bibkey: gaddejainkrishnakulkarnisharma2024multientropy
authors: A. Gadde, S. Jain, V. Krishna, H. Kulkarni, and T. Sharma
year: 2024
title: Monotonicity conjecture for multi-party entanglement. Part I
doi: 10.1007/JHEP02(2024)025
url: https://arxiv.org/abs/2308.16247v1
claim: Section 7 asks whether the replica invariant E_n^(q) is positive on all normalized pure states.
strata_touched:
  - D5/S3/Quantum/Information/MultiEntropyPositivityRefutation
license: citation-only
triage: anchor

Positivity of the multi-entropy replica invariant

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DOI: 10.1007/JHEP02(2024)025 Source: https://arxiv.org/abs/2308.16247v1

JHEP 02 (2024) 025. Section 1.2, printed pp. 6–7, equations (1.10)–(1.11), defines the replica invariant and the Rényi multi-entropy. The source states:

“Consider m = n^{q−1} replicas, arranged as a q−1 dimensional regular hyper-cubical lattice.”

“The permutation σ_q is taken to be identity and the rest of the permutations are taken to be the generators of independent Z_n symmetries i.e. σ_a = g_a for a = 1, …, q − 1.”

Footnote 2 states:

“Note that in defining the Renyi multi-entropy S_n in equation (1.11), we have tacitly assumed that the quantity E_n^(q) is positive.”

Section 7, printed p. 32, states:

“The definition (1.11) of S_n requires that the quantity E_n be positive definite. In this paper, we did not prove or even discuss this and phrased the monotonicity conjecture in terms of E_n instead. Although we found that E_n is positive in the examples we studied, it would be nice to prove this in general and to see if there is any relation between the positivity of E_n and the coarse-graining monotonicity.”

The scalar positivity question is encoded as positive real part and zero imaginary part, for every n,q ≥ 2 and every normalized pure state on finite local bases. The final party is unshifted. The coarse-graining inequalities (1.13)–(1.15) are statements directly about E and are distinct from this question.