Multipartite Gaussian steering: monogamy constraints and quantum cryptography applications
We derive laws for the distribution of quantum steering among different parties in multipartite Gaussian states under Gaussian measurements. We prove that a monogamy relation akin to the generalized Coffman-Kundu-Wootters inequality holds quantitatively for a recently introduced measure of Gaussian steering. We then define the residual Gaussian steering, stemming from the monogamy inequality, as an indicator of collective steering-type correlations. For pure three-mode Gaussian states, the residual acts a quantifier of genuine multipartite steering, and is interpreted operationally in terms of the guaranteed key rate in the task of secure quantum secret sharing. Optimal resource states for the latter protocol are identified, and their possible experimental implementation discussed. Our results pin down the role of multipartite steering for quantum communication.
pacs:03.67.Dd, 03.65.Ud 42.50.Dv, 42.50.Ex
With the imminent debacle of Moore’s law, and the constant need for faster and more reliable processing of information, quantum technologies are set to radically change the landscape of modern communication and computation. A successful and secure quantum network relies on quantum correlations distributed and shared over many sites Kimble (2008). Different kinds of multipartite quantum correlations have been considered as valuable resources for various applications in quantum communication tasks. Multipartite entanglement van Loock and Furusawa (2003); van Loock and Braunstein (2000); Horodecki et al. (2009); Guhne and Tóth (2009); Adesso and Illuminati (2006); Adesso et al. (2006); Bancal et al. (2011) and multipartite Bell nonlocality Svetlichny (1987); Collins et al. (2002); Seevinck and Svetlichny (2002); Adesso and Piano (2014) are two well known instances and have received extensive attention in recent developments of quantum information theory, as well as in other branches of modern physics. There has been substantial experimental progress in engineering and detection of both such correlations, by using e.g. photons Shalm et al. (2013); Huang et al. (2011); Yao et al. (2012); Lavoie et al. (2009); Erven et al. (2014), ions Monz et al. (2011), or continuous variable (CV) systems Armstrong et al. (2012); Jing et al. (2003); Su et al. (2007); Yokoyama et al. (2013). However, as an intermediate type of quantum correlation between entanglement and Bell nonlocality, multipartite quantum steering Li et al. (2015); Armstrong et al. (2015) still defies a complete understanding. In consideration of the intrinsic relevance of the notion of steering to the foundational core of quantum mechanics, it has become a worthwhile objective to deeply explore the characteristics of multipartite steering distributed over many parties, and to establish what usefulness to multiuser quantum communication protocols can such a resource provide, where bare entanglement is not enough and Bell nonlocality may not be accessible.
The concept of quantum steering was originally introduced by Schrödinger Schrödinger (1935) to describe the “spooky action-at-a-distance” effect noted in the Einstein-Podolsky-Rosen (EPR) paradox Einstein et al. (1935); Reid (1989); Reid et al. (2009), whereby local measurements performed on one party apparently adjust (steer) the state of another distant party. Recently identified as a distinct type of nonlocality Wiseman et al. (2007); Jones et al. (2007), quantum steering is thus a directional form of quantum correlations, characterized by its inherent asymmetry between the parties Walborn et al. (2011); Händchen et al. (2012); Bowles et al. (2014); Quintino et al. (2015); He et al. (2015a); Kogias et al. (2015a, b). Additionally, steering allows verification of entanglement, without assumptions of the full trust of reliability of equipment at all of the nodes of a communication network Cavalcanti et al. (2009). Steering is then a natural resource for one-sided device-independent quantum key distribution Branciard et al. (2012); Walk et al. (2016). For bipartite systems, a comprehensive quantitative investigation of quantum steering has been recently proposed Skrzypczyk et al. (2014); Kogias and Adesso (2015); Costa and Angelo (2016); He et al. (2015b) and tested in several systems Saunders et al. (2010); Bennet et al. (2012); Wittmann et al. (2012); Sun et al. (2016, 2014); Kocsis et al. (2015); Wollmann et al. (2016). Comparatively little is known about steering in multipartite scenarios. For instance, Refs. He and Reid (2013); Teh and Reid (2014); Cavalcanti et al. (2015) derived criteria to detect genuine multipartite steering, and Ref. Reid (2013) presented some limitations on joint quantum steering in tripartite systems.
