bibkey: zozor2014entropylikeuncertainty
authors: Steeve Zozor; Gustavo Martín Bosyk; Mariela Portesi
year: 2014
title: "General entropy-like uncertainty relations in finite dimensions"
doi: 10.1088/1751-8113/47/49/495302
url: https://arxiv.org/abs/1311.5602v3
claim: "For two nondegenerate observables with eigenbasis overlap c, the paper proves a lower bound on the sum of two generalized (h,φ)-entropies, shows it is c-optimal for c > 1/√2 when N = 2 or N ≥ 4, and suspects that it is also c-optimal when N = 3."
strata_touched:
  - D5/S3/Quantum/Information/QutritOverlapBoundNonOptimality
license: citation-only
triage: anchor

General entropy-like uncertainty relations in finite dimensions

Steeve Zozor, Gustavo Martín Bosyk and Mariela Portesi, arXiv:1311.5602v3 (quant-ph), 2014-10-07; published in J. Phys. A 47 (2014) 495302. Quotations are from the v3 TeX source.

Tsallis entropies (Section 2):

S_\lambda(p) = \frac{\displaystyle 1 -\sum_k p_k^{,\lambda}}{\lambda-1}

The overlap of two nondegenerate observables (Section 3):

T_{ij} = \langle b_j | a_i \rangle,

c(T) = \max_{i,j} | \langle b_j | a_i \rangle |,

The c-optimal bound (Section 3):

\widetilde{\B}{(h_A,\phi_A),(h_B,\phi_B);N}(c) :: = \min{T \in \U(N):: c(T) = c} \overline{\B}_{(h_A,\phi_A),(h_B,\phi_B)}(T)

The bound in the nondegenerate case (Section 3):

\B_{(h_A,\phi_A),(h_B,\phi_B)}(c) = \min_{\theta \in [0, \gamma]} \left( \D_{(h_A,\phi_A)}(\theta) + \D_{(h_B,\phi_B)}(\gamma - \theta) \right)

Corollary 2 and the open case (Section 3):

When $c>\frac{1}{\sqrt{2}}$ and $N=2$ or $N\ge 4$, the bound~\eref{Cabnondeg:eq} is $c$-optimal,

We suspect that this corollary is also valid when $N=3$, but we have not been able to prove it yet.

Verified locator

• URL: https://arxiv.org/abs/1311.5602v3 (v3, 2014-10-07; source Uncertainty_qunit_R1.tex, md5 5a6f1537cc4f0901472e506d6addbd85): Tsallis entropies (l. 220–226), the overlap (l. 375–393), the c-optimal bound (l. 407–415), the function D (l. 740–743), the bound (22) (l. 823–828), Corollary 2 and the suspicion (l. 838–855).
• DOI: 10.1088/1751-8113/47/49/495302 (J. Phys. A 47, 495302). The published text was not read.