Classical Representability for Partial Boolean Structures in Quantum Mechanics

The quantum mechanical description of a physical system provides a set of predictions for all possible experiments that can be performed on it. Given an observable $$A$$ , representing a physical quantity, e.g., energy, its expectation value $$\langle A\rangle$$ , or the probability for the observation of a certain outcome, can be obtained from the knowledge of the quantum state through the Born rule 7.1 $$\langle A\rangle_{\rho}=\sum_{i}\lambda_{i}{\rm Prob}(\lambda_{i})=\sum_{i}\lambda_{i}\mbox{tr}[\rho P_{i}]=\mbox{tr}[\rho A],\;\text{ with }\;{\rm Prob}(\lambda_{i})=\mbox{tr}[\rho P_{i}],$$ where $$A$$ has spectral decomposition $$A=\sum_{i}\lambda_{i}P_{i}$$ . A remarkable property of quantum mechanics (QM) is that not all physical observable can be jointly measured, the most famous example being the position and momentum of a particle. Such observables are said to be incompatible 19, 40, 43. In the most general experiment on a physical system, then, each round consists in the joint measurement of a set of compatible observables. For each set of compatible observables, a quantum state defines a classical probability distribution over all possible outcomes of their measurement. For projective measurements, the textbook QM observables, this is a direct consequence of the spectral theorem applied to a set of mutually commuting observables. The same structure arises for set of jointly measurement generalized measurements, i.e., positive operator-valued measures (POVMs); see e.g., 44. In the following, when we speak about observables we refer to projective measurements, sometimes called , unless explicitly stated otherwise.