In this Rapid Communication we focus on steerability of multipartite Gaussian states of CV systems by Gaussian measurements, a physical scenario which closely aligns with the traditional EPR paradox, and which is of primary relevance for experimental implementations Adesso and Illuminati (2007a); Weedbrook et al. (2012); Cerf et al. (2007). In order to investigate the shareability of Gaussian steering from a quantitative perspective Kogias et al. (2015a), we establish monogamy relations imposing constraints on the degree of bipartite EPR steering that can be shared among -mode CV systems in pure Gaussian states, in analogy with the Coffman-Kundu-Wootters (CKW) monogamy inequality for entanglement Coffman et al. (2000); Adesso and Illuminati (2006); Adesso et al. (2006); Hiroshima et al. (2007); Osborne and Verstraete (2006); Adesso et al. (2012); Lancien et al. (2016). We further propose an indicator of collective steering-type correlations, the residual Gaussian steering (RGS), stemming from the laws of steering monogamy, that is shown to act as a quantifier of genuine multipartite steering for pure three-mode Gaussian states. Finally, we show how the RGS acquires an operational interpretation in the context of a partially device-independent quantum secret sharing (QSS) protocol Hillery et al. (1999); Lau and Weedbrook (2013); Kogias et al. (2016); not (a); Cleve et al. (1999). Specifically, taking into account arbitrary eavesdropping and potential cheating strategies of some of the parties Kogias et al. (2016), the achievable key rate of the protocol is shown to admit tight lower and upper bounds which are simple linear functions of the RGS. This in turn allows us to characterize optimal resources for CV QSS in terms of their multipartite steering.
Monogamy of Gaussian steering.
A fundamental property of entanglement, that has profound applications in quantum communication, is known as monogamy Coffman et al. (2000); Terhal (2004); Lancien et al. (2016). Any two quantum systems that are maximally entangled with each other, cannot be entangled (or, even, classically correlated) with any other third system. Therefore, entanglement cannot be freely shared among different parties. In their seminal paper Coffman et al. (2000), CKW derived a monogamy inequality that quantitatively describes this phenomenon for any finite entanglement shared among arbitrary three-qubit states : , where is the squared concurrence, quantifying the amount of bipartite entanglement across the bipartition . Osborne and Verstraete later generalized the CKW monogamy inequality to qubits Osborne and Verstraete (2006). For CV systems, however, both the quantification and the study of the distribution of entanglement constitute in general a considerably harder problem. Remarkably, if one focuses on the theoretically and practically relevant class of Gaussian states, various results similar to the qubit case have been derived, using different entanglement measures Adesso and Illuminati (2006); Adesso et al. (2006); Hiroshima et al. (2007); Adesso et al. (2012); Adesso and Illuminati (2007b, a). Of particular interest to us will be the fact that the Gaussian Rényi-2 entanglement monotone , which quantifies entanglement of bipartite Gaussian states , has been shown to obey a CKW-type monogamy inequality for all -mode Gaussian states with covariance matrix (CM) Adesso et al. (2012),
where each comprises one mode only. Recall that the CM of a -mode state has elements , where is the vector collecting position and momentum operators of each mode, satisfying canonical commutation relations , with and being the single-mode symplectic form Adesso and Illuminati (2007a).
Quantum steering is a type of correlation that allows for entanglement certification in a multi-mode bipartite state even when one of the parties’ devices, say Bob’s, are completely uncharacterized (untrusted). In this case, we say that Bob can steer Alice’s local state Wiseman et al. (2007); Jones et al. (2007). Keeping our focus on Gaussian states and measurements Kogias et al. (2015a), the question, thus, naturally arises: is steering monogamous? Intuitively one would expect that there should exist limitations on the distribution of steering-type correlations, since steering is only a stronger form of the already monogamous entanglement. A first answer to this question was recently given by Reid Reid (2013), who showed that, under restrictions to measurements and detection criteria involving up to second order moments, if a single-mode party can be steered by a single-mode party then no other single-mode party can simultaneously steer . This was recently generalized to the case of parties and comprising an arbitrary number of modes Ji et al. (2015a); Adesso and Simon (2016). Ref. Reid (2013) also discussed other monogamy relations for steering and nonlocality both in discrete and CV systems.
