Certifying quantum state and dimension via phase-space Bell tests for continuous variable systems

Abstract

We investigate generalized Bell tests in phase space for continuous variables that can be used to certify quantum states and witness the dimension of a given state. Our so-called s-parametrized observables in the Bell tests correspond to measuring excitation (photon)-number distribution in conjunction with phase -space displacements. Specifically, we study three practically important classes of states broadly, i.e. two-mode squeezed states, entangled coherent states and general entangled state with bounded photon numbers. We obtain the degree of optimal violation of Bell tests for each class, which enables us to certify a quantum state yielding a maximal violation. Furthermore, our analysis also makes it possible to witness the dimension of a quantum state, i.e. the photon number, by obtaining the bound of Bell violation for a different total photon number.