Measurements Incompatible in Quantum Theory Cannot Be Measured Jointly in Any Other No-Signaling Theory

Abstract

It is well known that jointly measurable observables cannot lead to a violation of any Bell inequalityindependent of the state and the measurements chosen at the other site. In this letter we prove the converse: every pair of incompatible quantum observables enables the violation of a Bell inequality and therefore must remain incompatible within any other no-signaling theory. While in the case of von Neumann measurements it is sufficient to use the same pair of observables at both sites, general measurements can require different choices. The main result is obtained by showing that for arbitrary dimension the CHSH inequality provides the Lagrangian dual of the characterization of joint measurability. This leads to a simple criterion for joint measurability beyond the known qubit case."While [...] More than seventy years after Einstein, Podolsky and Rosen (EPR) raised this puzzle we know, as a consequence of Bell's argument [2], that a complete theory in the sense of EPR would force us to pay a high price-such as giving up Einstein locality. Could there, however, be a theory which provides more information than quantum mechanics but still is 'incomplete enough' to circumvent such fundamental conflicts? In this work we address a particular instance of this question, in the context of which the answer is clearly negative.We consider observables which are not jointly measurable, i.e., incompatible within quantum mechanics and show that they all enable the violation of a Bell inequality. That is, there exists a bipartite quantum state and a set of observables for an added site together with which the given observables violate a Bell inequality. As a consequence the observed probabilities do not admit a joint distribution [3] unless this depends on the observable chosen at the added site, which conflicts with Einstein locality, i.e., the no-signaling condition (see appendix). So, if a hypothetical no-signaling theory is a refinement of quantum mechanics (but otherwise consistent with it [19]), it can not render possible the joint measurability of observables which are incompatible within quantum mechanics-even if these observables are already almost jointly measurable in quantum theory.An enormous amount of work has been done in related directions: Bell inequalities [4] and no-signaling theories [5] are lively fields of research. It is well known (and used in constructing quantum states admitting a local hidden variable description [6,7]) that jointly measurable quantum observables can never lead to a violation of a Bell inequality [3]. The converse, however, has hardly been addressed. For generalized measurements (POVMs) this might partly be due to the fact that no criterion for joint measurability is known beyond twolevel systems, for which it was derived only recently [8]. A first indication of the present result can be found in [9] where it has been observed that for particular two-level observables the border of joint measurability [10] coincides with the one for the violation of the CHSH Bell inequality [11]. ...

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

“…Similar results in terms of operators in case of measurements on states were obtained in [2,8]. Now we will proceed by deriving some conditions on the incompatibility of two-outcome measurements based on the results of Prop.…”

“…We will formulate the problem of compatibility of two two-outcome measurements as a problem of linear programming [16] similar to the one obtained in [2]. We will start with the results of Prop.…”

“…The degree of compatibility (also called robustness of incompatibility) has been studied for measurements on channels and combs [2,7,8].…”

“…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].…”

Probabilistic theories with purification

“…An intense work on non-locality in general non-signaling theories has followed this observation, opening a very active line of research (see e.g. [2][3][4][5]). On the other hand, the authors of Refs.…”