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arXiv:1310.2690 · Physical Review A

Criteria for genuine N -partite continuous-variable entanglement and Einstein-Podolsky-Rosen steering

Run Yan Teh, M. D. Reid

Published Dec 23, 2014• 88 citations
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Abstract

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.

DOI
10.1103/physreva.90.062337

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Following previous work, we distinguish between genuine NNN-partite entanglement and full NNN-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 NNN-partite inseparability. We explain how the criteria can be used to detect the genuine NNN-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=9N=9N=9. This makes our work accessible to experiment. For N=3N=3N=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 develop the concept of genuine N-partite Einstein-Podolsky-Rosen (EPR) steering. This nonlocality is the natural multipartite extension of the original EPR paradox. Useful properties emerge that are not guaranteed for genuine multipartite entangled states. In particular, there is a close link with the task of one-sided, device-independent quantum secret sharing. We derive inequalities to demonstrate multipartite EPR steering for Greenberger-Horne-Zeilinger and Gaussian continuous variable states in loophole-free scenarios.
We derive inequalities sufficient to detect the genuine NNN-partite steering of NNN distinct systems. Here, we are careful to distinguish between the concepts of full NNN-partite steering inseparability (where steering is confirmed individually for all bipartitions of the NNN systems, thus negating the bilocal hidden state model for each bipartition) and genuine NNN-partite steering (which excludes all convex combinations of the bilocal hidden state models). Other definitions of multipartite steering are possible and we also derive inequalities to detect a stricter genuine NNN-partite steering where only one site needs to be trusted. The inequalities are expressed as variances of quadrature phase amplitudes and thus apply to continuous-variable systems. We show how genuine NNN-partite steerable states can be created and detected for the nodes of a network formed from a single-mode squeezed state passed through a sequence of $N\ensuremath{-}1$ beam splitters. A stronger genuine NNN-partite steering is created, if one uses two squeezed inputs or NNN squeezed inputs. We are able to confirm that genuine tripartite steering (by the above definition and by the stricter definition) has been realized experimentally. Finally, we analyze how bipartite steering and entanglement are distributed among the systems in the tripartite case, illustrating with monogamy inequalities. While we use Gaussian states to benchmark the criteria, the inequalities derived in this paper are not based on the assumption of Gaussian states, which gives an advantage for quantum communication protocols.
We derive conditions in the form of inequalities to detect the genuine NNN-partite entanglement of NNN systems. The inequalities are expressed in terms of variances of spin operators and can be tested by local spin measurements performed on the individual systems. Violation of the inequalities is sufficient (but not necessary) to certify the multipartite entanglement and occurs when a type of spin squeezing is created. The inequalities are similar to those derived for continuous-variable systems, but instead are based on the Heisenberg spin-uncertainty relation $\mathrm{\ensuremath{\Delta}}{J}_{x}\mathrm{\ensuremath{\Delta}}{J}_{y}\ensuremath{\ge}|\ensuremath{\langle}{J}_{z}\ensuremath{\rangle}|/2$. We also extend previous work to derive spin-variance inequalities that certify the full tripartite inseparability or genuine multipartite entanglement among systems with fixed spin JJJ, as in Greenberger--Horne--Zeilinger (GHZ) states and W states where J=1/2J=1/2J=1/2. These inequalities are derived from the planar spin-uncertainty relation ${(\mathrm{\ensuremath{\Delta}}{J}_{x})}^{2}+{(\mathrm{\ensuremath{\Delta}}{J}_{y})}^{2}\ensuremath{\ge}{C}_{J}$ where CJ{C}_{J}CJ​ is a constant for each JJJ. Finally, it is shown how the inequalities detect multipartite entanglement based on Stokes operators. We illustrate with experiments that create entanglement shared among separated atomic ensembles, polarization-entangled optical modes, and the clouds of atoms of an expanding spin-squeezed Bose-Einstein condensate. For each example, we give a criterion to certify the mutual entanglement.
Einstein-Podolsky-Rosen (EPR) steering is a form of quantum nonlocality which is intermediate between entanglement and Bell nonlocality. EPR steering is a resource for quantum key distribution that is device independent on only one side in that it certifies bipartite entanglement when one party's device is not characterized while the other party's device is fully characterized. In this work, we introduce two types of genuine tripartite EPR steering, and derive two steering inequalities to detect them. In a semi-device-independent scenario where only the dimensions of two parties are assumed, the correlations which violate one of these inequalities also certify genuine tripartite entanglement. It is known that Alice can demonstrate bipartite EPR steering to Bob if and only if her measurement settings are incompatible. We demonstrate that quantum correlations can also detect tripartite EPR steering from Alice to Bob and Charlie, even if Charlie's measurement settings are compatible.
Collective Einstein-Podolsky-Rosen (EPR) steering is a unique multipartite resource that enables the realization of more secure communication protocols. However, due to the lack of suitable experimental criteria, research on collective EPR steering has so far been limited to continuous variable systems. Here, we employ the coherence-based collective EPR steering criteria proposed by Mondal et al. [Phys. Rev. A 99, 012312 (2019)] and develop a semidefinite programming method to find the optimal measurement settings. Experimentally, we take the three-qubit generalized Greenberger-Horne-Zeilinger states and generalized WWW states as examples, demonstrating that collective EPR steering can be observed by detecting the average l1{l}_{1}l1​ norm of coherence of the steered subsystem. We also explored the effect of measurement settings on complementarity relations between different collective steering criteria, and established a connection between collective steering and coherence complementarity. Our paper fills a gap in the verification of collective steerability in discrete variable systems and reveals a profound connection between EPR steering and quantum coherence.
Monogamy inequalities for the way bipartite Einstein-Podolsky-Rosen (EPR) steering can be distributed among NNN systems are derived. One set of inequalities is based on witnesses with two measurement settings, and may be used to demonstrate correlation of outcomes between two parties, that cannot be shared with more parties. It is shown that the monogamy for steering is directional. Two parties cannot independently demonstrate steering of a third system, using the same two-setting steering witness, but it is possible for one party to steer two independent systems. This result explains the monogamy of two-setting Bell inequality violations and the sensitivity of the continuous variable (CV) EPR criterion to losses on the steering party. We generalize to mmm settings. A second type of monogamy relation gives the quantitative amount of sharing possible, when the number of parties is less than or equal to mmm, and takes a form similar to the Coffman-Kundu-Wootters relation for entanglement. The results enable characterization of the tripartite steering for CV Gaussian systems and qubit Greenberger-Horne-Zeilinger and WWW states.