Abstract
Quantum correlations can be stronger than anything achieved by classical systems, yet they are not reaching the limit imposed by relativity. The principle of information causality offers a possible explanation for why the world is quantum and why there appear to be no even stronger correlations. Generalizing the nosignaling condition it suggests that the amount of accessible information must not be larger than the amount of transmitted information. Here we study this principle experimentally in the classical, quantum and postquantum regimes. We simulate correlations that are stronger than allowed by quantum mechanics by exploiting the effect of polarizationdependent loss in a photonic Belltest experiment. Our method also applies to other fundamental principles and our results highlight the special importance of anisotropic regions of the nosignalling polytope in the study of fundamental principles.
Introduction
Quantum mechanics is one in a large class of theories which are consistent with relativity in the sense that they do not allow signals to be sent faster than the speed of light. Many of these theories exhibit strong nonlocal correlations between distant particles that cannot be explained by the properties of the individual particles alone. Surprisingly, quantum mechanics is not the most nonlocal among them, which raises the question about the physical principle that singles out quantum mechanics and sets the limit on the possible strength of correlations in nature.
Here we experimentally address this fundamental question by testing the principle of information causality in the classical, quantum and postquantum regime. While the nosignaling principle limits the speed with which distant parties can communicate, information causality states that the accessible information cannot be more than the information content of a communicated message, no matter what other shared resources are used. Both classical and quantum mechanics satisfy this principle, while it is violated by most postquantum theories^{1}.
We experimentally emulate correlations of various strengths from classical to almost maximally nonlocal and demonstrate a violation of the principle of information causality in the case where the simulated correlations are beyond the quantum regime. Apparent superquantum correlations are, in our approach, a consequence of the nonunitary evolution of quantum states when subjected to polarizationdependent loss with postselection^{2}. For moderate loss, we find that initially entangled states can result in superquantum correlations, while unentangled states still appear classical. For higher loss on the other hand we observe superquantum correlations even for classical input states.
Nosignaling resources can formally be treated as pairs of black boxes shared between arbitrarily separated Alice and Bob^{3}, see Fig. 1a). Each box has a single input and output and the correlation between them is only restricted by the nosignaling principle. This means that the local outcome only depends on the local input, such that Alice cannot learn anything about Bob's input from only her output.
A typical quantum example of such a resource is a pair of entangled particles, shared between Alice and Bob, where inputs correspond to measurement settings and outputs to measurement outcomes. Since the work of John Bell—and numerous subsequent confirming experiments—it is now widely accepted that these particles exhibit nonlocal correlations, which have no classical explanation. Under the nosignaling constraint alone, however, there are even stronger nonlocal correlations than quantum entanglement^{4}. The maximum that is compatible with relativity is achieved by the socalled PopescuRohrlich (PR)box^{4}, characterized by perfect correlations of the form A⊕B = ab, between Alice's and Bob's inputs a and b and outputs A and B, respectively. Here ⊕ denotes addition modulo 2, equivalent to the logical XOR, where A⊕B = 0 when A = B and 1 otherwise.
A convenient operational way of quantifying nonlocality is the ClauserHorneShimonyHolt (CHSH) inequality^{5}. This experimentally testable reformulation of Bell's inequality is satisfied by any correlation that can be described by a local hidden variable model. Such models are a description of correlations that can arise in classical systems, but cannot describe nonlocal correlations obtained from e.g. entangled quantum states. Written in terms of correlations of the form A ⊕ B = ab the inequality takes the form
Here P(A ⊕ B = ab  a, b) denotes the probability for obtaining outputs A, B, which satisfy A ⊕ B = ab given the inputs a for Alice and b for Bob. While this inequality is satisfied by any classical correlations, it can be violated in the quantum case. This violation, however, is bounded to a value of , known as Tsirelson's bound^{6}. Note, that inequality (1) is presented here in a slightly different form than conventionally^{5}, where the classical bound is 2 and Tsirelson's bound is . They are, however, linearly related and the difference is a simple rescaling of S.
Despite being a simple consequence of the mathematical formalism of quantum mechanics, it is unclear what the physical motivation is for this seemingly suboptimal limit on the strength of quantum correlations. In fact even the algebraic maximum S = 4 can be achieved (by the PRbox) without violating the nosignaling principle.
