Joint measurability of quantum effects and the matrix diamond

Bluhm, Andreas; Nechita, Ion

doi:10.1063/1.5049125

Cited by 32 publications

(60 citation statements)

References 51 publications

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“…The noise set depends on the specific measurements, which makes this measure different than all the other measures considered in this work. 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].…”

Section: Incompatibility Depolarising Robustness 311 Definition Anmentioning

confidence: 99%

“…the measurement generating a uniform distribution of outcomes regardless of the state. It has been investigated in many works [7,28,31,34,35,40], and also in the framework of general probabilistic theories [41,42].…”

Section: Incompatibility Random Robustness 321 Definition and Propmentioning

confidence: 99%

“…where p a a { } and q b b { } are probability distributions. This measure has been investigated in many works [12,16,29,31,34,35,[42][43][44][45], and also in the framework of general probabilistic theories [46,47].…”

Section: Incompatibility Probabilistic Robustness 331 Definition Amentioning

confidence: 99%

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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%

Section: Incompatibility Random Robustness 321 Definition and Propmentioning

confidence: 99%

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Incompatibility robustness of quantum measurements: a unified framework

2019

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“…Recently, the authors have found that the inclusion constants for the free spectrahedral relaxation of the 1 -ball, the matrix diamond [DDOSS17], are relevant for the joint measurability of binary quantum measurements [BN18]. The fact that not all observables can be measured at the same time is one of the most remarkable properties of quantum mechanics, the observables of position and momentum providing the best-known example of this behavior [Hei27,Boh28].…”

mentioning

confidence: 99%

Compatibility of Quantum Measurements and Inclusion Constants for the Matrix Jewel

Bluhm¹,

Nechita²

2020

SIAM J. Appl. Algebra Geometry

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In this work, we establish the connection between the study of free spectrahedra and the compatibility of quantum measurements with an arbitrary number of outcomes. This generalizes previous results by the authors for measurements with two outcomes. Free spectrahedra arise from matricial relaxations of linear matrix inequalities. A particular free spectrahedron which we define in this work is the matrix jewel. We find that the compatibility of arbitrary measurements corresponds to the inclusion of the matrix jewel into a free spectrahedron defined by the effect operators of the measurements under study. We subsequently use this connection to bound the set of (asymmetric) inclusion constants for the matrix jewel using results from quantum information theory and symmetrization. The latter translate to new lower bounds on the compatibility of quantum measurements. Among the techniques we employ are approximate quantum cloning and mutually unbiased bases.

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“…(See Ref. [BN18] for other recent connections between free spectrahedra and quantum information.) The context of this work is a connection between decompositions of quantum states and decompositions of nonnegative matrices presented in [DN19].…”

Section: Introductionmentioning

confidence: 99%

Separability for mixed states with operator Schmidt rank two

Cuevas

Drescher²,

Netzer³

2019

Quantum

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The operator Schmidt rank is the minimum number of terms required to express a state as a sum of elementary tensor factors. Here we provide a new proof of the fact that any bipartite mixed state with operator Schmidt rank two is separable, and can be written as a sum of two positive semidefinite matrices per site. Our proof uses results from the theory of free spectrahedra and operator systems, and illustrates the use of a connection between decompositions of quantum states and decompositions of nonnegative matrices. In the multipartite case, we prove that any Hermitian Matrix Product Density Operator (MPDO) of bond dimension two is separable, and can be written as a sum of at most four positive semidefinite matrices per site. This implies that these states can only contain classical correlations, and very few of them, as measured by the entanglement of purification. In contrast, MP-DOs of bond dimension three can contain an unbounded amount of classical correlations.

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Joint measurability of quantum effects and the matrix diamond

Cited by 32 publications

References 51 publications

Incompatibility robustness of quantum measurements: a unified framework

Incompatibility robustness of quantum measurements: a unified framework

Compatibility of Quantum Measurements and Inclusion Constants for the Matrix Jewel

Separability for mixed states with operator Schmidt rank two

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