Multipartite Einstein-Podolsky-Rosen steering and genuine tripartite entanglement with optical networks
The Einstein-Podolsky-Rosen (EPR) paradox epr () established
a link between entanglement steering-1 (); ent () and nonlocality in
quantum mechanics nonlocality (). EPR steering eprsteereric (); Wiseman (); wise2 (); steering ()
is the nonlocality associated with the EPR paradox and has traditionally
only been investigated between two parties ou epr (); hann group (); smithsteerxp (); bwzeil (); hw-1 (); rrmp-1 (); boyd ().
Here, we present the first experimental observations of multipartite
EPR steering, and of the genuine tripartite continuous variable entanglement of three mesoscopic optical systems threeent (); aokicv (); eiscv (). We explore different linear optics networks - each
one with optimised asymmetries - that create multipartite steerable
states containing different numbers of quantised optical modes (qumodes). By introducing asymmetric loss on a 7-qumode state, we characterize 8 regimes of directional steering, showing that regimes exist for an -qumode state. Further, we reveal the directional monogamy of steering, and experimentally demonstrate continuous variable one-sided semi device-independent quantum secret sharing secretsh (). Our methods establish principles for the development of multiparty quantum communication protocols with asymmetric observers, and can be extended to qubits, whether photonic svetexp (); bwzeil (); hw-1 (); boyd (); threeent (); smithsteerxp (), atomic 14blatt (), superconducting supercondqubits (), or otherwise.
Schrödinger introduced the term “steering” to describe the nonlocality apparent in the EPR paradox, and pointed out that these states involve a quantum property called “entanglement” steering (); steering-1 (). Wiseman et al Wiseman (); wise2 () have formalised the meaning of steering in terms of violations of local hidden state models, and revealed that the EPR paradox is a manifestation of quantum steering. In simple terms, quantum steering dictates that measurements made by one observer can apparently “steer” (alter) the state of another observer at a different location.
The observation of multipartite EPR steering has not been possible until recently as the framework necessary to understand the concept has only just been developed eprsteereric (); Wiseman (); wise2 (); genuineEPR (). The motivation to expand this framework arises from considerations of real-world quantum networks, such as the quantum internet KimbleNet (), for which security and privacy are of paramount importance. Here, we expand on the theoretical framework and derive optimised criteria to detect multipartite EPR steering using linear optical circuits. The criteria involve the canonical position and momentum observables, which are realised in our experiment as highly efficient quadrature phase amplitude measurements. Following the criteria, we present the first experimental investigation of multipartite EPR steering, including demonstration of directional monogamy relations which give bounds on the way steering is distributed among the different parties. Further, we demonstrate the principle of one-sided device-independent quantum secret sharing and in doing so confirm for the first time the continuous variable genuine tripartite entanglement of three optical modes. For bipartite EPR states, there are 3 different regimes: 2-way, 1-way, and no-way steering onewaysteer (); steermurray (). In general, for each qumode state, regimes of steering are possible. Here, we create 7 different quantum networks, each producing a multipartite EPR steerable state, with different levels of correlations. By introducing asymmetry into the network, we manipulate the 7-qumode state, to experimentally achieve all 8 different regimes of directional steering.
Underpinning the idea of multipartite EPR steering is the quantum concept of entanglement ent (); steering-1 (). systems are genuinely -partite entangled if and only if the entanglement of the -party system cannot be produced by mixing quantum states with fewer than systems entangled threeent (); eiscv (). Suppose three observers (Alice, Bob and Charlie) each make measurements on three respective quantum systems, labelled and . We show in the Supplementary Materials that genuine tripartite entanglement of the three systems is confirmed if:
Here , () are the position and momentum observables of the system , scaled in such a way that the Heisenberg uncertainty relation becomes . Genuine tripartite entanglement has a different meaning to full tripartite inseparability threeent (); eiscv (). The latter occurs when the entanglement cannot be produced by entangling any (fixed) two parties, and does not eliminate that entanglement is created by mixing different bipartite entangled states.
To understand tripartite EPR steering, we again consider that Alice’s measurements are the observables and of a quantum system. However, this assumption is no longer applied to Bob and Charlie, who need not report the results of quantum observables. We suppose that Bob and Charlie can collaborate to give a prediction for the outcome of Alice’s (or ) measurement and denote the average uncertainty in their inferences by (and ). If , then we realise an EPR steering paradox mreidepr (); Wiseman (). In that case, it is as though Alice’s and values were predetermined to an accuracy that contradicts quantum mechanics epr (), or else that there is an actual “steering” of Alice’s system by Bob and Charlie’s actions steering (). We symbolise this directional EPR steering by the notation . Confirming steering of A by the group BC amounts to confirming entanglement between the two groups, but with fewer assumptions made about group BC. Steering is therefore a greater experimental challenge than entanglement.
