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arXiv:2512.02749 · Physical Review A

Minimal decomposition entropy and optimal representations of absolutely maximally entangled states

N Ramadas

quant-phSubmitted Dec 2, 2025• Announced Dec 3, 2025• 0 citations• Revised Mar 10, 2026 (v2)
v1 · Dec 2, 2025v2 · Mar 10, 2026
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Abstract

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 $S_q$ 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 $q>1$ and compare AME and Haar-random states for \( q = 2 \) and \( q = \infty \) in qubit, qutrit, and ququad systems. For \( q = 2 \), AME states of four qutrits and ququads show lower minimal entropy than generic states, indicating sparser optimal forms. For \( q = \infty \) -- 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.

DOI
10.1103/cvdl-1jw6

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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 ) and ( q = \infty ) in qubit, qutrit, and ququad systems. For ( q = 2 ), AME states of four qutrits and ququads show lower minimal entropy than generic states, indicating sparser optimal forms. For ( q = \infty ) -- 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.
Ordering and classifying multipartite quantum states by their entanglement content remains an open problem. One class of highly entangled states, useful in quantum information protocols, the absolutely maximally entangled (AME) ones, are especially hard to compare as all their subsystems are maximally random. While it is well known that there is no AME state of four qubits, many analytical examples and numerically generated ensembles of four-qutrit AME states are known. However, we prove the surprising result that there is truly only one AME state of four qutrits up to local unitary equivalence. In contrast, for larger local dimensions, the number of local unitary classes of AME states is shown to be infinite. Of special interest is the case of local dimension 6, where it was established recently that a four-party AME state does exist, providing a quantum solution to the classically impossible Euler problem of 36 officers. Based on this, an infinity of quantum solutions are constructed, and we prove that these are not equivalent. The methods developed can be usefully generalized to multipartite states of any number of particles.
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.
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.
Multipartite entanglement is a fundamental aspect of quantum mechanics, crucial to advancements in quantum information processing and quantum computation. Within this field, Genuinely Multipartite Entanglement (GME), being entangled in all bipartitions, and Absolutely Maximally Entanglement (AME), maximally entangled in all bipartitions, represent two significant types of entanglement with diverse applications. In this work, we introduce a new measure called the GME-AME multipartite entanglement measure, with a non-zero value representing the GME states and the maximum value is reached only by the AME states. The measure is applied to study the multipartite entanglement of four partite systems using the operator to state mapping, and the four partite permutation qutrit states are classified according to the measure. With various examples, we show that our measure is robust in classifying the four partite entangled states.
Abstract The classification of multipartite entanglement is essential as it serves as a resource for various quantum information processing tasks. This study concerns a particular class of highly entangled multipartite states, the so-called absolutely maximally entangled (AME) states. These are characterized by maximal entanglement across all possible bipartitions. In particular we analyze the local unitary (LU) equivalence among AME states using invariants. One of our main findings is that the existence of special irredundant orthogonal arrays implies the existence of an infinite number of equivalence classes of AME states constructed from these. In particular, we show that there are infinitely many LU inequivalent three-party AME states for local dimension d > 2 and five-party AME states for d ⩾ 2 .
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.