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
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.
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 obtain a formal characterization of the compatibility or otherwise of a set of positive-operator-valued measures (POVMs) via their Naimark extensions. We show that a set of POVMs is jointly measurable if and only if there exists a single Naimark extension, specified by a fixed ancilla state on the same ancilla Hilbert space, that maps them to a set of commuting projective measurements (PVMs). We use our result to obtain an easily checkable sufficient condition for the compatibility of a pair of dichotomic observables in any dimension. This in turn leads to a characterization of the compatibility regions for some important classes of observables including a pair of unsharp qubit observables. Finally, we also outline as to how our result provides an alternate approach to quantifying the incompatibility of a general set of quantum measurements.
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 consisting of a finite number of POVMs and then prove that the joint measurability of
MS 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 is jointly measurable if and only if the corresponding “row stochastic operator matrix"
MS composed of
MS can be expressed as an “operator convex combination" of
{0,IS}—the row stochastic operator matrices. In particular, it is proved that the POVM measurement assemblage
M(n0 ,n1) induced by two three-dimensional real unit vectors
ni(i=0,1) is jointly measurable if and only if
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).
Measurement incompatibility and the negativity of quasi-probability distributions both demonstrate the signature of nonclassicality. However, their relations largely remained qualitative, and no explicit operational connection has been established. We provide an operational link between measurement incompatibility and the Margenau-Hill(MH) quasi-probability distribution associated with two dichotomic observables. { We first derive the joint measurability (measurement compatibility) condition for any pair of dichotomic observables in arbitrary finite dimension
d.} We then rigorously prove that for any pair of unsharp dichotomic measurements in dimension
d, the positivity of the MH distribution is equivalent to joint measurability \emph{i.e.}, the MH distribution is positive \emph{if and only if} the measurements are jointly measurable. We further introduce the MH-like quasi-probability distribution for
n dichotomic unsharp observables in arbitrary finite dimension and derive a sufficient condition of joint-measurability when the observables are mutually anticommuting. Finally, we propose an interferometric setup, inspired by quantum-switch architectures, that directly reconstructs MH quasi-probability which in turn provides a test of incompatibility.