Abstract
A physical theory is called nonlocal when observers can produce instantaneous effects over distant systems. Nonlocal theories rely on two fundamental effects: local uncertainty relations and steering of physical states at a distance. In quantum mechanics, the former one dominates the other in a wellknown class of nonlocal games known as XOR games. In particular, optimal quantum strategies for XOR games are completely determined by the uncertainty principle alone. This breakthrough result has yielded the fundamental open question whether optimal quantum strategies are always restricted by local uncertainty principles, with entanglementbased steering playing no role. In this work, we provide a negative answer to the question, showing that both steering and uncertainty relations play a fundamental role in determining optimal quantum strategies for nonlocal games. Our theoretical findings are supported by an experimental implementation with entangled photons.
Introduction
The uncertainty principle is a fundamental feature of quantum theory, which postulates the existence of incompatible observables, the results of whose measurements on identically prepared systems cannot be predicted simultaneously with certainty. Recently, the traditional formulation of uncertainty relations in terms of standard deviations and commutators has been eschewed in favor of the entropic uncertainty relations^{1} and the even more fundamental finegrained uncertainty relations^{2}. These finegrained uncertainty relations are formulated in terms of the basic entities of the theory, namely the probabilities of particular sets of outcomes for given sets of measurements, and are thus able to capture the uncertainty of these measurements in a more general manner than the entropic measures or the statistical standard deviations. Moreover, the uncertainty bounds are expressed in a manner independent of the specific underlying quantum state, an advantage over the traditional formulation in terms of average values of commutators on fixed states. Another fundamental feature of quantum theory is steering, identified by Schrödinger in ref. ^{3}. This property determines, for two systems in a shared (entangled) state, which states can be prepared on one system by a measurement on the other. Quantum steering can be used as a resource to generate ensambles of quantum systems incompatible with a local hidden variable (LHV) model^{4}. For twoqubit states, all states that Alice can steer are restricted to an ellipsoid within the Bloch sphere of Bob^{5}.
The results of measurements on distant quantum systems can be correlated in a way that defies classical local realistic description. This nonlocality of quantum theory is evidenced in the violation of Bell inequalities by spatially separated quantum systems. Quantum correlations are restricted to some extent by the nosignaling principle, i.e., the measurement results cannot allow for signaling between the distant locations. Nevertheless, there exist nonlocal correlations allowed by the nosignaling principle that cannot be realized in quantum theory^{6,7}.
The fundamental question why quantum correlations are nonlocal yet not as strong as allowed by the nosignaling principle is an intriguing one that has stimulated the formulation of many striking new informationtheoretic principles. So far none of the known principles have been able to capture the set of quantum correlations in its entirety^{8}, thus a comprehensive answer to this question is still lacking. The testbeds for these principles are a special class of Bell inequalities based on socalled quantum nonlocal games which extract purely probabilistic aspects of the nonlocality test, independent of the physical realization. Consideration of nonlocal games lead to a significant breakthrough in ref. ^{2} where two fundamental concepts of quantum theory, the strength of nonlocal correlations and the uncertainty principle, were shown to be inextricably quantitatively linked with each other.
Moreover it was shown that in a large class of nonlocal games for which optimal quantum strategies were explicitly known (the class of XOR games for which an explicit characterization of the optimal quantum strategy was provided by Tsirelson^{9}) these are not only just linked, but one of them—uncertainty—fully determines the nonlocality of quantum theory with steering playing no role. An important question left open in ref. ^{2} was whether such a phenomenon holds in general. If it did, this would constitute a defining property of quantum mechanics: that something fully local (the uncertainty principle for a singleparty’s measurements) governs something nonlocal (the Bell violation on a shared system).
