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arXiv:2411.12208 · Physical Review A

Extremal maximal entanglement

Wanchen Zhang, Yu Ning, Fei Shi, Xiande Zhang

Published May 7, 2025• 3 citations
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Abstract

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 $n$-qubit absolutely maximally entangled state exists only when $n$ equals 2, 3, 5, and 6. A natural question arises when it does not exist: Which $n$-qubit pure state has the largest number of maximally mixed $\ensuremath{\lfloor}n/2\ensuremath{\rfloor}$-party reductions? We denote this number by $\mathrm{Qex}(n)$. It was shown that $\mathrm{Qex}(4)=4$ by Higuchi and Sudbery [Phys. Lett. A 273, 213 (2000)] and $\mathrm{Qex}(7)=32$ by Huber et al. [Phys. Rev. Lett. 118, 200502 (2017)]. In this paper, we give a general upper bound of $\mathrm{Qex}(n)$ 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 $\mathrm{Qex}(8)=56$, which is the third known value for this problem.

DOI
10.1103/physreva.111.052410

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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⌊n/2\ensuremath⌋\ensuremath{\lfloor}n/2\ensuremath{\rfloor}\ensuremath⌊n/2\ensuremath⌋-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 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 DA=2∣A∣Tr⁡(ρA2)−1=2∣A∣∥ρA−IA2∣A∣∥22, D_A=2^{|A|}\operatorname{Tr}(ρ_A^2)-1 =2^{|A|}\left\|ρ_A-\frac{I_A}{2^{|A|}}\right\|_2^2, DA​=2∣A∣Tr(ρA2​)−1=2∣A∣​ρA​−2∣A∣IA​​​22​, 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: ∑i∈TDT∖{i}≥1(∣T∣=2m+1). \sum_{i\in T}D_{T\setminus\{i\}}\geq1 \qquad (|T|=2m+1). ∑i∈T​DT∖{i}​≥1(∣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 Qex,εD(8,4)=56,0≤ε<15. Q_{\mathrm{ex},\varepsilon}^{D}(8,4)=56, \qquad 0\leq\varepsilon<\frac15. Qex,εD​(8,4)=56,0≤ε<51​. 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.
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 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.
Abstract A pure multipartite quantum state is called absolutely maximally entangled (AME), if all reductions obtained by tracing out at least half of its parties are maximally mixed. Maximal entanglement is then present across every bipartition. The existence of such states is in many cases unclear. With the help of the weight enumerator machinery known from quantum error correction and the shadow inequalities, we obtain new bounds on the existence of AME states in dimensions larger than two. To complete the treatment on the weight enumerator machinery, the quantum MacWilliams identity is derived in the Bloch representation. Finally, we consider AME states whose subsystems have different local dimensions, and present an example for a 2 × 3 × 3 × 3 system that shows maximal entanglement across every bipartition.
Pure multipartite quantum states of n parties and local dimension q are called k-uniform if all reductions to k parties are maximally mixed. These states are relevant for our understanding of multipartite entanglement, quantum information protocols and the construction of quantum error correcting codes. To our knowledge, the only known systematic construction of these states is based on a class of classical error correction codes known as maximum distance separable. We present a systematic method to construct other examples of k-uniform states and show that the states derived through our construction are not equivalent to any k-uniform state constructed from maximum distance separable codes. Furthermore, we used our method to construct several examples of absolutely maximally entangled states whose existence was open so far.
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.