bibkey: buniy2024maximallyentangledclasses
authors: Roman V. Buniy, Robert P. Feger, Thomas W. Kephart
year: 2024
title: "Decoherence and the classes of maximally entangled states"
doi: 10.1142/S0219749924500357
url: https://arxiv.org/abs/2210.07618v1
claim: "A maximally entangled state of an n-partite system is a state at which every algebraic entanglement invariant (kernel dimensions of the flattenings and contraction maps built from its coordinates; for n = 3 the four invariants of Appendix A) takes its least value, and these states form the maximally entangled class C_D. The length of a state is its number of nonzero coordinates in a product basis, and L_D is the least length in C_D. From exact values for 3-partite dimensions up to 5 (Table XII) the authors conjecture L_(d₁,d₂,d₃) = d₁ + d₂ + d₃ − 2 when min(d₁, d₂, d₃) ≥ 3 (Conjecture III.4)."
strata_touched:
  - D5/S3/Quantum/Entanglement/TripartiteMaximalClassLengthRefutation
license: citation-only
triage: anchor

Buniy–Feger–Kephart, decoherence and the classes of maximally entangled states

R. V. Buniy, R. P. Feger and T. W. Kephart, Decoherence and the classes of maximally entangled states, Int. J. Quantum Inf. 22, 2450035 (2024), DOI 10.1142/S0219749924500357; arXiv:2210.07618 [quant-ph] (v1, the only arXiv version).

Verified locator

DOI: 10.1142/S0219749924500357 (as listed by Crossref, published July 2024). Primary version read: https://arxiv.org/abs/2210.07618v1. The journal text was not read. Source: https://arxiv.org/pdf/2210.07618v1 (text read from the PDF):

• §II.A, Definition II.1 (maximally entangled states and the maximally entangled class);
• §III, Eq. (2) (the length L(v) and L_D);
• §III.C, Table XII and Conjecture III.4;
• Appendix A, Theorem A.1 and Eqs. (A1)–(A6) (the invariants n₁, n₂, n₃, n₁,₂,₃ for n = 3).

Source statements

Definition II.1: “A maximally entangled state (MES) of an n-partite system with the dimensions D = (d₁, …, d_n) is any state v for which every invariant in the set N_D(v) takes its least possible value. The set of all maximally entangled states is the maximally entangled class (MEC) C_D.”

Appendix A: “The algebraic invariants are N = (n₁, n₂, n₃, n₁,₂,₃), where n_i = dim K_i, 1 ≤ i ≤ 3 and n₁,₂,₃ = dim K₁,₂,₃.”

Conjecture III.4: “The shortest length of states in an MEC for n = 3 is L_(2,d₂,d₂) = 2d₂ − 2, L_(2,d₂,d₃) = 2d₂ for d₂ < d₃ and L_(d₁,d₂,d₃) = Σ_{i=1}^{3} d_i − 2 for min(d₁, d₂, d₃) ≥ 3.”

Reading used by the repository

A state is a coefficient array v_{ijk} in the product basis.

• K₁ = {w : Σ_i v_{ijk}w_i = 0 for all j, k}, and K₂, K₃ likewise for the second and third factors (Eqs. (A1)–(A3); the printed (A3) writes w_j for w_k).
• K₁,₂,₃ is the space of arrays z with Σ_{j,k} v_{ajk}z_{bjk} = 0, Σ_{i,k} v_{iak}z_{ibk} = 0 and Σ_{i,j} v_{ija}z_{ijb} = 0 for all indices a, b (Eqs. (A4)–(A6)).
• The class C_D is the set of states at which all four dimensions are minimal over all states.