The intriguing result of ref. ^{2} is that while the degree of nonlocality in any theory is generally determined by a combination of two factors—the strength of the uncertainty principle and the degree of steering allowed in the theory, in quantum theory the degree of nonlocality for the wellknown class of twoplayer XOR games is purely determined by the strength of the uncertainty principle alone. More precisely^{2}, shows that in a twoparty Bell scenario, the strength of nonlocality in any theory is determined by the uncertainty relations for Bob’s measurements acting on the states that Alice can steer to. On the other hand in quantum theory, for all XOR games (aka bipartite correlation Bell inequalities)^{9}, the states which Alice can steer to are identical to the most certain states, so that only the uncertainty relations of Bob’s local measurements determine the outcome.
In this paper, we show that the onetoone correspondence between the uncertainty principle and the degree of nonlocality in quantum theory (referred hereafter as the Uncertainty Principle—Quantum Game Value correspondence, or UPQGV correspondence) observed for XOR games in ref. ^{2} does not hold in general, by presenting an explicit counterexample of a nonlocal game violating the correspondence. We provide an intuitive explanation in terms of the Schrodinger–Hughston–Jozsa–Wootters theorem^{10} for when the UPQGV correspondence breaks down. To show that the game does not have other optimal strategies that could obey the correspondence and to facilitate experimental testing of our result, we prove a selftesting property of the game, namely that there is a unique state and measurements (up to local unitaries and attaching irrelevant ancillae) that achieves the optimal quantum value. Furthermore, the game is not an isolated example, we extend it to show that every twoparty nonmaximally entangled state \(\left \psi \right\rangle\) is the optimal state for a game G_{ψ} for which the correspondence does not hold. The tradeoff existing between steering and uncertainty is conclusively shown by means of an experimental implementation, in which the steered states manifestly are seen to be distant from the maximally certain state even after the experimental errors are taken into account.
Results
Uncertainty principle—quantum game value correspondence
Let us first recall the precise correspondence between the finegrained uncertainty relations and the strength of nonlocality established in ref. ^{2}. Consider a twoplayer nonlocal game G, in which Alice and Bob receive questions x,y from respective input sets X,Y according to some input distribution π_{X,Y}(x, y). They return answers a, b from some output sets A, B, respectively. The winning constraint is specified by a predicate V(a, bx, y) ∈ {0, 1}. The success probability in the game ω_{s}(G) is thus written as
where \({\cal S}\) refers to a set of conditional probability distributions (boxes) P_{A,B}_{X,Y}. One considers boxes taken from sets C,Q,NS corresponding to the set of classical, boxes and general nosignaling boxes, with corresponding values ω_{c}(G), ω_{q}(G), and ω_{ns}(G) respectively. One may also restrict attention to the free games for which the input distributions are independent, i.e., π_{X,Y}(x, y) = π_{X}(x)π_{Y}(y).
We will in particular be interested in ω_{q}(G), i.e., the value obtained from those boxes for which there exists a state ρ on a Hilbert space \({\cal H}_d\) and sets of measurement operators (POVMs) \(\left\{ {M_a^x} \right\},\left\{ {M_b^y} \right\}\) such that P_{A,B}_{X,Y}(a,bx,y) = \({\mathrm{Tr}}\left( {\rho M_a^x \otimes M_b^y} \right)\). The idea in ref. ^{2} is to rewrite the game expression in Eq. (1) as
Let \(P_{{\mathrm{BY,X,A}}}(b{\mathrm{}}y,x,a)_{\hat \sigma _{ax}^B}\) be Bob’s marginal probability distribution when his state is steered by Alice to \(\hat \sigma _{ax}^B\). Now, observe that for each (x, a), the expression
constitutes a finegrained uncertainty relation on Bob’s system with \(\xi _B^{(x,a)}\) denoting the maximum over all possible states \(\hat \sigma _{ax}^B\) of Bob’s system. When the optimal value \(\xi _B^{(x,a)}\) equals unity, we refer to the corresponding uncertainty relation as trivial, i.e., while the probabilities are bounded below unity for some states, there exist states for which the outcomes (for each of Bob’s inputs y) can be fixed with certainty. On the other hand, when \(\xi _B^{(x,a)} < 1\), we infer that one cannot obtain a measurement outcome with certainty for all measurements simultaneously.
