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International Conference on Signal Image Processing and Communication (ICSIPC)

An explicit construction of minimal support AME states

Guanmin Guo, Ruihu Li

Published Oct 9, 2022• 0 citations
Publisher

Abstract

Absolutely maximally entangled (abbreviation AME) states are pure multipartite states that, when reduced to half or less of the sites, the maximum entropy mixed state is obtained, which have found significant applications in teleportation and quantum secret sharing. In this paper, we consider the sufficient conditions for the existence of the minimal support AME (n, q) state, and present a systemically explicit construction about minimal support AME states, then several minimal support AME states are drawn by employing GRS codes, Singleton arrays and their deformed arrays.

DOI
10.1117/12.2643577

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Absolutely maximally entangled (abbreviation AME) states are pure multipartite states that, when reduced to half or less of the sites, the maximum entropy mixed state is obtained, which have found significant applications in teleportation and quantum secret sharing. In this paper, we consider the sufficient conditions for the existence of the minimal support AME (n, q) state, and present a systemically explicit construction about minimal support AME states, then several minimal support AME states are drawn by employing GRS codes, Singleton arrays and their deformed arrays.
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Absolutely maximally entangled, AME, states are pure multipartite states that give rise to the maximally mixed states when half or more of the parties are traced out. AME states find applications in fields like teleportation or quantum secret sharing, and the problem of finding conditions on their existence has been considered in a number of papers. We consider here AME states of minimal support, that are simpler to analyse. An equivalence with coding theory gives a sufficient condition for their existence, that the number of sites be equal to the local dimension plus one, when the local dimension ddd is a power of a prime number. In this paper, an equivalence with latin hypercubes is used to prove that the above sufficient condition fails in the first case in which the local dimension is not a prime power, d=6d=6d=6. Results for other values of ddd are also given.
In this paper we prove that no absolutely maximally entangled, AME, state with minimal support exists with 7 sites and 5 levels. General AME states are pure multipartite states that, when reduced to half or less of the sites, the maximum entropy mixed state is obtained. They have found applications in teleportation and quantum secret sharing, and finding conditions for their existence is a well known open problem. We consider the version of this problem for minimally supported AME states. We single out known both sufficient and necessary conditions in that case. From our negative result, we show that the necessary condition is not sufficient. The proof uses a recent result on the theory of general, nonlinear, classical codes.
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.
Understanding and classifying multipartite entanglement is fundamental to quantum information processing. This work focuses on absolutely maximally entangled (AME) states, a class of highly entangled states characterized by their maximal entanglement across any bipartitions. To analyze and classify AME states, we employ the minimal decomposition entropy, defined as the minimum Rényi entropy SqS_qSq​ associated with the state's decomposition over all local product bases. This quantity identifies the product bases in which the state is maximally localized, thereby yielding optimal representations for analyzing properties of AME states. We develop an efficient algorithm for computing the minimal decomposition entropy for finite q>1q>1q>1 and compare AME and Haar-random states for q=2 q = 2 q=2 and q=∞ q = \infty q=∞ in qubit, qutrit, and ququad systems. For q=2 q = 2 q=2, AME states of four qutrits and ququads show lower minimal entropy than generic states, indicating sparser optimal forms. For q=∞ q = \infty q=∞ -- related to the geometric measure of entanglement -- AME states exhibit higher entanglement. The algorithm also simplifies known AME states into sparser representations, aiding in distinguishing genuinely quantum AME states from those constructible from classical combinatorial designs. Our results advance the classification of AME states and demonstrate the utility of minimal decomposition entropy as both a local unitary invariant and a tool for state simplification.
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.