Optics and Spectroscopy

Continuous variable tripartite entanglement and Einstein-Podolsky-Rosen correlations from triple nonlinearities

M. K. Olsen, Ashton S. Bradley, M. D. Reid

Published Aug 1, 2007• 1 citation

Abstract

We develop three-mode generalizations of the Einstein-Podolsky-Rosen paradox, allowing us to define inequalities which may be used to indicate continuous-variable tripartite entanglement. We use these inequalities and an appropriate version of the previously developed van Loock-Furusawa inequalities to theoretically compare the continuous-variable tripartite entanglement available from the use of three concurrent x (2) nonlinearities and from three independent squeezed states mixed on beamsplitters.

DOI: 10.1134/s0030400x07080036

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Optics and Spectroscopy

Continuous variable tripartite entanglement and Einstein-Podolsky-Rosen correlations from triple nonlinearities

M. K. Olsen, Ashton S. Bradley, M. D. Reid

Published Aug 1, 2007• 1 citation

Publisher

Abstract

DOI: 10.1134/s0030400x07080036

Related papers

Continuous variable tripartite entanglement and Einstein–Podolsky–Rosen correlations from triple nonlinearities

M. K. Olsen, Ashton S. Bradley, M. D. Reid

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We consider three modes $A$ , $B$ , and $C$ and derive monogamy inequalities that constrain the distribution of bipartite continuous variable Einstein-Podolsky-Rosen entanglement amongst the three modes. The inequalities hold without the assumption of Gaussian states, and are based on measurements of the quadrature phase amplitudes ${X}_{i}$ and ${P}_{i}$ at each mode $i=A,B,C$ . The first monogamy inequality involves the well-known quantity ${D}_{IJ}$ defined by Duan-Giedke-Cirac-Zoller as the sum of the variances of $({X}_{I}\ensuremath{-}{X}_{J})/2$ and $({P}_{I}+{P}_{J})/2$ where $[{X}_{I},{P}_{J}]={\ensuremath{\delta}}_{IJ}$. Entanglement between $I$ and $J$ is certified if ${D}_{IJ}<1$ . A second monogamy inequality involves the more general entanglement certifier ${\text{Ent}}_{IJ}$ defined as the normalized product of the variances of ${X}_{I}\ensuremath{-}g{X}_{J}$ and ${P}_{I}+g{P}_{J}$ , where $g$ is a real constant. The monogamy inequalities give a lower bound on the values of ${D}_{BC}$ and ${\text{Ent}}_{BC}$ for one pair, given the values ${D}_{BA}$ and ${\text{Ent}}_{BA}$ for the first pair. This lower bound changes in the absence of two-mode Gaussian steering of $B$ . We illustrate for a range of tripartite entangled states, identifying regimes of saturation of the inequalities. The monogamy relations explain without the assumption of Gaussianity the experimentally observed saturation at ${D}_{AB}=0.5$ where there is symmetry between modes $A$ and $C$ .We consider three modes $A$, $B$, and $C$ and derive monogamy inequalities that constrain the distribution of bipartite continuous variable Einstein-Podolsky-Rosen entanglement amongst the three modes. The inequalities hold without the assumption of Gaussian states, and are based on measurements of the quadrature phase amplitudes ${X}_{i}$ and ${P}_{i}$ at each mode $i=A,B,C$. The first monogamy inequality involves the well-known quantity ${D}_{IJ}$ defined by Duan-Giedke-Cirac-Zoller as the sum of the variances of $({X}_{I}\ensuremath{-}{X}_{J})/2$ and $({P}_{I}+{P}_{J})/2$ where $[{X}_{I},{P}_{J}]={\ensuremath{\delta}}_{IJ}$. Entanglement between $I$ and $J$ is certified if ${D}_{IJ}<1$. A second monogamy inequality involves the more general entanglement certifier ${\text{Ent}}_{IJ}$ defined as the normalized product of the variances of ${X}_{I}\ensuremath{-}g{X}_{J}$ and ${P}_{I}+g{P}_{J}$, where $g$ is a real constant. The monogamy inequalities give a lower bound on the values of ${D}_{BC}$ and ${\text{Ent}}_{BC}$ for one pair, given the values ${D}_{BA}$ and ${\text{Ent}}_{BA}$ for the first pair. This lower bound changes in the absence of two-mode Gaussian steering of $B$. We illustrate for a range of tripartite entangled states, identifying regimes of saturation of the inequalities. The monogamy relations explain without the assumption of Gaussianity the experimentally observed saturation at ${D}_{AB}=0.5$ where there is symmetry between modes $A$ and $C$.

Physical Review AAug 14, 20178 citations

We compare theoretically the tripartite entanglement available from the use of three concurrent

χ^{(2)}

nonlinearities and three independent squeezed states mixed on beamsplitters, using the van Loock-Furusawa inequalities. We also define three-mode generalisations of the Einstein-Podolsky-Rosen paradox which are an alternative for demonstrating the inseparability of the density matrix.

Following previous work, we distinguish between genuine

N

-partite entanglement and full

N

-partite inseparability. Accordingly, we derive criteria to detect genuine multipartite entanglement using continuous-variable (position and momentum) measurements. Our criteria are similar but different to those based on the van Loock--Furusawa inequalities, which detect full

N

-partite inseparability. We explain how the criteria can be used to detect the genuine

N

-partite entanglement of continuous variable states generated from squeezed and vacuum state inputs, including the continuous-variable Greenberger-Horne-Zeilinger state, with explicit predictions for up to

N=9

. This makes our work accessible to experiment. For

N=3

, we also present criteria for tripartite Einstein-Podolsky-Rosen (EPR) steering. These criteria provide a means to demonstrate a genuine three-party EPR paradox, in which any single party is steerable by the remaining two parties.

We consider three modes

A

B

, and

C

and derive monogamy inequalities that constrain the distribution of bipartite continuous variable Einstein-Podolsky-Rosen entanglement amongst the three modes. The inequalities hold without the assumption of Gaussian states, and are based on measurements of the quadrature phase amplitudes

{X}_{i}

and

{P}_{i}

at each mode

i=A,B,C

. The first monogamy inequality involves the well-known quantity

{D}_{IJ}

defined by Duan-Giedke-Cirac-Zoller as the sum of the variances of $({X}_{I}\ensuremath{-}{X}_{J})/2$ and

({P}_{I}+{P}_{J})/2

where $[{X}_{I},{P}_{J}]={\ensuremath{\delta}}_{IJ}$. Entanglement between

I

and

J

is certified if

{D}_{IJ}<1

. A second monogamy inequality involves the more general entanglement certifier

{\text{Ent}}_{IJ}

defined as the normalized product of the variances of ${X}_{I}\ensuremath{-}g{X}_{J}$ and

{P}_{I}+g{P}_{J}

, where

g

is a real constant. The monogamy inequalities give a lower bound on the values of

{D}_{BC}

and

{\text{Ent}}_{BC}

for one pair, given the values

{D}_{BA}

and

{\text{Ent}}_{BA}

for the first pair. This lower bound changes in the absence of two-mode Gaussian steering of

B

. We illustrate for a range of tripartite entangled states, identifying regimes of saturation of the inequalities. The monogamy relations explain without the assumption of Gaussianity the experimentally observed saturation at

{D}_{AB}=0.5

where there is symmetry between modes

A

and

C