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arXiv:2205.03129 · Journal of Physics A Mathematical and Theoretical

Incompatibility of effects in general probabilistic models*

Roberto Beneduci, Leon Loveridge

Published May 13, 2022• 2 citations
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Abstract

Abstract We give a necessary and sufficient condition for the incompatibility of a pair of effects in a general probabilistic model in which the state space is a total convex space, which can be obtained by minimising a real parameter. This has an interpretation in terms of the least noise that must be included to make the given pair compatible.

DOI
10.1088/1751-8121/ac6f9d

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Abstract We give a necessary and sufficient condition for the incompatibility of a pair of effects in a general probabilistic model in which the state space is a total convex space, which can be obtained by minimising a real parameter. This has an interpretation in terms of the least noise that must be included to make the given pair compatible.
We formulate the necessary and sufficient conditions for the existence of a pair of maximally incompatible two-outcome measurements in a finite-dimensional general probabilistic theory. The conditions are on the geometry of the state space; they require the existence of two pairs of parallel exposed faces with an additional condition on their intersections. We introduce the notion of discrimination measurement and show that the conditions for a pair of two-outcome measurements to be maximally incompatible are equivalent to requiring that a (potential, yet nonexisting) joint measurement of the maximally incompatible measurements would have to discriminate affinely dependent points. We present several examples to demonstrate our results.
We quantify the fuzziness of observables in general probabilistic theories and derive a noise content inequality for incompatible observables. We apply the derived inequality to the standard quantum theory, quantum theory of processes, and polytope state spaces. The noise content for positive operator-valued measures takes a particularly simple form and equals the sum of minimal eigenvalues of all the effects. We illustrate our findings by a number of examples including the introduced notion of reverse observables.
We quantify the intrinsic noise content of an observable in a general probabilistic theory and derive a noise content inequality for incompatible observables. We apply the derived inequality to standard quantum theory, the quantum theory of processes, and polytope state spaces. The noise content for positive operator-valued measures takes a particularly simple form and equals the sum of minimal eigenvalues of all the effects. We illustrate our findings with a number of examples including the introduced notion of reverse observables.
It is well known that when two or more quantum measurements suffer from imperfections they may lose their incompatibility. For a quantum system of finite dimension ddd we study the incompatibility of all projective measurements subjected to white noise and loss. We derive a necessary and sufficient condition for this set of measurements to become compatible in terms of their visibility ppp and efficiency $\ensuremath{\eta}$.
Incompatibility of quantum measurements is of fundamental importance in quantum mechanics. It is closely related to many nonclassical phenomena such as Bell nonlocality, quantum uncertainty relations, and quantum steering. We study the necessary and sufficient conditions of quantum compatibility for a given collection of n measurements in d-dimensional space. From the compatibility criterion for two-qubit measurements, we compute the incompatibility probability of a pair of independent random measurements. For a pair of unbiased random qubit measurements, we derive that the incompatibility probability is exactly 3/5. Detailed results are also presented in figures for pairs of general qubit measurements.