In the following we provide general quantitative CKW-type limitations to the distribution of Gaussian steering among many parties. For our purposes, we will focus on a recently proposed Gaussian steering measure Kogias et al. (2015a), , which quantifies how much party can steer party in a Gaussian state with CM by Gaussian measurements. In particular, we now show that the Gaussian steering measure is monogamous, hence satisfies a CKW-type monogamy inequality in direct analogy with entanglement. Consider an arbitrary (pure or mixed) -mode Gaussian state with CM , where each party comprises a single mode . Then, the following inequalities hold, :
For pure states with CM , the proof is straightforward. Namely, recall from Kogias et al. (2015a) that the leftmost terms of (2), (3) and (1) all coincide on pure states. On the other hand, for the marginal states of any two modes and one has Kogias et al. (2015a). Inequalities (2) and (3) then follow readily from the monogamy inequality (1) for Gaussian entanglement. The full proof of the above inequalities for general mixed states is deferred to the Appendix.
The monogamy relations just derived in this work impose fundamental restrictions to the distribution of Gaussian steering among multiple parties in fully quantitative terms. To analyze these in more detail, let us focus on a tripartite scenario, in which the monogamy inequalities take the simpler form,
As in the original CKW inequality, these inequalities enjoy a very appealing interpretation: the degree of steering (by Gaussian measurements) exhibited by the state when all three parties are considered (i.e., , or, can be larger that the sum of the degrees of steering exhibited by the individual pairs. On a more extreme level, there exist quantum states where parties and cannot individually steer party , i.e., , but collectively they can, i.e., . We will see the importance of this type of correlations later when we discuss applications to QSS. We remark that the monogamy inequality (4) realizes a crucial nontrivial strengthening of Result 5 in Reid (2013), which can be recast as in our notation. On the other hand, the reverse monogamy relation (3) settles an open question raised in the same work Reid (2013).
The residuals of the subtractions in (4), (5) quantify steering-type correlations that correspond to a collective property of the three parties, not reducible to the properties of the individual pairs. We proceed by investigating this quantitatively in a mode-invariant way. In analogy with what done for entanglement Adesso and Illuminati (2006); Adesso et al. (2006, 2012), we can calculate the residuals from the monogamy inequalities (4) or (5) and minimise them over all mode permutations. It turns out that, in the paradigmatic case of pure three-mode Gaussian states with CM (), we obtain the same quantity (RGS) from either (4) or (5), regardless of the steering direction. Explicitly, denoting by any cyclic permutation of , the RGS for three-mode pure Gaussian states with CM is defined as
The RGS is a monotone under Gaussian local operations and classical communication, as one can prove analogously to the case of the residual entanglement of Gaussian states Adesso and Illuminati (2006); Adesso et al. (2006, 2012); Kogias et al. (2015a); Kogias and Adesso (2015). Furthermore, finding a non-zero value of the RGS certifies genuine tripartite steering, as defined by He and Reid He and Reid (2013), since a sufficient requirement to violate the corresponding biseparable model for pure states is the demonstration of steering in all directions: and . We can then regard the RGS as a meaningful quantitative indicator of genuine tripartite steering for pure three-mode Gaussian states under Gaussian measurements.
In Fig. 1(a) we plot the RGS as a function of and for a given . An elementary analysis reveals that the RGS is maximized on bisymmetric states with , i.e., when the states are steerable across any global split of the three modes and also steerable, but no other steering exists between any two parties. In this case, the genuine tripartite steering reduces to the collective steering . This quantitative analysis completes the existing picture of quantum correlations in pure three-mode Gaussian states, together with the cases of tripartite Bell nonlocality in terms of maximum violation of the Svetlichny inequality Adesso and Piano (2014) and genuine tripartite entanglement in terms of Gaussian Rényi-2 entanglement Adesso and Piano (2014). Bisymmetric states maximize all three forms of nonclassical correlations; compare e.g. our Fig. 1(a) with Fig. 1(a)–(b) in Adesso and Piano (2014).