This principle is physically motivated by the fact that, according to special relativity, fasterthanlight information transfer would allow information to be sent backwards in time and thus violate causality. Nevertheless, it does not explain why superquantum correlations such as the PRbox are incompatible with quantum mechanics and seem not to exist in nature. A possible explanation is offered by the principle of information causality—a generalization of nosignaling—which states that there cannot be more information available than was transmitted^{7}.
This can be understood on the basis of the following elementary informationtheoretic protocol: Bob tries to gain information from a set of data that is only known to Alice. The parties are allowed to use an arbitrary amount of shared nosignaling resources, but may not communicate more than m classical bits. In this case, the information causality principle states that the amount of information accessible to Bob should be limited to m classical bits^{7}.
In the simplest instance, Alice has a set of two bits {a_{0}, a_{1}} and Bob wants to guess one of them, which we denote a_{b}^{8}, see Fig. 1a). Alice and Bob then input a_{0} ⊕ a_{1} and b into their respective black box and obtain outputs A and B. From this output Alice computes an m = 1bit message M = A⊕a_{0} and sends it to Bob, who calculates his guess for Alice's bit as G = M⊕B = a_{0} ⊕ A ⊕ B. In the case of a shared PRbox, Bob can guess either one of Alice's bits perfectly, since in that case A ⊕ B = ab and thus G = a_{0} ⊕ b(a_{0} ⊕ a_{1}).
In the more general case considered here, Alice has a dataset {a_{0}, … a_{N}_{−1}} of N = 2^{n} bits and Bob wants to guess the bit with index . As discussed in Ref. 7, Alice and Bob can achieve this task by using a nested version of the protocol outlined above, with N − 1 black boxes on n levels and 1 bit of classical communication.
The protocol is illustrated in Fig. 2b) for the case n = 2. From every output Alice computes a temporary message M_{k}_{,i}, where k denotes the level and i the number of the box on that level. Since she is only allowed 1 bit of communication, she uses these temporary messages as the inputs for the boxes on the nextlower level and only sends the final message to Bob. Depending on b_{n} Bob then decodes either M_{n}_{−1,1} or M_{n}_{−1,2} and then moves on to the nexthigher level until he reaches the bit of interest.
Bob's success can then be quantified by
where I(a_{k} : G  b = k) is the Shannon mutual information between the k'th bit of Alice's list and Bob's guess for it^{7}. This quantity can further be bounded as
where h(P_{k}) is the binary entropy of the success probability P_{k} for guessing the k'th bit.
Results
Experimentally, we generate apparent superquantum correlations based on the effect of polarizationdependent loss in a postselected Belltest experiment^{2}, see Fig. 2a). We use photon pairs created from a continuouswave pumped spontaneous parametric downconversion source in a polarization Sagnac design^{9,10}, as illustrated in Fig. 2b). Using this approach we obtain photon pairs with very high efficiency and in a continuously tunable fashion that enables us to produce any bipartite quantum state^{11}.
In particular, we used the maximally entangled state as the initial state, where H/V〉 represent horizontal and vertical polarization, respectively. For comparison, we also considered the corresponding fully decohered and thus separable state ρ_{sep} = (HV〉〈HV + V H〉〈V H)/2. This state was produced as a mixture of the two pure state components HV〉 and V H〉 by probabilistically mixing the respective coincidence counts.
The initial state is then subjected to polarizationdependent loss, introduced to the system by means of a JaminLebedev polarizationinterferometer, which allows individual control of the degree of loss for each polarization mode for both Alice and Bob, see Fig. 2c). In the symmetric case considered here the loss was parametrized by a single parameter κ, where κ = 0 corresponds to the lossfree scenario and κ = 1 means complete loss of one polarization. With this setup we simulated correlations of increasing strength, ranging from classical to quantum and close to maximal nonsignaling as discussed in detail in the methods section.
Using these correlations we investigated the information causality protocol on up to four levels (corresponding to a 16bit dataset for Alice) with 1bit of communication. Crucially, we implemented the protocol in Fig. 1b) on a shotbyshot basis, rather than estimating the performance from coincidence probabilities. For this we used an AITTTM8000 timetagging module with a temporal resolution of 82 ps to register the single photon counts for all possible outcomes. From this data, using passive feedforward, i.e. at the processing stage, we were able to reconstruct over 10^{5} individual trials of the protocol for each of the 21 settings of uniformly increasing κ.