An example situation of the uncertainty relation is shown in Fig. 1 and steering to the maximally certain states is exemplified in Fig. 2.
Let \(\{ {\tilde \sigma _{ax}^B} \}\) denote the set of states of Bob’s system that achieve the maximum value \(\xi _B^{(x,a)}\) of the uncertainty expressions for each (x, a) for given optimal measurement operators \(\left\{ {M_b^y} \right\}\). The question then arises whether Alice is able to steer Bob’s system to these maximally certain states and thus achieve the bound set by the uncertainty principle for the game G. We are thus lead to consider the effect of steering. For any bipartite nosignaling box shared by Alice and Bob, any measurement on Alice’s system creates a set of singleparty boxes on Bob’s side {P_{B}_{Y}(by)_{x,a}} = {P_{B}_{Y,X,A}(by, x, a)}. We say that with this particular input–output pair (x, a), Alice has steered the state of Bob’s system to the set of boxes {P_{B}_{Y}(by)_{x,a}} with probability P_{A}_{X}(ax).
We see therefore in Eq. (2) the separation of the game expression into two components, one where Bob’s (optimal) measurements define a set of uncertainty relations one for each (x, a) and a second component wherein Alice tries to steer Bob’s system to the maximally certain states for these relations. The strength of nonlocality in any theory is thus seen as a tradeoff between the strength of the uncertainty relations and the amount of steering allowed in the theory.
In ref. ^{2}, it was shown that for the wellknown class of twoplayer XOR games for which the optimal measurements are known, the strength of nonlocality is purely determined by the uncertainty relation with steering not constraining the value in any way. In other words, the optimal measurements and the state share the property that in all these known instances, Alice is able to steer Bob’s system to the most certain states corresponding to the set of uncertainty relations of his system for each input–output pair (x, a).
Note that the restriction to nonlocal games rather than all Bell inequalities is crucial for the correspondence to be meaningful. Indeed, for general Bell inequalities, where one is allowed to scale the Bell expression with arbitrary multiples of the normalization and nosignaling equalities, it is possible to show that the correspondence can always be made to hold up to arbitrary high accuracy. This general observation inspired by recent results in ref. ^{11} is explained in detail in the Supplementary Note 3.
Twoplayer XOR games are nonlocal games with an arbitrary number of inputs and binary outputs, where the winning constraint of the game only depends on the xor of the parties’ outputs. Building on a breakthrough theorem by Tsirelson^{9}, it was shown in refs. ^{12,13} that the quantum value of twoparty xor games can be calculated precisely by means of a semidefinite program, and the Tsirelson theorem allows to recover the optimal state and measurement operators for any such game. In effect, apart from the pseudotelepathy games^{14} and a few other isolated instances, these are the games where the optimal measurements are known and for which the relation between the uncertainty principle and nonlocality was established in ref. ^{2}. The difficulty in establishing the relationship for general nonlocal games is due to the fact that the problem of finding the quantum strategy of arbitrary nonlocal games is hard^{15}; one usually uses a hierarchy of semidefinite programs^{16,17} which converge to the true quantum value.
Note that it is natural to ask about the relation of the steeringtype representation of the Eq. (2) to the wellknown Schrodinger–Hughston–Jozsa–Wootters theorem which defines all the ensembles Alice may steer to. It is tempting to expect that the result^{2} is due to application of that theorem. This is, however, not the case because of the crucial fact it is not guaranteed that the maximally certain states together with the optimal local probabilities P_{A}_{X}(ax) obey the nosignaling condition of the SHJW theorem (see Supplementary Note 2 for more discussion), viz.
