Entanglement and nonclassicality for multi-mode radiation field states

TL;DR

This paper explores how nonclassicality in multi-mode quantum optical states can lead to entanglement through classical operations like beamsplitters, introducing new criteria and tests for detecting nonclassicality and entanglement.

Contribution

It develops new methods to identify nonclassicality and entanglement in multi-mode states, including a unified test for nonclassicality and NPT entanglement after beamsplitter interactions.

Findings

Single-mode nonclassicality conditions become entanglement signatures after beamsplitter action.

A combined test for nonclassicality and NPT entanglement in two-mode states.

Extension of Mandel matrix to classify levels of nonclassicality in two-mode systems.

Abstract

Nonclassicality in the sense of quantum optics is a prerequisite for entanglement in multi-mode radiation states. In this work we bring out the possibilities of passing from the former to the latter, via action of classicality preserving systems like beamsplitters, in a transparent manner. For single mode states, a complete description of nonclassicality is available via the classical theory of moments, as a set of necessary and sufficient conditions on the photon number distribution. We show that when the mode is coupled to an ancilla in any coherent state, and the system is then acted upon by a beamsplitter, these conditions turn exactly into signatures of NPT entanglement of the output state. Since the classical moment problem does not generalize to two or more modes, we turn in these cases to other familiar sufficient but not necessary conditions for nonclassicality, namely the Mandel parameter criterion and its extensions. We generalize the Mandel matrix from one-mode states to the two-mode situation, leading to a natural classification of states with varying levels of nonclassicality. For two--mode states we present a single test that can, if successful, simultaneously show nonclassicality as well as NPT entanglement. We also develop a test for NPT entanglement after beamsplitter action on a nonclassical state, tracing carefully the way in which it goes beyond the Mandel nonclassicality test. The result of three--mode beamsplitter action after coupling to an ancilla in the ground state is treated in the same spirit. The concept of genuine tripartite entanglement, and scalar measures of nonclassicality at the Mandel level for two-mode systems, are discussed. Numerous examples illustrating all these concepts are presented.