The compatibility dimension of quantum measurements

Loulidi, Faedi; Nechita, Ion

doi:10.48550/arxiv.2008.10317

Cited by 2 publications

(2 citation statements)

References 22 publications

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“…While completing this work, we became aware of the recent and independent work in Ref. [47]. In this Appendix, we give a criterion for two rank-one projective measurements in a d-dimensional space to be incompatible in every (d − 1)-dimensional subspace.…”

Section: Note Addedmentioning

confidence: 99%

Quantum measurement incompatibility in subspaces

Uola,

Kraft,

Designolle

et al. 2020

Preprint

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We consider the question of characterising the incompatibility of sets of high-dimensional quantum measurements. We introduce the concept of measurement incompatibility in subspaces. That is, starting from a set of measurements that is incompatible, one considers the set of measurements obtained by projection onto any strict subspace of fixed dimension. We identify three possible forms of incompatibility in subspaces: (i) incompressible incompatibility: measurements that become compatible in every subspace, (ii) fully compressible incompatibility: measurements that remain incompatible in every subspace, and (iii) partly compressible incompatibility: measurements that are compatible in some subspace and incompatible in another. For each class we discuss explicit examples. Finally, we present some applications of these ideas. First we show that joint measurability and coexistence are two inequivalent notions of incompatibitliy in the simplest case of qubit systems. Second we highlight the implications of our results for tests of quantum steering.

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Section: Note Addedmentioning

confidence: 99%

Quantum measurement incompatibility in subspaces

Uola,

Kraft,

Designolle

et al. 2020

Preprint

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“…Furthermore, it has been argued that the optimal cloning fidelity for a set of non-commuting density operators on a d-dimensional space gives a lower bound on the degree of incompatibility possible for a d-dimensional quantum state [21]. More recently, the results of [20] have been generalised by making use of approximate cloning machines to characterize the idea of compatibility dimension [22].…”

Section: Introductionmentioning

confidence: 99%

On optimal cloning and incompatibility

Mitra¹,

Mandayam²

2021

J. Phys. A: Math. Theor.

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We investigate the role of symmetric quantum cloning machines (QCMs) in quantifying the mutual incompatibility of quantum observables. Specifically, we identify a cloning-based incompatibility measure whereby the incompatibility of a set of observables maybe quantified in terms of how well a uniform ensemble of their eigenstates can be cloned via a symmetric QCM. We show that this new incompatibility measure is faithful since it vanishes only for commuting observables. We prove an upper bound for for any set of observables in a finite-dimensional system and show that the upper bound is attained if and only if the observables are mutually unbiased. Finally, we use our formalism to obtain the optimal quantum cloner for a pair of qubit observables. Our work marks an important step in formalising the connection between two fundamental concepts in quantum information theory, namely, the no-cloning principle and the existence of incompatible observables in quantum theory.

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The compatibility dimension of quantum measurements

Cited by 2 publications

References 22 publications

Quantum measurement incompatibility in subspaces

Quantum measurement incompatibility in subspaces

On optimal cloning and incompatibility

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