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arXiv:1410.1148 · Physical Review A

Joint measurability, steering, and entropic uncertainty

H. S. Karthik, A. R. Usha Devi, A. K. Rajagopal

Published Jan 22, 2015• 13 citations
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Abstract

There has been a surge of research activity recently on the role of joint measurability of unsharp observables in nonlocal features, viz., violation of Bell inequality and EPR steering. Here, we investigate the entropic uncertainty relation for a pair of noncommuting observables (of Alice's system) when an entangled quantum memory of Bob is restricted to record outcomes of jointly measurable positive operator valued measures. We show that with this imposed constraint of joint measurability at Bob's end, the entropic uncertainties associated with Alice's measurement outcomes---conditioned by the results registered at Bob's end---obey an entropic steering inequality. Thus, Bob's nonsteerability is intrinsically linked to his inability to predict the outcomes of Alice's pair of noncommuting observables with better precision, even when they share an entangled state. As a further consequence, we prove that in the joint measurability regime, the quantum advantage envisaged for the construction of security proofs in quantum key distribution is lost.

DOI
10.1103/physreva.91.012115

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There has been a surge of research activity recently on the role of joint measurability of unsharp observables in nonlocal features, viz., violation of Bell inequality and EPR steering. Here, we investigate the entropic uncertainty relation for a pair of noncommuting observables (of Alice's system) when an entangled quantum memory of Bob is restricted to record outcomes of jointly measurable positive operator valued measures. We show that with this imposed constraint of joint measurability at Bob's end, the entropic uncertainties associated with Alice's measurement outcomes---conditioned by the results registered at Bob's end---obey an entropic steering inequality. Thus, Bob's nonsteerability is intrinsically linked to his inability to predict the outcomes of Alice's pair of noncommuting observables with better precision, even when they share an entangled state. As a further consequence, we prove that in the joint measurability regime, the quantum advantage envisaged for the construction of security proofs in quantum key distribution is lost.
We give an overview of joint unsharp measurements of non-commuting observables using positive operator valued measures (POVMs). We exemplify the role played by joint measurability of POVMs in entropic uncertainty relation for Alice’s pair of non-commuting observables in the presence of Bob’s entangled quantum memory. We show that Bob should record the outcomes of incompatible (non-jointly measurable) POVMs in his quantum memory so as to beat the entropic uncertainty bound. In other words, in addition to the presence of entangled Alice–Bob state, implementing incompatible POVMs at Bob’s end is necessary to beat the uncertainty bound and hence predict the outcomes of non-commuting observables with improved precision. We also explore the implications of joint measurability to validate a moment matrix constructed from average pairwise correlations of three dichotomic non-commuting qubit observables. We prove that a classically acceptable moment matrix – which ascertains the existence of a legitimate joint probability distribution for the outcomes of all the three dichotomic observables – could be realized if and only if compatible POVMs are employed.
We give an overview of joint unsharp measurements of non-commuting observables using positive operator valued measures (POVMs). We exemplify the role played by joint measurability of POVMs in entropic uncertainty relation for Alice's pair of non-commuting observables in the presence of Bob's entangled quantum memory. We show that Bob should record the outcomes of incompatible (non-jointly measurable) POVMs in his quantum memory so as to beat the entropic uncertainty bound. In other words, in addition to the presence of entangled Alice-Bob state, implementing incompatible POVMs at Bob's end is necessary to beat the uncertainty bound and hence, predict the outcomes of non-commuting observables with improved precision. We also explore the implications of joint measurability to {\em validate} a moment matrix constructed from average pairwise correlations of three dichotomic non-commuting qubit observables. We prove that a classically acceptable moment matrix -- which ascertains the existence of a legitimate joint probability distribution for the outcomes of all the three dichotomic observables -- could be realized if and only if compatible POVMs are employed.
We give an overview of joint unsharp measurements \nof non-commuting observables using positive operator \nvalued measures (POVMs). We exemplify the role \nplayed by joint measurability of POVMs in entropic \nuncertainty relation for Alice’s pair of non-commuting \nobservables in the presence of Bob’s entangled quantum \nmemory. We show that Bob should record the \noutcomes of incompatible (non-jointly measurable) \nPOVMs in his quantum memory so as to beat the entropic \nuncertainty bound. In other words, in addition \nto the presence of entangled Alice–Bob state, implementing \nincompatible POVMs at Bob’s end is necessary \nto beat the uncertainty bound and hence predict \nthe outcomes of non-commuting observables with \nimproved precision. We also explore the implications \nof joint measurability to validate a moment matrix \nconstructed from average pairwise correlations of \nthree dichotomic non-commuting qubit observables. \nWe prove that a classically acceptable moment matrix \n– which ascertains the existence of a legitimate \njoint probability distribution for the outcomes of all \nthe three dichotomic observables – could be realized if \nand only if compatible POVMs are employed.
Quantum entanglement is a key resource for foundational quantum information and communication tasks, such as teleportation, entanglement swapping and quantum key distribution [1]. Quantum steering is a quantum correlation that is distinct from other nonclassical correlations such as Bell nonlocality [2] and entanglement. Steering describes the effect of measuring one system, let's say Alice, and affecting the measurement results on the other system, let's say Bob. Moving through the hierarchy of nonlocal classes, from Bell nonlocality towards nonsepa-rability, requires an increasing number of parties and apparatus that need to be trusted but gives access to protocols which are more robust to noise for projective measurements [3,4]. In a quantum steering task [5] the correlations of Alice and Bob's measurement outcomes are tested against a classical probability bound which can be only violated if the state shared between the parties is quantum steerable.
We establish a general connection between entropic uncertainty relations, Einstein-Podolsky-Rosen steering, and joint measurability. Specifically, we construct steering inequalities from any entropic uncertainty relation, given that the latter satisfies two natural properties. We obtain steering inequalities based on R\'enyi entropies. These turn out to be tight in many scenarios using max- and min-entropies. Considering steering tests with two noisy measurements, our inequalities exactly recover the noise threshold for steerability. This is the case for any pair of qubit two-outcome measurements as well as for pairs of mutually unbiased bases in any dimension. This shows that easy-to-evaluate quantities, such as entropy, can optimally witness steering, despite the fact that they are coarse-grained representations of the underlying statistics.
We discuss the connection between the incompatibility of quantum measurements, as captured by the notion of joint measurability, and the violation of Bell inequalities. Specifically, we explicitly present a given set of non-jointly-measurable positive-operator-value measures (POVMs) MA{\mathcal{M}}_{A}MA​ with the following property. Considering a bipartite Bell test where Alice uses MA{\mathcal{M}}_{A}MA​, then for any possible shared entangled state $\ensuremath{\rho}$ and any set of (possibly infinitely many) POVMs NB{\mathcal{N}}_{B}NB​ performed by Bob, the resulting statistics admits a local model and can thus never violate any Bell inequality. This shows that quantum measurement incompatibility does not imply Bell nonlocality in general.