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arXiv:2108.06926 · Physical Review A

Full multipartite steering inseparability, genuine multipartite steering, and monogamy for continuous-variable systems

Run Yan Teh, Manuel Gessner, M. D. Reid, Matteo Fadel

Published Jan 3, 2022• 26 citations
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Abstract

We derive inequalities sufficient to detect the genuine $N$-partite steering of $N$ distinct systems. Here, we are careful to distinguish between the concepts of full $N$-partite steering inseparability (where steering is confirmed individually for all bipartitions of the $N$ systems, thus negating the bilocal hidden state model for each bipartition) and genuine $N$-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 $N$-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 $N$-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 $N$-partite steering is created, if one uses two squeezed inputs or $N$ 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.

DOI
10.1103/physreva.105.012202

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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−1N\ensuremath{-}1N\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.
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.
The multipartite entangled state has drawn broad attention for both foundations of quantum mechanics and applications in quantum information processing. Here, we study the spatially separated NNN-partite continuous-variable Greenberger-Horne-Zeilinger--like states, which can be produced by a linear optical network with squeezed light and $N\ensuremath{-}1$ beamsplitters. We investigate the properties of multipartite Einstein-Podolsky-Rosen steering possessed by those states, and find that the steering of a given quantum mode is allowed when not less than half of the modes within the states take part in the steering group. This is certified by the detection of the correlation between position and momentum quadratures of the steered mode and a combination of quadratures of other modes inside the steering group. The steering is evidenced by the high correlation where the steering group can infer the quadratures of the steered mode to high precision, i.e., below the quantum limit for the position and momentum quadratures of the steered quantum mode. We also examine the influence of inefficiency on the multipartite steering, and derive the threshold of the loss tolerance. Furthermore, we discuss the collective NNN-partite steering induced by the asymmetric loss on beams, which exists when a given quantum mode can only be steered by all the remaining $N\ensuremath{-}1$ modes collaboratively. The present multipartite steering correlation may have potential applications in certain quantum information tasks where the issue of trust is important, such as one-sided device-independent quantum secret sharing.
We consider three modes AAA, BBB and CCC 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 XiX_{i}Xi​ and PiP_{i}Pi​ at each mode i=A,Bi=A,Bi=A,B. The first monogamy inequality is DBA+DBC≥1D_{BA}+D_{BC}\geq1DBA​+DBC​≥1 where DBA<1D_{BA}<1DBA​<1 is the widely used symmetric entanglement criterion, for which DBAD_{BA}DBA​ is the sum of the variances of (XA−XB)/2(X_{A}-X_{B})/2(XA​−XB​)/2 and (PA+PB)/2(P_{A}+P_{B})/2(PA​+PB​)/2. A second monogamy inequality is EntBAEntBC≥1(1+(gBA(sym))2)(1+(gBC(sym))2)Ent_{BA}Ent_{BC}\geq\frac{1}{\left(1+(g_{BA}^{(sym)})^{2}\right)\left(1+(g_{BC}^{(sym)})^{2}\right)}EntBA​EntBC​≥(1+(gBA(sym)​)2)(1+(gBC(sym)​)2)1​ where EntBA<1Ent_{BA}<1EntBA​<1 is the EPR variance product criterion for entanglement. Here EntBAEnt_{BA}EntBA​ is a normalised product of variances of XB−gBA(sym)XAX_{B}-g_{BA}^{(sym)}X_{A}XB​−gBA(sym)​XA​ and PB+gBA(sym)PAP_{B}+g_{BA}^{(sym)}P_{A}PB​+gBA(sym)​PA​, and gBA(sym)g_{BA}^{(sym)}gBA(sym)​ is a parameter that gives a measure of the symmetry between the moments of AAA and BBB. We also show that the monogamy bounds are increased if a standard steering criterion for the steering of BBB 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 DAB=0.5D_{AB}=0.5DAB​=0.5 for the entanglement between AAA and BBB when both modes have 50\% losses, and may be useful to establish rigorous bounds of correlation for the purpose of quantum key distribution protocols.
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.
Quantum steering is considered one of the most well-known nonlocal phenomena in quantum mechanics. Unlike entanglement and Bell nonlocality, the asymmetry of quantum steering makes it vital for one-sided device-independent quantum information processing. Although there has been much progress on steering detection for bipartite systems, the criterion for Einstein-Podolsky-Rosen steering in tripartite systems remains challenging and inadequate. In this paper we first derive a promising steering criterion for any three-qubit states via correlation matrix. Furthermore, we propose the monogamy relation between the tripartite steering of system and the bipartite steering of subsystems based on the derived criterion. Finally, as illustrations, we demonstrate the performance of the steering criterion and the monogamy relation by means of several representative examples. We believe that the results and methods presented in this work could be beneficial to capture genuine multipartite steering in the near future.
We consider quantum steering certification in multipartite networks, with a focus on minimal trust scenarios: all-except-one parties are untrusted and treated device-independently. We show that it is always possible to lift steering certification to the measurement-device-independent regime, in which even the (last) trusted party can treat their local hardware as a black-box, except for a set of fiduciary quantum states used as the inputs to the experiment. This holds both for finite-dimensional systems as well as for bosonic continuous-variable systems, which we characterize in terms of semi-quantum games in the bipartite case. Additionally, we introduce measurement-device-independent network steering protocols based entirely on Gaussian operations -- which cannot be used for fully device-independent protocols, and thus become instead a viable option for minimal trust certification as soon as a single trusted input is inserted in the network. Our results present a basis for steering-based applications (such as randomness generation) with minimal trust beyond full nonlocality and with feasible experimental requirements.