Computable genuine multimode entanglement measure: Gaussian vs. non-Gaussian

TL;DR

This paper develops a measure for genuine multimode entanglement in continuous variable systems, deriving formulas for Gaussian states and analyzing how photon addition and subtraction affect entanglement in non-Gaussian states, revealing new enhancement and freezing phenomena.

Contribution

It introduces a computable geometric measure of multimode entanglement applicable to Gaussian and non-Gaussian states, with explicit formulas and comparative analysis of photon addition and subtraction effects.

Findings

Photon addition enhances GGM in three-mode states, while photon subtraction is more effective in four-mode states.

Photon subtraction from four-mode squeezed vacuum states generally yields higher entanglement than photon addition.

A novel freezing of GGM occurs during specific photon subtraction scenarios.

Abstract

Genuine multimode entanglement in continuous variable systems can be quantified by exploring the geometry of the state-space, namely via the generalized geometric measure (GGM) which is defined as the shortest distance of a given multimode state from a nongenunely multimode entangled state. For multimode Gaussian states, we derive a closed form expression of GGM in terms of the symplectic invariants of the reduced states. Following that prescription, the characteristics of GGM for typical three- and four-mode Gaussian states are investigated. In the non-Gaussian paradigm, we compute GGM for photon-added as well as -subtracted states having three- and four-modes and find that both addition and subtraction of photons lead to enhancement of the genuine multimode entanglement content of the state compared to its Gaussian counterpart. Our analysis reveals that when an initial three-mode vacuum state is evolved according to an interacting Hamiltonian, photon addition is more beneficial in increasing GGM compared to photon subtraction while the picture is opposite for four-mode case. Specifically, subtracting photons from four-mode squeezed vacuum states almost always results in higher multimode entanglement content than that of photon addition with both single as well as multimode and constrained as well as unconstrained operations. Furthermore, we observe a novel freezing feature of GGM during some specific cases of photon subtraction with respect to the number of photons subtracted.