States in the Hilbert space formulation and the phase space formulation of quantum mechanics

TL;DR

This paper develops practical criteria for verifying whether matrices or functions in the Hilbert space and phase space formulations of quantum mechanics represent valid quantum states, focusing on positivity conditions.

Contribution

It introduces new, adaptable criteria for recognizing quantum states in both formulations, applicable to operators and functions with Weyl-type products.

Findings

Criteria effectively distinguish quantum states from non-states.

Applicable to operators and functions with minimal modifications.

Enhances practical state verification in quantum mechanics.

Abstract

W consider the problem of testing if a given matrix in the Hilbert space formulation of quantum mechanics or a function in the phase space formulation of quantum theory represent a quantum state. We propose several practical criteria to recognise states in these both versions of quantum physics. After minor modifications they can be applied to check positivity of any operators acting in a Hilbert space or positivity of any functions from an algebra with a Weyl type * -- product.