bibkey: bosyketal2016unifiedcorrelations
authors: G. M. Bosyk, G. Bellomo, S. Zozor, M. Portesi and P. W. Lamberti
year: 2016
title: "Unified entropic measures of quantum correlations induced by local measurements"
doi: 10.1016/j.physa.2016.06.131
url: "https://arxiv.org/abs/1604.00329v1"
claim: "Quantum correlation measures D^K_(q,s), K∈{A,B,AB}, are defined as the minimal change of the unified (q,s)-entropy under local rank-one projective measurements, divided by (Tr ρ^q)^s; the triangle-like inequality D^A+D^B≥D^AB is proved for von Neumann and Tsallis-2 entropies and left open for other entropic measures."
strata_touched:
  - D5/S3/Entropy/UnifiedEntropyTriangleRefutation
license: citation-only
triage: anchor

Bosyk, Bellomo, Zozor, Portesi and Lamberti, Unified entropic measures of quantum correlations induced by local measurements

G. M. Bosyk, G. Bellomo, S. Zozor, M. Portesi and P. W. Lamberti, Physica A 462 (2016) 930–939; arXiv:1604.00329v1. Quotations are from the v1 TeX source.

The quantum unified entropies are $S_{(q,s)}(\rho )=((\mathrm{Tr}{\rho }^q{)}^s-1)/((1-q)s)$ for $q>0$, $q\ne 1$, $s\ne 0$ (Tsallis at $s=1$; von Neumann and Rényi as the limits $q\to 1$ and $s\to 0$). For a bipartite state and the local rank-one projective measurements ${\Pi }^A=\{P_i^A\otimes I^B\}$, ${\Pi }^B=\{I^A\otimes P_j^B\}$, ${\Pi }^{AB}=\{P_i^A\otimes P_j^B\}$ without postselection, the disturbance is $D_{(q,s)}^{{\Pi }^K}(\rho )=[S_{(q,s)}({\Pi }^K(\rho ))-S_{(q,s)}(\rho )]/(\mathrm{Tr}{\rho }^q{)}^s$ and the correlation measures are $D_{(q,s)}^K(\rho )={\min }_{{\Pi }^K}{D}_{(q,s)}^{{\Pi }^K}(\rho )$. The triangle-like inequality (38), $D_{(q,s)}^A+D_{(q,s)}^B\ge D_{(q,s)}^{AB}$, is linked to a local contractivity property of the unilocal disturbances, which the paper shows numerically to fail for many Rényi and Tsallis indices. The conclusions state:

In the general case, we only proof this for the von Neumann and Tsallis with entropic index of order $2$, which follows from the contractivity property under a projective measurement of quantum relative entropy and Hilbert Schmidt distance, respectively. We provide numerical counterexamples where the local contractivity property of unilocal disturbances fails in a wide range of the entropic index of Rényi and Tsallis entropies, but it remains open if the triangle-like inequality is fulfilled for other entropic measures.

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Source: https://arxiv.org/abs/1604.00329v1

• DOI: https://doi.org/10.1016/j.physa.2016.06.131 (Physica A: Statistical Mechanics and its Applications 462, 930–939, 2016; title, authors, journal, volume and pages from Crossref).
• URL: https://arxiv.org/abs/1604.00329v1. The e-print member Unifiedarxiv.tex has 1159 lines and md5 39dedcceee781b4711c8b1a124b08dca. Locations:
  • the unified entropies, l. 187–212;
  • the local measurements, l. 371–394;
  • the disturbance (16) and the measures (17), l. 418–433;
  • the triangle-like inequality (38) and its link to local contractivity, l. 697–756;
  • the open question, l. 940–951.