At a correlation strength of S = 3.874(5), the information available to Bob is at least I ≥ 1.86(2) bits, despite only receiving 1 bit from Alice. For four nesting levels of the protocol we establish lower bounds as high as I ≥ 7.47(11) bit, which violates the information causality inequality I ≤ 1 by almost 60 standard deviations. Similarly for weaker correlations, Bob has more information available than contained in Alice's message for all nesting levels as soon as the correlation strength surpasses S ≈ 3.5. The fact that this value is significantly higher than Tsirelson's bound of S_{Q} ≈ 3.41 emphasizes that the quantity I only recovers this bound in the asymptotic limit n → ∞.
In the following we therefore consider an alternative figure of merit, motivated by identifying the protocol in Fig. 1b) as a special case of a socalled random access code^{12}. Using similar ideas as in Ref. 7, the efficiency of this task can be bounded by
which thus also encompasses the principle of information causality^{12}. This bound, however, can indeed be saturated by quantum states for any size of Alice's dataset, as illustrated in Fig. 3. Note that our data violates the bound before the correlations surpass Tsirelson's bound. This is a result of a slight anisotropy in the simulated correlations due to experimental imperfections and a resulting bias for certain datasets. It is not present when considering isotropic correlations, see Fig. 3b). Crucially, this highlights the dependence of both figures of merit (3) and (4) to the specific random choice of Alice's dataset.
In particular, the separable state used in the simulation produces entanglementlike correlations for one measurement choice of Alice and uncorrelated outputs for the other, see Supplementary Figure S1. Hence, depending on the choice of dataset the figures of merit η and I might resemble the behavior expected for an entangled state, for a completely mixed state or, for higher nesting level, anything inbetween. Only when averaging over all possible datasets, {a_{i}}, for a given level or employing the “depolarization” protocol introduced in Ref. 13 to make the correlations isotropic without changing the CHSH value, can the quantities (3) and (4) be used as reliable figures of merit, see Supplementary Fig. S1 and S2. Note, however, that anisotropic superquantum correlations (averaged over all datasets) do not necessary violate Tsirelson's bound. In this case the principle of information causality cannot be probed using the depolarization approach, since it would result in isotropic correlations and information causality would not be violated.
Discussion
In contrast to the full set of nosignaling correlations and the set of classical correlations, which both have the form of a wellcharacterized polytope, much less is known about the quantum set^{3,14}. Understanding the set of quantum correlations theoretically and characterizing it experimentally should thus be a primary aim from a practical as well as a fundamental perspective. Information causality, which has been proposed as a physical principle to reconstruct the set of quantum correlations, has already proven successful in recovering the famous Tsirelson bound. This limit of quantum correlations, however, is only one extremal point on the continuous boundary and there exist correlations below it, which nevertheless do not admit a quantum description^{1}. Information causality also rules out such correlations for some 2dimensional slices of the full (8dimensional) nosignaling polytope, while it does not for other slices^{1}. This shortcoming, nevertheless is not definite and might just be a result of a suboptimal protocol in Fig. 1b).
A violation of information causality would in particular imply that the tested theory does not admit a suitable measure of one of the most elementary information theoretic quantities: entropy^{12,15}. Such a measure is assumed to be consistent with the classical limit and such that the entropy change ΔH of a composite system XY satisfies ΔH(XY) ≥ ΔH(X) + ΔH(Y) under local evolution of the subsystems X and Y. Hence, a failure of these requirements could be interpreted as allowing for the generation of nonlocal correlations via local transformations. Similar consequences might also arise from the violation of alternatives to information causality, which are more or less successful in recovering part of the quantum boundary. Examples include the principles of local orthogonality^{16}, the requirement that the theory has a suitable classical limit^{17} or that certain communication^{18,19,20} or computational tasks^{21} are nontrivial.
Our method of simulating superquantum correlations could be adapted to explore some of these alternative principles as well. Of particular interest, however, would be a test of information causality in the multipartite case, since most of the above principles are formulated in the bipartite setting, which is bound to fail in recovering the full quantum boundary due to the existence of multipartite superquantum correlations, which obey every bipartite principle^{22,23}. While there are studies of information causality for higherdimensional systems, which strengthen its position as a physical principle that determines quantum correlations^{24}, a suitable generalization to the multipartite case is still an open problem.
As highlighted by our experiment, special focus has to be put on anisotropic regions of the nosignaling polytope. Specifically we find that the introduced figures of merit are not valid in a single instance of the protocol and have to be averaged over all possible datasets or estimated from the depolarized, isotropic data. This subtle, but very important detail is clearly highlighted by our experimental results, where we show how even a small amount of imbalance can result in a violation of the principle by quantum states for a specific choice of parameters, while obeying the principle on average.