Figure 1(b) presents the RGS measure for Gaussian states generated by three squeezed vacuum fields (one in momentum, two in position) with experimentally feasible squeezing parameter (i.e., dB of squeezing) Jing et al. (2003); Zhou et al. (2015); Eberle et al. (2013) injected at two beamsplitters with reflectivities and as depicted in the inset of Fig. 1(b), setting so that , . When , one can generate a permutationally invariant Greenberger-Horne-Zeilinger (GHZ)-like state with van Loock and Braunstein (2000). As one might expect, the latter states maximize the RGS in this case.
Operational connections to quantum secret sharing.
Secret sharing Shamir (1979); Blakley (1899) is a conventional cryptographic protocol in which a dealer (Alice) wants to share a secret with two players, Bob and Charlie, but with one condition: Bob and Charlie should be unable to individually access the secret (which may involve highly confidential information) and their collaboration would be required in order to prevent wrongdoings.
QSS schemes Hillery et al. (1999); Karlsson et al. (1999); not (a) have been proposed to securely accomplish this task, by exploiting multipartite entanglement to secure and split the classical secret among the players in a single go. Very recently, we provided an unconditional security proof for entanglement-based QSS protocols in a companion paper Kogias et al. (2016). In our scheme, the goal of the dealer is to establish a secret key with a joint degree of freedom of the players. The players can only retrieve Alice’s key and decode the classical secret by collaborating and communicating to each other their local measurements to form the joint variable. The unconditional security of these schemes stems from the utilized partially device-independent setting, treating the dealer as a trusted party with characterized devices, and the (potentially, dishonest) players as untrusted parties whose measuring devices are described as black boxes. Given this intrinsically asymmetric separation of roles, one would expect that multipartite steering be closely related to the security figure of merit of QSS. Here we prove such a connection quantitatively.
To start with, let us assume that the dealer, Alice, and the players, Bob and Charlie, all perform homodyne measurements of the quadratures with outcomes , with , on the shared tripartite state. Following Kogias et al. (2016), a guaranteed (asymptotic) secret key rate for the QSS protocol (extracted from the correlations of Alice’s momentum detection and a joint variable for Bob and Charlie) to provide security against external eavesdropping is given by , while the key rate providing unconditional security against both eavesdropping and dishonest actions of the players is
Here, is the minimum inference variance of Alice’s momentum outcome given the players’ joint outcome , and similarly for the other variances. A tripartite shared state whose correlations result in nonzero values of the right-hand side of (7) can be regarded a useful resource for unconditionally secure QSS.
We focus on pure three-mode Gaussian states with CM in standard form, fully specified by the local invariants as before. Our first observation is that is directly quantified by the collective steering, . For the considered class of states, one has indeed , where the joint variables were chosen to have the linear form and , with the real constants optimized as to minimize the inferred variances ; see also Walk et al. (2016); Kogias et al. (2015a). Putting everything together, we get:
We can now define a mode-invariant QSS key rate bound that takes into account eavesdropping and potential dishonesty of the players, by minimizing the right-hand side of Eq. (7) over the choice of the dealer, i.e., over permutations of , , and . A nonzero value of the figure of merit on a tripartite Gaussian state with CM guarantees the usefulness of the state for unconditionally secure QSS, for any possible assignment of the roles. For pure three-mode Gaussian states, the mode-invariant key rate can be evaluated explicitly (although its lengthy expression is omitted here) and analyzed in the physical space of the parameters . We find that admits exact linear upper and lower bounds as a function of the RGS , for all states with standard form CM :
The bounds are illustrated in Fig. 2 together with a numerical exploration of randomly generated pure three-mode Gaussian states. Remarkably, the bounds are tight, and families of states saturating them can be readily provided. Specifically, the lower (dotted blue) boundary is spanned by states with , ; conversely, the upper (solid black) boundary is spanned by states with , . While these cases are clearly extremal, GHZ-like states (dashed red), specified by and producible as discussed in Fig. 1(b), nearly maximize the QSS key rate at fixed RGS, thus arising as convenient practical resources for the considered task, independently of the distribution of trust. Indeed, a squeezing level of dB, referring to the scheme of Fig. 1(b), is required to ensure a nonzero key rate using these states. This is well within the current experimental feasibility, since up to dB of squeezing has been demonstrated Zhou et al. (2015); Eberle et al. (2013). In general, by imposing nonnegativity of the lower bound in (8), we find that for all pure three-mode Gaussian states with RGS . Our analysis reveals that partially device-independent QSS is empowered by multipartite steering, yielding a direct operational interpretation for the RGS in terms of the guaranteed key rate of the protocol.
