Abstract
The concept of ‘steering’ was introduced in 1935 by Schrödinger^{1} as a generalization of the EPR (Einstein–Podolsky–Rosen) paradox. It has recently been formalized as a quantuminformation task with arbitrary bipartite states and measurements^{2}, for which the existence of entanglement is necessary but not sufficient. Previous experiments in this area^{3,4,5,6} have been restricted to an approach^{7} that followed the original EPR argument in considering only two different measurement settings per side. Here we demonstrate experimentally that EPRsteering occurs for mixed entangled states that are Bell local (that is, that cannot possibly demonstrate Bell nonlocality). Unlike the case of Bell inequalities^{8,9,10,11}, increasing the number of measurement settings beyond two—we use up to six—significantly increases the robustness of the EPRsteering phenomenon to noise.
Main
It was Einstein, Podolsky and Rosen who first highlighted the fact that ‘as a consequence of two different measurements performed upon the first system, the second system may be left in states with two different [kinds of] wavefunctions’^{12}. For them, this spooky action at a distance was unacceptable, and proved that the Copenhagen interpretation of quantum mechanics was incomplete. In the example they used to illustrate this ‘paradox’, the two different kinds of wavefunction were position and momentum eigenstates, which are clearly incompatible because ‘precise knowledge of [Q] precludes such a knowledge of [P]’ (ref. 12). In this paradigm, Reid^{7} first developed quantitative criteria for the experimental demonstration of the EPR paradox on the basis of Heisenberg’s uncertainty relation (ΔP)(ΔQ)≥ℏ/2; see also ref. 13.
In the same year as the EPR paper, Schrödinger introduced the term ‘steering’^{1} to describe the EPR paradox, and generalized it to more than two measurements, saying ‘Since I can predict either [Q] or [P] without interfering with [the second] system,…[it] must know both answers; which is an amazing knowledge. [The second system] does not only know these two answers but a vast number of others’. It is only very recently that general EPRsteering inequalities, allowing for measurements of an arbitrary number of different observables by the two parties, have been developed^{14}. This followed the first formal definition of steering in refs 2, 15, which proved that demonstrating EPRsteering is strictly easier than demonstrating Bell nonlocality (that is, violating a Bell inequality) but strictly harder than demonstrating nonseparability (by quantum state tomography or entanglement witnesses^{16}). The existence of this hierarchy is a logical consequence of the definitions, and its strictness was proved by consideration of twoqubit Werner states and restriction to projective measurements. Although nonprojective measurements could, in theory, be more powerful^{17}, this is a remote possibility for such a simple state, and we restrict ourselves to projective measurements here also. Finally, we emphasize that none of these forms of quantum nonlocality allows for the possibility of fasterthanlight signalling.
Here we exploit the modern formulation of EPRsteering for the first time, demonstrating EPRsteering with discrete binaryoutcome measurements on Werner states^{18} of a pair of photonpolarization qubits. This family of states (singlets with isotropic noise) is well studied, and it is proven that some of the states we use to demonstrate EPRsteering violate no Bell inequality. This is not the case for any of the states used in previous demonstrations^{3,4,5,6} of the EPR paradox, which relied on the EPR–Reid inequalities, and used measurements with continuous outcomes.
EPRsteering is a form of quantum nonlocality that is logically intermediate^{2,15} between Bell nonlocality^{19} and nonseparability. The second party, Bob, trusts quantum mechanics to describe his own measurements, but makes no assumptions about the distant first party, Alice, who has to convince him that she can affect the nature of his quantum state by her choice of measurement setting. So termed in analogy to Bell inequalities, EPRsteering inequalities^{14} are a superset of the former, and a subset of entanglement witnesses. Steering inequalities are, in principle, easier to violate experimentally than Bell inequalities because of the asymmetry between the parties; see Fig. 1. Instead of considering correlation functions for classical variables (measurement outcomes) on the two sides, in EPRsteering we consider correlations between classical variables declared by Alice but quantum expectation values measured by Bob.
Here we consider linear EPRsteering inequalities^{14} involving statistics collected from an experiment with n measurement settings for each side. For qubits, we can take Bob’s kth measurement setting to correspond to the Pauli observable , along some axis u_{k}. Denoting Alice’s corresponding declared result (we make no assumption that it is derived from a quantum measurement) by the random variable A_{k}∈{−1,1} for all k, the EPRsteering inequality is of the form
We call the quantity S_{n} the steering parameter for n measurement settings. The bound C_{n} is the maximum value S_{n} can have if Bob has a preexisting state known to Alice, rather than half of an entangled pair shared with Alice. It is easy to see that this bound is
where denotes the largest eigenvalue of .
To derive useful inequalities we consider measurement settings based around the four Platonic solids that have vertices that come in antipodal pairs (Fig. 2). Each pair defines a measurement axis u_{k}, giving us an arrangement for n=3, 4, 6 and 10 settings. For n=2 settings, we use a square arrangement. For each measurement scheme (except for n=10, which we did not implement experimentally) we do the following: (i) Derive the bound C_{n} in the inequality (1). (ii) Experimentally demonstrate EPRsteering by violating the inequality using Werner states. (iii) Theoretically show that Alice can saturate the inequality by sending Bob pure states drawn by her from a particular ensemble. (iv) Experimentally demonstrate (iii) above by nearly saturating the EPRsteering inequality in that way.
