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arXiv:1907.11253 · Physical Review A

Absolutely maximally entangled states, quantum-maximum-distance-separable codes, and quantum repeaters

Daniel Alsina, Mohsen Razavi

Published Feb 1, 2021• 17 citations
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Abstract

We address the relation between absolutely maximally entangled (AME) states and quantum- maximum-distance-separable (QMDS) codes by constructing whole families of QMDS codes from AME states that have stabilizer representations. We introduce a generic reduction-friendly form for the generator set of the stabilizer representation of an AME state, from which the stabilizer form for children codes, all QMDS, can be obtained. Our method would work for relevant high-dimensional codes as well as qubit-based codes. We then relate this to optimal codes for one-way quantum repeaters, by minimizing the short-term infrastructure cost as well as the long-term running cost of such quantum repeaters. This would allow us to obtain the optimal QMDS code, derived from an AME parent state, that can be used in such quantum repeaters.

DOI
10.1103/physreva.103.022402

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We address the relation between absolutely maximally entangled (AME) states and quantum- maximum-distance-separable (QMDS) codes by constructing whole families of QMDS codes from AME states that have stabilizer representations. We introduce a generic reduction-friendly form for the generator set of the stabilizer representation of an AME state, from which the stabilizer form for children codes, all QMDS, can be obtained. Our method would work for relevant high-dimensional codes as well as qubit-based codes. We then relate this to optimal codes for one-way quantum repeaters, by minimizing the short-term infrastructure cost as well as the long-term running cost of such quantum repeaters. This would allow us to obtain the optimal QMDS code, derived from an AME parent state, that can be used in such quantum repeaters.
Abstract 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 n parties with local dimension q a power of a prime for all q ⩾ 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 − 1 prime, we show how to construct stabilizer 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 for n even with the highest distance allowed by the quantum Singleton bound.
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.
We present new bounds on the existence of general quantum maximum distance separable codes (QMDS): the length n of all QMDS codes with local dimension D and distance d >= 3 is bounded by n <= D-2 + d - 2. We obtain their weight distribution and present additional bounds that arise from Rains' shadow inequalities. Our main result can be seen as a generalization of bounds that are known for the two special cases of stabilizer QMDS codes and absolutely maximally entangled states, and confirms the quantum MDS conjecture in the special case of distance-three codes. As the existence of QMDS codes is linked to that of highly entangled subspaces (in which every vector has uniform r-body marginals) of maximal dimension, our methods directly carry over to address questions in multipartite entanglement.
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.
We present new bounds on the existence of general quantum maximum distance separable codes (QMDS): the length n of all QMDS codes with local dimension D and distance d ≥ 3 is bounded by n ≤ D 2 + d − 2 . We obtain their weight distribution and present additional bounds that arise from Rains' shadow inequalities. Our main result can be seen as a generalization of bounds that are known for the two special cases of stabilizer QMDS codes and absolutely maximally entangled states, and confirms the quantum MDS conjecture in the special case of distance-three codes. As the existence of QMDS codes is linked to that of highly entangled subspaces (in which every vector has uniform r -body marginals) of maximal dimension, our methods directly carry over to address questions in multipartite entanglement.
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.