Monogamy inequalities for entanglement using continuous variable measurements

TL;DR

This paper derives universal continuous variable monogamy inequalities for tripartite entanglement that do not assume Gaussian states, providing bounds on entanglement distribution applicable to noisy and lossy quantum systems.

Contribution

It introduces new monogamy inequalities for continuous variable entanglement that are valid for all tripartite states, including non-Gaussian states, and explores their implications.

Findings

Monogamy inequalities hold for all tripartite states without Gaussian assumptions.

The inequalities are saturated under certain loss and noise conditions.

Steering criteria influence the bounds of entanglement distribution.

Abstract

We consider three modes $A$, $B$ and $C$ and derive continuous variable monogamy inequalities that constrain the distribution of bipartite entanglement amongst the three modes. The inequalities hold for all such tripartite states, without the assumption of Gaussian states, and are based on measurements of two conjugate quadrature phase amplitudes $X_{i}$ and $P_{i}$ at each mode $i=A,B$. The first monogamy inequality is $D_{BA}+D_{BC}\geq1$ where $D_{BA}<1$ is the widely used symmetric entanglement criterion, for which $D_{BA}$ is the sum of the variances of $(X_{A}-X_{B})/2$ and $(P_{A}+P_{B})/2$. A second monogamy inequality is $Ent_{BA}Ent_{BC}\geq\frac{1}{\left(1+(g_{BA}^{(sym)})^{2}\right)\left(1+(g_{BC}^{(sym)})^{2}\right)}$ where $Ent_{BA}<1$ is the EPR variance product criterion for entanglement. Here $Ent_{BA}$ is a normalised product of variances of $X_{B}-g_{BA}^{(sym)}X_{A}$ and $P_{B}+g_{BA}^{(sym)}P_{A}$, and $g_{BA}^{(sym)}$ is a parameter that gives a measure of the symmetry between the moments of $A$ and $B$. We also show that the monogamy bounds are increased if a standard steering criterion for the steering of $B$ is not satisfied. We illustrate the monogamy for continuous variable tripartite entangled states including the effects of losses and noise, and identify regimes of saturation of the inequalities. The monogamy relations explain the experimentally observed saturation at $D_{AB}=0.5$ for the entanglement between $A$ and $B$ when both modes have 50\% losses, and may be useful to establish rigorous bounds of correlation for the purpose of quantum key distribution protocols.