uniform states in $(\C^q)^{\otimes n}$ are central objects in multipartite entanglement theory, with applications to quantum secret sharing, quantum masking, and quantum error correction. In the extremal case k=⌊n/2⌋k=\lfloor n/2\rfloork=⌊n/2⌋, Scott (2004) proved a sharp nonexistence bound showing that AME states cannot exist once the number of parties nnn exceeds a threshold of order 2q22q^{2}2q2 (with a parity dependence on nnn), where qqq is the local dimension. Recently, Ning et al.\ studied \emph{defective} AME states (i.e., k=⌊n/2⌋−lk=\lfloor n/2\rfloor-lk=⌊n/2⌋−l with l>0l>0l>0), gave explicit Scott-type bounds for defects l=1,2l=1,2l=1,2 and conjectured a general (2l+2)q2+o(q2)(2l+2)q^{2}+o(q^{2})(2l+2)q2+o(q2) behavior. In this paper, we solve this conjecture and establish a fully explicit Scott-type upper bound for AME states with arbitrary defect l≥0l\ge 0l≥0, yielding Scott's bound for l=0l=0l=0 and Ning et al.'s bounds for l=1,2l=1,2l=1,2 as special cases. Equivalently, this gives nonexistence bounds for one-dimensional pure quantum error-correcting codes near the quantum Singleton regime. The proof uses a truncated MacWilliams linear-programming system and an explicit infeasibility certificate. As a direct application, we derive explicit asymptotic upper bounds on k/nk/nk/n for fixed local dimension qqq, improving the implicit upper bounds given by Ning et al.
We propose a straightforward method to determine the maximal entanglement of pure states using the criterion of maximal I-concurrence, a measure of entanglement. The square of concurrence for a bipartition X∣X′X|X^\primeX∣X′ of a pure state is defined as EX∣X′2=2[1−tr(ρX2)]E^2_{X| X ^\prime}=2[1-tr({ρ_X}^2)]EX∣X′2​=2[1−tr(ρX​2)]. From this, we can infer that the concurrence EX∣X′E_{X| X ^\prime}EX∣X′​ reaches its maximum when tr(ρX2)tr({ρ_X}^2)tr(ρX​2) is minimized. Using this approach, we identify numerous Absolutely Maximally Entangled (AME) pure states that exhibit maximal entanglement across all possible bipartitions. Conditions are derived for pure states to achieve maximal mixedness in all bipartitions, revealing that any pure state with an odd number of subsystem coefficients does not meet the AME criterion. Furthermore, we obtain equal maximal multipartite entangled pure states across all bipartitions using our maximal concurrence criterion.