Methods
Examining the results of a CHSHinequality test make it clear where our data crosses the boundary of the quantum set. In our investigation we focused on the scenario of a fixed maximally entangled state ψ_{+}〉 in situations with different amounts of loss, as shown in Fig. 4. We further considered the state ρ_{sep}, which resembles the state ψ_{+}〉 after full decoherence as might happen during propagation between Alice and Bob. This allows for an intuitive comparison between the entangled and unentangled case.
The tested inequality has the form of a CHSHinequality with measurements in the yzplane of the Bloch sphere. For the lossless case κ = 0, Alice's and Bob's measurements can be viewed as the application of appropriate basisrotations (around the Xaxis) followed by projective measurements in the H/V〉basis. These rotations can also be seen as phasegates in the diagonal polarization basis . In the case where polarization dependent loss is present, these phasegates become nonunitary. They act as the identity on the state and impose a phase on the nonorthogonal state , where κ = 〈uw〉. The precise relation between κ and the degree of loss is discussed in Ref. 2. As nonunitary operations can only be performed nondeterministically, postselection on success is required, which results in the observation of apparent superquantum correlations. Finally, we use the first step of the depolarization procedure in Ref. 13 to symmetrize the simulated correlations, while preserving their possible anisotropy.
Curiously, we note that moderate polarizationdependent loss can lead to superquantum correlations for entangled states without invalidating the CHSH inequality for separable states, as suggested in Ref. 2. This observation even holds when optimizing the separable state for maximal CHSHvalue, for each degree of loss^{2}. Note, however, that these results were obtained using the same measurements for both separable and entangled states, whereas arbitrary hidden variable theories would allow arbitrary measurements.
Figure 2 illustrates the obtained values of the CHSH parameter S and compares them to the ideal case, which, for the initially entangled state, is described by
Here Θ is a function of κ, which can be analytically approximated by Θ = π(17+ cos(πκ))/12, as discussed in Ref. 2.
We experimentally violate Tsirelson's bound by more than 7 standard deviations, S = 3.423(1) at a loss parameter of κ = 0.3. At this point, the achieved value for the unentangled state, S = 2.821(2), is indeed well below the classical bound of 3 and even the optimal unentangled state does not violate the inequality until κ ≈ 0.37. In the region 0.3 ≤ κ ≤ 0.37 it is therefore possible to exploit superquantum correlations from entangled states while unentangled states still appear classical. With increasing loss, both states eventually violate Tsirelson's bound and approach the numerical maximum of S = 4, with experimental values of S = 3.9341(6) and S = 3.929(2) for the entangled and separable state, respectively. The increasing deviation from the theoretical predictions in Fig. 4 is a result of the decreasing signaltonoise ratio in the singlephoton detectors for highloss settings.
Related experiments have observed apparent violations of Tsirelson's bound as a consequence of explicit violations of the detection loophole^{25} or the fairsampling assumption^{26}. The latter is in fact typically violated if the quantum system of interest has more (possibly) correlated degrees of freedom than those tested in the Bellinequality^{27}. Violation of Tsirelson's bound has also been considered as an intermediate step in deriving threequbit inequalities^{28}.
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Acknowledgements
We thank Tim Ralph and Cyril Branciard for helpful discussions. We also thank the team from the Austrian Institute of Technology for kindly providing the timetagging modules for this experiment. We acknowledge financial support from the ARC Centres of Excellence for Engineered Quantum Systems (CE110001013) and Quantum Computation and Communication Technology (CE110001027). D.W.B. is funded by an Australian Research Council Future Fellowship (FT100100761) and A.F. by an Australian Research Council Discovery Early Career Researcher Award (DE130100240). A.G.W. acknowledges support from a UQ ViceChancellor's Senior Research Fellowship.
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M.R., A.F. and D.W.B. developed the concepts. M.R. and A.F. designed the experiment and analysed the data. M.R. performed the experiments and analysed data. A.G.W. supervised the project. All authors wrote the manuscript.
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Ringbauer, M., Fedrizzi, A., Berry, D. et al. Information Causality in the Quantum and PostQuantum Regime. Sci Rep 4, 6955 (2014). https://doi.org/10.1038/srep06955
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DOI: https://doi.org/10.1038/srep06955
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