Discussion and conclusion.
We have proven that a recently proposed measure of quantum steering under Gaussian measurements Kogias et al. (2015a); Kogias and Adesso (2015) obeys CKW-type monogamy inequalities for all Gaussian states of any number of modes. We remark that monogamy extends in fact to arbitrary non-Gaussian states under Gaussian measurements, as it is established solely at the level of covariance matrices. Notice however that resorting to non-Gaussian measurements can lead to extra steerability even for Gaussian states Wollmann et al. (2016); Ji et al. (2015b), and might allow circumventing some monogamy constraints Reid (2013); Ji et al. (2015a); Adesso and Simon (2016).
In the important case of pure three-mode Gaussian states, we demonstrate that the residual steering emerging from the laws of monogamy can act as a quantifier of genuine tripartite steering. The latter measure is endowed with an operational interpretation, as it is shown to provide tight bounds on the mode-invariant key rate of a partially device-independent QSS protocol, whose unconditional security has been very recently investigated Kogias et al. (2016). Our study, combined with Kogias et al. (2016), provides practical recipes demonstrating that an implementation of QSS secure against eavesdropping and potentially dishonest players is feasible with current technology using tripartite Gaussian states and Gaussian measurements not (b).
Acknowledgements.Note added. After completion of this work, monogamy inequalities for multipartite Gaussian steering in the case of more than one mode per party have been investigated in Lami et al. (2016). Acknowledgments. YX and QH acknowledge the support of the National Natural Science Foundation of China under Grants No. 11274025, No. 61475006, and No. 61675007. IK and GA acknowledge funding from the European Research Council under Starting Grant No. 637352 (GQCOP). GA thanks R. Simon and L. Lami for useful discussions.
Appendix A: Proof of (2).
It suffices to prove the inequality for tripartite states as in (4), with being a single mode and , comprising arbitrary number of modes. One can then apply iteratively this inequality to obtain the corresponding -partite one (2).
To do so, recall that from Reid (2013); Ji et al. (2015a); Adesso and Simon (2016) it is impossible for and to simultaneously steer the one-mode party , that is, implies (and vice versa). Therefore, the monogamy relation (4) reduces to (or the analogous expression with swapped ), which holds true because the Gaussian steering measure (for one-mode steered party ) is nonincreasing under local Gaussian operations on the steering party Kogias and Adesso (2015), which include discarding (or ). This proves Eq. (2) for any -mode CM .
Appendix B: Proof of (3).
In this case we have to recall the explicit expression of the Gaussian steering measure Kogias and Adesso (2015), defined for a bipartite ()-mode state with CM as
where denote the symplectic eigenvalues of the Schur complement of in . By definition of the Schur complement, and observing that for any valid CM , notice that we can write:
Applying to both sides we get, for any CM with , the bound (tight when Kogias and Adesso (2015))
where is the log-determinant of the CM Adesso and Simon (2016), and is the conditional log-determinant, which — in analogy to the standard conditional quantum entropy — is concave on the set of CMs Adesso and Simon (2016) and subadditive with respect to the conditioned subsystems,
To prove (3), it suffices to consider the case in which the multimode term and all the pairwise terms are nonzero. Applying then (9) to the leftmost term in (3), and using repeatedly the negation of (10), we get: , where in the last step we used again (9) which holds with equality on each of the two-mode terms involving and any , provided as per assumption. This concludes the proof of Eq. (3) for any -mode CM .
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