Next, we extend to observers and consider genuine tripartite steering. Any party state demonstrates -partite EPR steering of the th site by the remaining set of sites denoted if
We define the the square of this product, , to be the EPR steering number. In our experiment, the steering measurements are optimised linear combinations of the () (). Thus
where and are optimised real numbers. The steering detected by equation (2) is genuinely -partite if it cannot be explained as arising from any steering limited to or fewer parties. In the tripartite case, let be the sites of Alice, Bob and Charlie. All fixed two-party steering is negated, if we can demonstrate each of genuineEPR ()
which implies steering across all bipartitions: , , and . The condition (3) confirms the full inseparability of the quantum density matrix and also of any three-party hidden state model that could describe the system Wiseman (). The negation does not however rule out that the steering has been created by mixing states with two-party steering across different bipartitions. To claim genuine tripartite steering, we need to eliminate this possibility genuineEPR (). We prove in the Suppplementary Materials that this is done if
which is a stricter form of inequality Eq. (1).
We investigate multipartite entanglement and steering by employing the programmable linear optics circuit developed in ref Armstrong2012 () in order to create various multi-partite quantum states from different networks. Independent qumodes are shaped in order to be multiplexed on the same beam. By programmatically changing the measurement basis, the scheme allows us to emulate linear optics networks in real time. The various quantum networks that we create for this demonstration can be visualised in the programmable circuit of Fig. 1. We input 2 quadrature-squeezed qumodes and 6 quantum-noise limited vacuum modes into the linear optics circuit, and we have the freedom to programmatically vary each beam-splitter’s reflectivity. We set a beam-splitter reflectivity to and set it to a perfect mirror when we choose not to mix a particular input qumode into the state.
Homodyne detection is employed to measure the inferred variances of each qumode in the state. This provides highly efficient detection, and our measurements of the amplitudes do not rely on fair sampling assumptions smithsteerxp (); bwzeil (). Each qumode is characterised by the conjugate quadrature operators and of the quantum harmonic oscillator mode of the light field. Optimal circuits are created that take into account asymmetries in squeezing values of the input qumodes.
In our experiment we input two mixed states with different magnitudes of squeezing, allowing us to explore how best to bias and optimise the beam-splitter networks,
The optimised linear optics networks that we create for our mixed inputs are detailed in the methods section. We first generate 7 different quantum states from 7 different networks, with each beam-splitter’s reflectivity given in Table 1.
The EPR correlations in each state can be visualised by parametrically plotting components of the inferred variance terms. In all plots of Fig. 2, the axis is the () quadrature amplitude of the first qumode in the state, and the axis is the linear combination of the () quadrature amplitudes of the remaining qumodes in the -qumode state.
Perfect correlations would correspond to a semi-major axis of infinite length, strictly on the diagonal. This would require infinite energy and is unphysical; the ellipticity of each ellipse in Fig. 2 is indicative of finite squeezing. We see that the higher the number of qumodes in the state, the weaker the correlations become. This is evident in the slight rotation off the strictly diagonal axis in both quadratures, as well as the diminishing ellipticity of the correlations. This is due to the additional vacuum contributions in our circuit as we go to higher mode numbers.
Figure 2b quantifies the correlations in terms of the EPR steering number, Eq. (2). For each -qumode state, there are EPR steering numbers that must be tested in order to confirm the multipartite EPR steering of each qumode. Each EPR steering number represents the steering from one partition of the state to the other, or the direction of steering within the state. For , we confirm steering across all bipartitions, to satisfy criterion (3), thus ruling out a large class of separable classical models.
We next demonstrate directional properties of multipartite EPR steering, which have implications for security in quantum communications. Firstly, the manifestation of steering by inequality (2) is monogamous (Fig. 3a). This means that if such steering is shared between two parties, Alice and Bob, it cannot be shared between Alice and a third party, Charlie. Mathematically, the monogamy is described by genuineEPR (). The EPR steering number therefore measures the directional security of the channel, since it gives a lower bound on the inferred uncertainty of Alice’s amplitudes for any eavesdropping parties “Eve” denoted by . As no assumptions are made about the measurements of the steering parties, the security is independent of the exact nature of Bob or Eve’s measurement devices (though Alice’s measurement station must be trusted) onesidedcry (). Another property of multipartite EPR steering is that for some regimes, the steering of cannot take place by Bob (or Charlie) alone, but requires both parties. In Fig. 3a we illustrate one-sided device-independent quantum secret sharing secretsh () the values of Alice’s amplitudes can only be unlocked with a low uncertainty if the steering parties collaborate.
