Quantum probabilities of composite events in quantum measurements with multimode states

TL;DR

This paper explores the definition of quantum probabilities for composite events in multimode quantum systems, highlighting the roles of entanglement and proposing a new approach to quantum joint probabilities.

Contribution

It introduces a novel framework for quantum joint probabilities in multimode states, addressing limitations of existing probability concepts in quantum measurement theory.

Findings

Luders probability is a transition probability between quantum states.

Wigner distribution cannot be regarded as a quantum joint probability.

Entanglement of the prospect and state is necessary for mode interference.

Abstract

The problem of defining quantum probabilities of composite events is considered. This problem is of high importance for the theory of quantum measurements and for quantum decision theory that is a part of measurement theory. We show that the Luders probability of consecutive measurements is a transition probability between two quantum states and that this probability cannot be treated as a quantum extension of the classical conditional probability. The Wigner distribution is shown to be a weighted transition probability that cannot be accepted as a quantum extension of the classical joint probability. We suggest the definition of quantum joint probabilities by introducing composite events in multichannel measurements. The notion of measurements under uncertainty is defined. We demonstrate that the necessary condition for the mode interference is the entanglement of the composite prospect together with the entanglement of the composite statistical state. As an illustration, we consider an example of a quantum game. A special attention is payed to the application of the approach to systems with multimode states, such as atoms, molecules, quantum dots, or trapped Bose-condensed atoms with several coherent modes.