Inequalities between entropy and index of coincidence derived from information diagrams

Harremoës, Peter; Topsøe, Flemming

doi:10.1109/18.959272

Cited by 65 publications

(97 citation statements)

References 22 publications

(33 reference statements)

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“…The problem of finding the minimum and maximum of the Shannon entropy under assumption that the index of coincidence is constant has been analyzed in [57] (some generalizations and related topics can be found also in [16,137]). By [57, Theorem 2.5.…”

Section: Remarkmentioning

confidence: 99%

Highly symmetric POVMs and their informational power

Słomczyński

Szymusiak

2015

Quantum Inf Process

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We discuss the dependence of the Shannon entropy of normalized finite rank-1 POVMs on the choice of the input state, looking for the states that minimize this quantity. To distinguish the class of measurements where the problem can be solved analytically, we introduce the notion of highly symmetric POVMs and classify them in dimension 2 (for qubits). In this case, we prove that the entropy is minimal, and hence, the relative entropy (informational power) is maximal, if and only if the input state is orthogonal to one of the states constituting a POVM. The method used in the proof, employing the Michel theory of critical points for group action, the Hermite interpolation, and the structure of invariant polynomials for unitary-antiunitary groups, can also be applied in higher dimensions and for other entropy-like functions. The links between entropy minimization and entropic uncertainty relations, the Wehrl entropy, and the quantum dynamical entropy are described.

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Section: Remarkmentioning

confidence: 99%

Highly symmetric POVMs and their informational power

Słomczyński

Szymusiak

2015

Quantum Inf Process

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“…Actually, this problem is equivalent to seek the maximal and minimal H( µ) = − i µ i log 2 µ i for a given τ = 2(1 − i µ 2 i ), and the later one is a classical problem which has been solved in Refs. [48,49]. We use…”

Section: A Calculation Of Fmentioning

confidence: 99%

Observable estimation of entanglement of formation and quantum discord for bipartite mixed quantum states

Zhang

Chen

et al. 2011

Phys. Rev. A

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We present observable lower and upper bounds for the entanglement of formation (EOF) and quantum discord (QD), which facilitates estimates of EOF and QD for arbitrary experimental unknown states in finitedimensional bipartite systems. These bounds can be easily obtained by a few experimental measurements on a twofold copy ̺ ⊗ ̺ of the mixed states. Based on our results, we use the experimental measurement data of the real experiment given by Schmid et al. [Phys. Rev. Lett. 101, 260505 (2008)] to obtain the lower and upper bounds of EOF and QD for the experimental unknown state.

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“…In Harremoës and Topsøe [1] the exact range of the map P (IC(P ), H(P )) with P varying over either M 1 + (n) or M 1 + (N) was determined. The ranges in question, termed IC/H-diagrams, were denoted ∆, respectively ∆ n :…”

Section: H(pmentioning

confidence: 99%

“…Figure 1 illustrates the situation for the restricted diagrams ∆ n . The key result of [1] states that ∆ n is the bounded region determinated by a certain Jordan curve determined by n smooth arcs, the "upper arc" connecting Q 1 and Q n and then n − 1 "lower arcs" connecting Q n with Q n−1 , Q n−1 with Q n−2 etc. until Q 2 which is connected with Q 1 .…”

Section: H(pmentioning

confidence: 99%