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arXiv:1904.01785 · Physical Review A

Necessary and sufficient condition for joint measurability

Jeongwoo Jae, Kyunghyun Baek, Junghee Ryu, Jinhyoung Lee

Published Sep 18, 2019• 12 citations
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Abstract

To analyze the joint measurability of given measurements, we introduce a Hermitian operator-valued measure, called $W$ measure, such that it has marginals of positive operator-valued measures. We prove that $W$ measure is a positive operator-valued measure (POVM) if and only if its marginal POVMs are jointly measurable. The proof suggests employing a negative $W$ measure as an indicator of nonjoint measurability. This translates the joint measurability problem to unconstrained convex minimization of nondifferentiable function, which is solvable using a subgradient method. By applying triangle inequalities to the negativity, we derive the analytic joint measurability criteria for dichotomic and trichotomic variables. Moreover, we propose an operational test for joint measurability in a sequential measurement scenario.

DOI
10.1103/physreva.100.032113

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arXiv:2003.00785•Physical Review Research•Oct 28, 2020•11 citations
To analyze the joint measurability of given measurements, we introduce a Hermitian operator-valued measure, called WWW measure, such that it has marginals of positive operator-valued measures. We prove that WWW measure is a positive operator-valued measure (POVM) if and only if its marginal POVMs are jointly measurable. The proof suggests employing a negative WWW measure as an indicator of nonjoint measurability. This translates the joint measurability problem to unconstrained convex minimization of nondifferentiable function, which is solvable using a subgradient method. By applying triangle inequalities to the negativity, we derive the analytic joint measurability criteria for dichotomic and trichotomic variables. Moreover, we propose an operational test for joint measurability in a sequential measurement scenario.
The POVM measurement assemblage consisting of a set of positive operator-value measurements (POVMs) is an important tool for describing and studying the non-locality, quantum steering and quantum correlation of composite quantum systems. In this paper, we first introduce the definition of joint measurability of the POVM measurement assemblage MS\mathcal{M}_SMS​ consisting of a finite number of POVMs and then prove that the joint measurability of MS\mathcal{M}_SMS​ is equivalent to its compatibility. We give a general form of a POVM measurement which is jointly measurable with the one-rank projection measurement. It is proved that the POVM measurement assemblage MS\mathcal{M}_SMS​ is jointly measurable if and only if the corresponding “row stochastic operator matrix" MS\textbf{M}_SMS​ composed of MS\mathcal{M}_SMS​ can be expressed as an “operator convex combination" of {0,IS}\{0,I_S\}{0,IS​}—the row stochastic operator matrices. In particular, it is proved that the POVM measurement assemblage M(n0 ,n1)\mathcal{M}(\textbf{n}_{0~},\textbf{n}_{1})M(n0 ​,n1​) induced by two three-dimensional real unit vectors ni(i=0,1)\textbf{n}_{i}(i=0,1)ni​(i=0,1) is jointly measurable if and only if n0=±n1\textbf{n}_{0}=\pm\textbf{n}_{1}n0​=±n1​. As an application, we obtain a necessary and sufficient condition under which a two-qubit quantum state is unsteerable with M(n0,n1)\mathcal{M}(\textbf{n}_{0},\textbf{n}_{1})M(n0​,n1​).
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.
Measurements in quantum theory exhibit incompatibility, i.e., they can fail to be jointly measurable.An intuitive way to represent the (in)compatibility relations among a set of measurements is via a hypergraph representing their joint measurability structure: its vertices represent measurements and its hyperedges represent (all and only) subsets of compatible measurements.Projective measurements in quantum theory realize (all and only) joint measurability structures that are graphs.On the other hand, general measurements represented by positive operator-valued measures (POVMs) can realize arbitrary joint measurability structures.Here we explore the scope of joint measurability structures realizable with qubit POVMs.We develop a technique that we term marginal surgery to obtain nontrivial joint measurability structures starting from a set of compatible measurements.We show explicit examples of marginal surgery on a special set of qubit POVMs to construct joint measurability structures such as N-cycle and N-Specker scenarios for any integer N 3.We also show the realizability of various joint measurability structures with N ∈ {4, 5, 6} vertices.In particular, we show that all possible joint measurability structures with N = 4 vertices are realizable.We conjecture that all joint measurability structures are realizable with qubit POVMs.This contrasts with the unbounded dimension required in R. Kunjwal et al. [Phys. Rev. A 89, 052126 (2014)].Our results also render this previous construction maximally efficient in terms of the required Hilbert space dimension.We also obtain a sufficient condition for the joint measurability of any set of binary qubit POVMs which powers many of our results and should be of independent interest.