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arXiv:0810.0083

Comment on "Uncertainty Relations for Positive Operator Valued Measures"

Alexey E. Rastegin

Published Oct 1, 2008• 1 citation
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Abstract

The principal aim of this Comment is to correct those entropic uncertainty relations that are presented in a paper by Massar [arXiv:quant-ph/0703036v2 (current version)], concerning two approaches to a study of the noise produced by POVM's. It is next emphasized that the first of the entropic bounds for POVM obtained by the above author has been already presented in Ref. [8]. Some exposition obscurity with equation (14) of the commented paper is elucidated. Finally, some more specific remarks on the paper content are given.

DOI
10.48550/arxiv.0810.0083

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The principal aim of this Comment is to correct those entropic uncertainty relations that are presented in a paper by Massar [arXiv:quant-ph/0703036v2 (current version)], concerning two approaches to a study of the noise produced by POVM's. It is next emphasized that the first of the entropic bounds for POVM obtained by the above author has been already presented in Ref. [8]. Some exposition obscurity with equation (14) of the commented paper is elucidated. Finally, some more specific remarks on the paper content are given.
How much unavoidable randomness is generated by a positive-operator-valued measure (POVM)? We address this question using two complementary approaches. First, we study the variance of a real variable associated with the POVM outcomes. In this context we introduce an uncertainty operator which measures how much additional noise is introduced by carrying out a POVM rather than a von Neumann measurement. We illustrate this first approach by studying the variances of joint estimates of ${\ensuremath{\sigma}}_{x}$ and ${\ensuremath{\sigma}}_{z}$ for spin-1∕21∕21∕2 particles. We show that for unbiased measurements the sum of these variances is lower bounded by 111. In our second approach we study the entropy of the POVM outcomes. In particular, we try to establish lower bounds on the entropy of the POVM outcomes. We illustrate this second approach by examples.
Entropic uncertainty relations are underpinning to compute the quantitative security bound in quantum cryptographic applications, such as quantum random number generation (QRNG) and quantum key distribution (QKD). All security proofs derive a relation between the information accessible to the legitimate group and the maximum knowledge that an adversary may have gained, Eve, which exploits entropic uncertainty relations to lower bound Eve's uncertainty about the raw key generated by one party, Alice. The standard entropic uncertainty relations is to utilize the smooth min- and max-entropies to show these cryptographic applications' security by computing the overlap of two incompatible measurements or positive-operator valued measures (POVMs). This paper draws one case of the POVM-versioned standard entropic uncertainty relation yielding the trivial bound since the maximum overlap in POVMs always produces the trivial value, "one." So, it fails to tie the smooth min-entropy to show the security of the quantum cryptographic application.
We address entropic uncertainty relations between time and energy or, more precisely, between measurements of an observableGand the displacementrof theG-generated evolution e(-irG). We derive lower bounds on the entropic uncertainty in two frequently considered scenarios, which can be illustrated as two different guessing games in which the role of the guessers are fixed or not. In particular, our bound for the first game improves the previous result by Coleset al[Phys. Rev. Lett.122100401 (2019)]. To derive our bounds, we extend a recently proposed novel algebraic method by Gaoet al[arXiv:1710.10038 [quant-ph]] which was used to derive both strong subadditivity and entropic uncertainty relations for measurements.