Collision entropy and optimal uncertainty

Bosyk, Gustavo Martín; Portesi, Mariela; Plastino, A.

doi:10.1103/physreva.85.012108

Cited by 50 publications

(61 citation statements)

References 46 publications

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“…Thus, the average disturbance function we seek to minimize is of the form Closely following the earlier work of Ghirardi et al [34] and Bosyk et al [35], we first argue that the minimizing vector r must be coplanar with a and b. Let P denote the plane determined by the vectors a and b.…”

Section: Discussionmentioning

confidence: 91%

“…Our result for T 2 assumes importance in the light of the fact that such optimal analytical bounds are known only for a handful of entropic functions, namely, the Rényi entropies H 2 [35], H 1/2 , and the Tsallis entropy T 1/2 [31]. For the Shannon entropy, there is in general only a numerical estimate of the bound [12,34].…”

Section: Optimal Disturbance Tradeoff Relation For a Pair Of Qubmentioning

confidence: 97%

“…A similar approach has been used to obtain optimal entropic uncertainty relations for a pair of qubit observables, both in the case of the Shannon entropy [12,34] and the collision entropy [35]. Further details of our proof can be found in the appendix (Sec.…”

Section: Optimal Disturbance Tradeoff Relation For a Pair Of Qubmentioning

confidence: 99%

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Disturbance trade-off principle for quantum measurements

Mandayam

Srinivas

2014

Phys. Rev. A

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We demonstrate a fundamental principle of disturbance tradeoff for quantum measurements, along the lines of the celebrated uncertainty principle: The disturbances associated with measurements performed on distinct yet identically prepared ensembles of systems in a pure state cannot all be made arbitrarily small. Indeed, we show that the average of the disturbances associated with a set of projective measurements is strictly greater than zero whenever the associated observables do not have a common eigenvector. For such measurements, we show an equivalence between disturbance tradeoff measured in terms of fidelity and the entropic uncertainty tradeoff formulated in terms of the Tsallis entropy (T2). We also investigate the disturbances associated with the class of nonprojective measurements, where the difference between the disturbance tradeoff and the uncertainty tradeoff manifests quite clearly.The uncertainty principle, which is one of the cornerstones of quantum theory, has had a long history. In its original formulation by Heisenberg for canonically conjugate variables [1], the uncertainty principle was stated as an effect of the disturbance caused due to a measurement of one observable on a succeeding measurement of another. However, the subsequent mathematical formulation due to Robertson [2] and Schrödinger [3] in terms of variances departed from this original interpretation. Rather, they obtained a non-trivial lower bound on the product of the variances associated with the measurement of a pair of incompatible observables, performed on distinct yet identically prepared copies of a given system. The Robertson-Schrödinger inequality thus expresses a fundamental limitation as regards the preparation of an ensemble of systems in identical states, and is therefore a manifestation of the so-called preparation uncertainty [4].Along the same lines, the more recent entropic formulations of the uncertainty principle [5] also demonstrate the existence of a fundamental tradeoff for the uncertainties associated with independent measurements of incompatible observables on identically prepared ensemble of systems. Entropic uncertainty relations (EURs) have been obtained for specific classes of observables for both the Shannon and Rényi entropies [6][7][8][9][10][11][12][13][14], as well as for the Tsallis entropies [15][16][17].Here, we prove the existence of a similar principle of tradeoff for the disturbances associated with the measurements of a set of observables. It is a fundamental feature of quantum theory that when an observable is measured on an ensemble of systems, the density operator of the resulting ensemble is in general different from that prior to the measurement. The distance between these two density operators is indeed a measure of the disturbance due to measurement. Different measures of distance between density operators [18] give rise to different measures of disturbance. We are thus lead to a class of disturbance measures which have been used recently in the context of quantifying incompatibilit...

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Section: Discussionmentioning

confidence: 91%

Section: Optimal Disturbance Tradeoff Relation For a Pair Of Qubmentioning

confidence: 97%

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Disturbance trade-off principle for quantum measurements

Mandayam

Srinivas

2014

Phys. Rev. A

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“…Previously, such relations were only known for single qudit measurements for α → 1, α = 2, and α → ∞ (see e.g. [1,17,18]). More precisely, we show that for measurements on n-qubit states ρ in BB84 bases, the minimum values of the conditional Rényi entropies for any…”

Section: Resultsmentioning

confidence: 99%

Min-entropy uncertainty relation for finite-size cryptography

Berta

Wehner

2012

Phys. Rev. A

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Apart from their foundational significance, entropic uncertainty relations play a central role in proving the security of quantum cryptographic protocols. Of particular interest are thereby relations in terms of the smooth min-entropy for BB84 and six-state encodings. Previously, strong uncertainty relations were obtained which are valid in the limit of large block lengths. Here, we prove a new uncertainty relation in terms of the smooth min-entropy that is only marginally less strong, but has the crucial property that it can be applied to rather small block lengths. This paves the way for a practical implementation of many cryptographic protocols. As part of our proof we show tight uncertainty relations for a family of Rényi entropies that may be of independent interest.

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“…However, if a POVM is merely symmetric, the global extrema of entropy functions may also occur in other (non-inert) points. An example of this phenomenon can be found in [52], see also [19,136]. Let us consider a symmetric (but non-highly symmetric) POVM generated by the set of four Bloch vectors forming a rectangle B = {v 1 …”

Section: Proposition 8 In the Situation Above Singular Points Of Typmentioning

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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Collision entropy and optimal uncertainty

Cited by 50 publications

References 46 publications

Disturbance trade-off principle for quantum measurements

Disturbance trade-off principle for quantum measurements

Min-entropy uncertainty relation for finite-size cryptography

Highly symmetric POVMs and their informational power

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