Simulating Positive-Operator-Valued Measures with Projective Measurements

Oszmaniec, Michał; Guerini, Leonardo; Wittek, Peter; Acín, Antonio

doi:10.1103/physrevlett.119.190501

Cited by 128 publications

(153 citation statements)

References 42 publications

(107 reference statements)

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“…It has been investigated in many works [12,28,30,[32][33][34][35][36], often in relation with Einstein-Podolsky-Rosen steering. This specific type of noise has also been considered in scenarios different from incompatibility [37]. The physical motivation is as follows: take a depolarising quantum channel .…”

Section: Incompatibility Depolarising Robustness 311 Definition Anmentioning

confidence: 99%

Incompatibility robustness of quantum measurements: a unified framework

2019

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In quantum mechanics performing a measurement is an invasive process which generally disturbs the system. Due to this phenomenon, there exist incompatible quantum measurements, i.e. measurements that cannot be simultaneously performed on a single copy of the system. It is then natural to ask what the most incompatible quantum measurements are. To answer this question, several measures have been proposed to quantify how incompatible a set of measurements is, however their properties are not well-understood. In this work, we develop a general framework that encompasses all the commonly used measures of incompatibility based on robustness to noise. Moreover, we propose several conditions that a measure of incompatibility should satisfy, and investigate whether the existing measures comply with them. We find that some of the widely used measures do not fulfil these basic requirements. We also show that when looking for the most incompatible pairs of measurements, we obtain different answers depending on the exact measure. For one of the measures, we analytically prove that projective measurements onto two mutually unbiased bases are among the most incompatible pairs in every dimension. However, for some of the remaining measures we find that some peculiar measurements turn out to be even more incompatible.performed simultaneously by performing the parent measurement. If such a parent measurement does not exist, we say that the measurements are incompatible (or not jointly measurable). We remark here that other notions of compatibility, such as commutativity, non-disturbance and coexistence, are also used in the literature [1,3]; let us for completeness briefly explain how they are related. Commutativity of a measurement pair implies nondisturbance, which in turn implies joint measurability, which then implies coexistence. Moreover, it is known that none of the converse implications hold in general, therefore these notions are strictly distinct [4]. In this work we focus solely on the notion of joint measurability, because the existence (or not) of a parent measurement has a clear operational meaning. Therefore, throughout the present paper we use the terms '(in) compatibility' and '(non-)joint measurability' interchangeably. It is important to notice that whenever two measurements are compatible, they cannot be used to produce quantum advantage in tasks like Bell nonlocality [5] or Einstein-Podolsky-Rosen steering [6,7]. Moreover, it was recently shown that joint measurability is equivalent to a specific notion of classicality, namely, preparation non-contextuality [8,9]. Hence, one may think of compatible measurements as 'classical' , and incompatible measurements as a resource for the above tasks. Therefore, it is of fundamental importance to characterise and understand the structure of incompatible measurements.What is particularly important is to go beyond the dichotomy of compatible and incompatible measurements, and quantify to what extent a pair of measurements is incompatible. A natural framework for this qua...

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Section: Incompatibility Depolarising Robustness 311 Definition Anmentioning

confidence: 99%

Incompatibility robustness of quantum measurements: a unified framework

2019

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“…Simulability. The post-processing relation on observables generalizes to a preorder on the respective power set 2 O , as discussed and used in various ways in [9,10,11]. Namely, suppose X, X ′ ⊆ O are two arbitrary subsets.…”

Section: 3mentioning

confidence: 99%

Noise–Disturbance Relation and the Galois Connection of Quantum Measurements

et al. 2019

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The relation between noise and disturbance is investigated within the general framework of Galois connections. Within this framework, we introduce the notion of leak of information, mathematically defined as one of the two closure maps arising from the observable-channel compatibility relation. We provide a physical interpretation for it, and we give a comparison with the analogous closure maps associated with joint measurability and simulability for quantum observables.

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“…We prove that one can choose U O A j = ( ) and U B  = in the state (6) without the loss of generality, where O j ( ) is the planar rotation (7)…”

Section: Appendix a Real-valued Unitariesmentioning

confidence: 93%

“…In this respect, one may ask whether (i) all entangled states lead to Bell violation. This turns out not to be true for projective measurements [3] and for the general case of positive-operatorvalued-measure (POVM) measurements as well [4] (see also [5,6] for more recent results). Similarly, one may ask whether (ii) all incompatible measurements lead to Bell violation.…”

Section: Introductionmentioning

confidence: 99%

Measurement incompatibility does not give rise to Bell violation in general

Bene

Vértesi

2018

New J. Phys.

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In the case of a pair of two-outcome measurements, incompatibility is equivalent to Bell nonlocality. Indeed, any pair of incompatible two-outcome measurements can violate the Clauser-HorneShimony-Holt Bell inequality, which has been proven by Wolf et al (2009 Phys. Rev. Lett. 103 230402). In the case of more than two measurements the equivalence between incompatibility and Bell nonlocality is still an open problem, though partial results have recently been obtained. Here we show that the equivalence breaks for a special choice of three measurements. In particular, we present a set of three incompatible two-outcome measurements, such that if Alice measures this set, independent of the set of measurements chosen by Bob and the state shared by them, the resulting statistics cannot violate any Bell inequality. On the other hand, complementing the above result, we exhibit a set of N measurements for any N 2 > that is N 1 -( )-wise compatible, nevertheless it gives rise to Bell violation.

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Simulating Positive-Operator-Valued Measures with Projective Measurements

Cited by 128 publications

References 42 publications

Incompatibility robustness of quantum measurements: a unified framework

Incompatibility robustness of quantum measurements: a unified framework

Noise–Disturbance Relation and the Galois Connection of Quantum Measurements

Measurement incompatibility does not give rise to Bell violation in general

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