Werner states^{18} are the bestknown class of mixed entangled states. For qubits, they can be written as
where is the singlet state and I is the identity, and where μ∈[0,1]. Werner states are entangled if and only if (iff) μ>1/3 (ref. 18). They can violate the Clauser, Horne, Shimony and Holt^{20} (Bell–CHSH) inequality only if , and cannot violate any Bell inequality if μ<0.6595 (ref. 9). Reference 2 showed that these states are also steerable, with settings, iff μ>1/2. With n=2 projective measurements they are steerable iff , no better than the Bell–CHSH inequality. Deriving analytical expressions for the bounds C_{n} is a simple exercise in geometry (see Methods). For the square, octahedron and cube we find and . For higher n the exact expressions are lengthy; the approximate numerical values are C_{6}≈0.5393 and C_{10}≈0.5236. For a Wernerstate experiment, the expected value of S_{n} is μ (shown below). Thus, using n≥3 enables us to demonstrate EPRsteering for some Belllocal states, that is, states with 0.6595>μ>1/2. Also, with n as small as 6, C_{n} is already within 8% of the limit.
Consider the EPRsteering experiment, Fig. 1, from the point of view of an honest Alice, who does share a suitable entangled state with Bob. She claims to be able to prepare different types of state for Bob by making different remote measurements on her half of the state. If the state is a Werner state, she would claim to be able to prepare mixed states aligned (or antialigned) along any Blochsphere axis u. They agree to test this along a specific set of axes {u_{k}}. To maximize the correlation S_{n} in equation (1), Alice measures , and announces her result A_{k}. The value of the correlation S_{n} thus obtained will be μ, independent of n, owing to the invariance of the Werner state. Thus, for a given n, it should be possible to demonstrate EPRsteering if μ>C_{n}.
We experimentally demonstrate EPRsteering with Werner states in a polarizationencoded twoqubit photonic system, as shown in Fig. 3 and detailed in the Methods section. In Fig. 4 we show data for a variety of Werner states; in each case we measured the Bell–CHSH parameter, B, following the method of ref. 21, and the EPRsteering S_{3} parameter. We clearly see states that violate both a Bell–CHSH inequality and an S_{3} inequality, states that violate an S_{3} inequality but not a Bell–CHSH inequality and states that violate neither inequality, but are still entangled.
The amount of entanglement required to demonstrate steering decreases as the number of equally spaced measurement axes increases, that is, for Platonic solids of increasing order (Fig. 2). We measure S_{n} for states near the various steering bounds; see Fig. 5. We compare the values of S_{2} and S_{3} for each of three particular states. These states experimentally show that there exist cases where a state violates both the S_{2} and S_{3} inequalities, violates neither inequality or—most interestingly—violates the S_{3} but not the S_{2} inequality. Similar behaviour is observed for states near the S_{4} and S_{6} bounds. The S_{3} and S_{4} comparison is not especially interesting as C_{3}=C_{4}.
Now consider the EPRsteering experiment, Fig. 1, from the point of view of a dishonest Alice, who shares no entanglement with Bob. Such an Alice can adopt the following ‘cheating’ strategy. Draw a state from some localhiddenstate (LHS) ensemble and send it to Bob. Then, when Bob announces the measurement u_{k}, announce a result A_{k}(j) on the basis of the selected local hidden state and knowledge of Bob’s measurement setting. Although we call this a cheating strategy, Alice cannot actually cheat; the bound C_{n} in equation (1) is defined exactly so that it is saturated by the optimal cheating LHS ensemble. That is, the bounds we have derived are tight; a value of S_{n} greater than C_{n} is necessary to demonstrate EPRsteering.
From the symmetry of Bob’s measurement scheme, there are two obvious candidate LHS ensembles E_{n}: the vertex ensemble and the dual ensemble. In the first case the states are oriented on the Bloch sphere in the directions of the vertices of the figure defining {u_{k}}. In the second, they are oriented in the direction of the face centres (that is, the vertices of the dual figure). Interestingly, both of these possibilities are optimal, but for different values of n; see Fig. 2. Given an optimal ensemble, Alice’s optimal ‘cheating’ strategy, having been told Bob’s measurement axis u_{k}, is to announce as A the more likely outcome (+1 or −1) of Bob’s measurement on the state she has sent.
The experimental realization is simple—Alice prepares a single qubit state using a polarizing beam splitter (PBS), a halfwave plate (HWP) and a quarterwave plate (QWP), and this state is measured by Bob as before. We experimentally demonstrate the nearsaturation of the bound C_{n} using the optimal cheating ensemble for Alice, achieving above 95% saturation for all tested C_{n}, as shown in Fig. 5. The small discrepancy from perfect saturation is due to imperfect state preparation and measurement, even though the prepared states were tomographically measured to overlap with the ideal states to >99% fidelity. The discrepancy increases with n as the cumulative effect of slight misalignments in a series of measurements or preparations tends to reduce the observed C_{n}.
