An invitation to quantum incompatibility

Abstract

In the context of a physical theory, two devices, A and B, described by the theory are called incompatible if the theory does not allow the existence of a third device C that would have both A and B as its components. Incompatibility is a fascinating aspect of physical theories, especially in the case of quantum theory. The concept of incompatibility gives a common ground for several famous impossibility statements within quantum theory, such as "no-cloning" and "no information without disturbance"; these can be all seen as statements about incompatibility of certain devices. The purpose of this paper is to give a concise overview of some of the central aspects of incompatibility.

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

“…Based on the analysis of compatibility presented in [1] we obtain the following: Proposition 6. For any two measurements m i : K → P(Ω i ), i ∈ {1, 2}, we have DegCom(m 1 , m 2 ) ≥ 1 2 .…”

“…Incompatibility lies deeply within quantum mechanics and many of the famous and key aspect of quantum theories have been traced to Heisenberg uncertainty principle, no cloning theorem, violations of Bell inequalities and other notions making use of compatibility, see [1] for recent review. 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.…”

“…Recently incompatibility of measurements on quantum channels and combs has been in question [7] as it potentially could be used as a resource in quantum theory in a similar ways as an incompatibility of measurements on quantum states [1]. The degree of compatibility (also called robustness of incompatibility) has been studied for measurements on channels and combs [2,7,8].…”

Measurement Uncertainty Relations for Position and Momentum: Relative Entropy Formulation

“…A Gaussian bi-observable M with parameters (µ M , V M , J M ) is in C u,v if and only if J M = J, where J is given by (47). In this case, the condition (49) is equivalent to…”

Wigner-Araki-Yanase theorem beyond conservation laws

“…Formally, our result states that, whenever a quantity commutes with the evolved pointer of the apparatus, a part of it also necessarily commutes with the measured observable. In other words, in our explanation the WAY-theorem can be understood as a consequence of quantum compatibility [13,14] of a given quantity with the evolved pointer partially inherited by the measured observable. We believe that the intuition behind this formalism is conceptually clearer than in the preceding formulations listed above.…”