Continuing for higher , we modified the 7-qumode network to allow the systematic introduction of asymmetric noise into the network. This is done by adding variable loss to one half of the network via a vacuum coupling beam-splitter acting on one output arm of VBS. The monogamy relation explains the impossibility of steering when the losses on the steering channel are or greater bowen (). Steering is also sensitive to the noise on the steered system. Thus, one can manipulate the asymmetries to successively disable the steering of each one of the parties (Fig. 3b). In this way, we find that all 8 separate steering regimes are accessible by introducing up to of asymmetric loss into the circuit. The imperfect mode matching and detection (over efficiencies) of each qumode together with the systematic loss allows for the separate regimes.
Finally, the beam-splitters were varied to switch from multipartite steering to genuine tripartite entanglement. In Fig. 4, we confirm the genuine tripartite entanglement of three qumodes. The difference between full tripartite inseparability aokicv (); Armstrong2012 () and genuine tripartite entanglement has been explained threeent (). While the latter has been recently realised for three spatially separated photons threeent (), our results demonstrate genuine tripartite entanglement in a very different scenario: namely, for the EPR observables of three fields consisting of many photons detected at very high efficiencies.
We have presented the first experimental evidence of multipartite EPR steering in various quantum states containing different levels of distributed squeezing. For the three-qumode state, we have established both the full inseparability of the hidden state model and genuine tripartite entanglement. Our work reveals properties of multipartite steering that link the amount of steering to the security of channels in the network, and further shows how the steering along different channels can be controlled. The framework developed opens up the possibility of demonstrations of multipartite EPR steering in various quantum network applications.
The familiar bipartite EPR state is generated by setting the first beam-splitter reflectivity to and all other beam-splitters to mirrors. The measurement returns an EPR state and 6 unmixed vacuum modes, which are discarded. In order to create an -qumode state we set of these to function as beam-splitters and the remaining to function as mirrors. We emphasise that although our qumodes are in principle spatially distinguishable, measurement events are not spacelike separated, and our work cannot address locality loopholes bwzeil ().
In all of our networks, when the two squeezed qumodes are pure states and equal in squeezing magnitude, the optimal beam-splitter reflectivity of the first beam-splitter is . While unsurprising for quantum states containing even qumode numbers, it is less intuitive for odd numbered states, as one might expect that an unbalanced beam-splitter will favour the unbalanced network. The asymmetry is balanced by the quadrature amplitude optimisation gains.
For two mixed states that are unequal in squeezing values, the optimal beam-splitter ratio for even numbered states remains . The symmetry arises from the EPR steering criteria being directional. The symmetry breaks down for mixed-state inputs when we consider odd-numbered quantum states. In this situation we benefit from biasing the beam-splitters.
It is only necessary to optimise the first beam-splitter of the network in odd numbered quantum states. We see that the reflectivities are , , and for the qumode, qumode, and qumode states, respectively. For any asymmetry in the inputs this will converge to for large .
We calculate the moments corresponding to the criteria above, using
the unitary transformation ,
the interaction of the qumodes at the beam splitter aokicv ().
Here, , are the two output qumodes and ,
are the two qumodes input to the beam splitter. The reflectivity
of the beam splitter is given by . The optimisation gain parameters are tabulated in the Supplementary materials.
S.A., P.K.L, Q.Y.H. and M.D.R. conceived of and designed the experiment. S.A. and J.J. constructed and performed the experiment with supervision from H.-A.B. and P.K.L. S.A., M.W., R.Y.T., Q.H.G., Q.Y.H., and M.D.R. contributed equally to the theory. S.A. designed and created the virtual networks, and conducted the data analysis. S.A., Q.Y.H., and M.D.R. wrote the manuscript, and all authors commented on drafts.
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This research was conducted by the Australian Research
Council Centre of Excellence for Quantum Computation and Communication
Technology (project number CE110001029) and has been supported by
the Australian Research Council DECRA and Discovery Project Grants
schemes. SA is grateful for funding from the Australia-Asia Prime
Minister‘s Award. RYT thanks Swinburne University for a Research SUPRA
Award, and QHG thank National Natural Science Foundation of China
under Grant No. 11121091 and 11274025.