Our demonstration of EPRsteering using states that violate no Bell inequality is possible only because we have broken the conceptual shackles of previous EPR experiments^{3,4,5,6}. These followed the approach of ref. 7 based on the uncertainty relation for two observables with continuous spectra. We used discrete measurements on entangled qubits, and used up to n=6 measurement settings, showing that that increasing n makes the EPRsteering inequality much more robust to noise. In our work we made the fairsampling assumption, that undetected photons are statistically identical to detected photons. Thus we were content with an experimental efficiency far below that necessary to demonstrate EPRsteering without this assumption. However, because the degree of correlation required for EPRsteering is smaller than that for violation of a Bell inequality, it should be correspondingly easier to demonstrate steering of qubits without making the fairsampling assumption. This would provide an important and exciting extension of the fundamental principles we have demonstrated, and open the door to the application of EPRsteering phenomena for quantum communication protocols.
Methods
Wernerstate production.
To realize the Werner states, we start by generating identical single photons through typeI spontaneous parametric downconversion (SPDC). These photon pairs are initially unentangled in polarization. We use a nondeterministic controlledZ gate^{22,23,24,25} to entangle them in polarization. Ideally, this creates the state , where H_{1} is the Hadamard gate^{22} acting on qubit 1. Mixture was controllably added, enabling a change of μ, using the depolarizer (DP) method of Puentes et al. ^{26}—see Fig. 3 and description below. This method produces ‘Wernerlike’ states—states equivalent up to local rotations to the Werner states of equation (3).
The Wernerlike states are described by , where is a singlequbit unitary operation. We can undo to retrieve a Werner state by incorporating a unitary transformation in the measurement settings of qubit 1. To find the optimal unitary operation, we first tomographically reconstruct ρ following ref. 27. We numerically search for by maximizing the fidelity of W_{μ} with . The maximizing is then used to rotate the measurement settings for qubit 1.
Photon source and controlledZ gate.
Source: a 60 mW, linearly polarized, continuouswave 410nmwavelength laser is used to pump a BiBO (bismuth borate) nonlinear crystal to produce pairs of 820 nm single photons through typeI SPDC. With a coincidence window of 3 ns, a coincidence rate of approximately 10,000 counts s^{−1} is achieved. ControlledZ gate: the gate is implemented using a passively stable beam displacer configuration, as in ref. 22. We are not concerned with the gate’s limited success probability in generating entangled states, as we make the fairsampling assumption. The effective efficiency (coincidencestosingles ratio) is ≈0.2% in this experiment.
Depolarizer method.
By varying the azimuthal angle between two quartzglass Hanle DPs^{26}, we create a tunable, variable depolarizing device. It couples the polarization degree of freedom to the spatial degree of freedom—tracing over spatial information induces mixture. By optimizing these procedures, highquality Werner states (fidelity ⩾93.5% in each case) were produced for a wide range of μ.
Measurement technique.
We implement singlequbit polarization measurements on each qubit using a QWP, an HWP, a PBS and fibrecoupled singlephoton counting modules (SPCMs), enabling us to measure along arbitrary axes on the Bloch sphere for each qubit. By choosing different combinations of measurement axes, we can carry out a variety of measurement tasks: evaluating the Bell–CHSH inequality, evaluating the steering parameter S_{n} for different n or tomographically reconstructing the Wernerlike states, ρ.
Calculating C_{n}.
In each case we search over the possible sets {A_{k}}, numerically evaluating the maximand in equation (2). Then, choosing one of the sets that attains the maximum, we use the geometry of the relevant Platonic solid to evaluate it analytically. The same analytical expressions are found from the optimal LHS ensembles E_{n} of Fig. 2. Those not given in the main text are:
Here and are the side lengths of an icosahedron and dodecahedron respectively, circumscribed by the Bloch sphere, and θ=π/5.
Change history
02 November 2011
The authors wish to point out that there were systematic errors in some of the demonstrations of Alice's optimal attempt to cheat (orange lines in Fig. 5, for n = 3, 4 and 6), due to misalignment of the waveplates. The data have been retaken with the corrected settings and are included in Fig. 5 (the plot for n = 2 is unchanged). The arguments of the paper are unaffected by this correction. These changes have been made in the PDF and HTML versions of this Letter.
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Acknowledgements
This work was supported by the Australian Research Council. We thank E. Cavalcanti, A. Fedrizzi, D. Kielpinski, and A. G. White for discussions. We also thank B. Higgins, M. Palsson and S. Kocsis for their contributions.
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H.M.W. conceived the theory. S.J.J. and H.M.W. developed the theory. G.J.P. developed the experiment. H.M.W. and G.J.P. supervised the project. D.J.S. built the experiment, and collected and analysed the data. D.J.S., S.J.J., H.M.W. and G.J.P. composed the manuscript.
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Saunders, D., Jones, S., Wiseman, H. et al. Experimental EPRsteering using Belllocal states. Nature Phys 6, 845–849 (2010). https://doi.org/10.1038/nphys1766
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DOI: https://doi.org/10.1038/nphys1766
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