Joint Measurability of Generalized Measurements Implies Classicality

Abstract

The fact that not all measurements can be carried out simultaneously is a peculiar feature of quantum mechanics and responsible for many key phenomena in the theory, such as complementarity or uncertainty relations. For the special case of projective measurements quantum behavior can be characterized by the commutator but for generalized measurements it is not easy to decide whether two measurements can still be understood in classical terms or whether they show already quantum features. We prove that generalized measurements which do not fulfill the notion of joint measurability are nonclassical, as they can be used for the task of quantum steering. This shows that the notion of joint measurability is, among several definitions, the proper one to characterize quantum behavior. Moreover, the equivalence allows to derive novel steering inequalities from known results on joint measurability and new criteria for joint measurability from known results on the steerability of states.

All measurements in a probabilistic theory are compatible if and only if the state space is a simplex

“…In light of these discoveries compatibility in the framework of general probabilistic theories has been studied [2][3][4] in order to show the difference between classical and non-classical probabilistic theories. Also the connection of compatibility and steering in general probabilistic theories have been studied [5,6].…”

Dynamics of incompatibility of quantum measurements in open systems

“…One should notice the following important observation [6,8,15]: the nonexistence of the hidden state model for the correlations P(i, j|a, b) is equivalent to incompatibility of Alice's noisy local observables…”

“…For certain scenarios based on correlation experiments, the existence of such a classical description is equivalent to joint measurability of suitable observables. In particular, it has been recently shown that for the CHSH-Bell inequality [5] and EPRsteering scenarios [6][7][8]10], the appropriate quantum resource can be formulated in terms of incompatibility of the available local observables. It is therefore important to aim for a better understanding of the incompatibility of observables.…”

Unsharp Measurements and Joint Measurability

“…While the right-hand side of the entropic uncertainty relation (14) reduces to zero in this case, the left-hand side can be simplified (by substituting (19) in (13) and simplifying) to obtain,…”

“…While sharp values for non-commuting observables can not be assigned jointly via their PV measurements, their unsharp joint values could be perceived when compatible POVMs are employed. Active research efforts are dedicated to investigate operationally significant criteria of joint measurability (compatibility) of two or more POVMs and also to develop a resource theory of measurement incompatibility [5][6][7][8][9][10][11][12][13][14][15][16][17] . In this paper, we review the notion of compatible POVMs in the generalized measurement setting.…”