Counterexamples to the correspondence
Let us now exhibit an example of a nonlocal game for which the UPQGV correspondence does not hold, i.e., one where the optimal quantum state and measurements are such that Alice is unable to steer Bob’s system to the maximally certain state for each (x,a). Before we proceed to the counterexample, let us mention that it is possible that the optimal quantum value of a nonlocal game can be achieved with different sets of states and measurement operators (even going beyond a trivial unitary equivalence), therefore one must check whether the relation could hold for at least one optimal quantum strategy. Thus, in order to give a counterexample to the UPQGV correspondence, it is necessary to prove that the relationship does not hold for all optimal quantum strategies for the game. We achieve this requirement by proving a selftesting property of the counterexample, i.e., that up to unitary equivalences there is a unique state and sets of measurements that achieves the optimal value of the game.
We consider the Bell scenario B(2, 2, 2) of two parties, each performing one of two measurements and obtaining one of two outputs. The Bell inequality corresponding to the game denoted G^{(7)} is explicitly given by
where we have assumed that each party chooses their inputs uniformly, i.e., π_{X}(x) = π_{Y}(y) = \({\textstyle{1 \over 2}}\) for x, y ∈ {0, 1} so that π_{X,Y}(x,y) = \({\textstyle{1 \over 4}}\) and the classical bound is ω_{c}(G^{(7)}) = \({\textstyle{3 \over 4}}\). The optimal strategy for the game G^{(7)} violates the UPQGV correspondence (the proof of the following Proposition 1 is given in the Supplementary Note 1).
Proposition 1: The optimal quantum strategy for the game G^{(7)}(achieving ω_{q}(G^{(7)}) ≈ 0.782) violates the uncertainty principle—quantum game value correspondence, i.e., Alice is unable to steer Bob’s system to the maximally certain states and vice versa.
The uncertainty relations for each input–output pairs (x, a) of Alice for the game G^{(7)} are given as
where the uncertainty bounds are \(\xi _B^{(0,0)} = \xi _B^{(0,1)} \approx 0.882\), and \(\xi _B^{(1,0)} \approx 0.823\). The optimal state and measurements achieving ω_{q}(G^{(7)}) ≈ 0.782 are given in the Supplementary Note 4, where it is shown explicitly that while for (x = 1, a = 0) Alice steers Bob’s system to the maximally certain state, for (x = 0, a = 0) and (x = 0, a = 1) Alice is unable to steer Bob’s system to the maximally certain states of the corresponding (nontrivial) uncertainty relations. Further, the trivial uncertainty relation for (x = 1, a = 1) also fails to be saturated. The value ω_{q}(G^{(7)}) achievable in quantum theory is thus strictly lower than what is allowed by the uncertainty principle, and therefore the game G^{(7)} violates the UPQGV correspondence.
Let us now see why the UPQGV correspondence breaks down for the particular game G^{(7)}, and establish conditions for the correspondence to hold. To do so, we examine the assemblage \(\{ {P_{{\mathrm{AX}}}(a{\mathrm{}}x),\tilde \sigma _{ax}} \}\) of maximally certain states. For the game G^{(7)} it can be readily verified that the corresponding assemblage of maximally certain states does not obey the nosignaling relation Eq. (4), so the SHJW theorem does not guarantee the existence of a shared entangled state and measurements on Alice’s side that would prepare the corresponding maximally certain states on Bob’s system. Formally, we may make the observation (which follows from wellknown demands on steerability^{4}) that the UPQGV correspondence holds when the probabilities P_{A}_{X}(ax) together with the maximally certain states \(\hat \sigma _{ax}^{\mathrm{B}}\) obey the nosignaling constraint in Eq. (4).
Observation 2: The uncertainty principle determines the nonlocality of quantum theory whenever the maximally certain states \(\hat \sigma _{ax}^{\mathrm{B}}\) of one party’s measurements together with the optimal local probabilities {P(ax)} of the other party, forms a nosignaling assemblage, i.e., when \(\{ {P(a{\mathrm{}}x),\hat \sigma _{ax}^{\mathrm{B}}} \}\) obeys Eq. ( 4 ).
The game G^{(7)} shows that this condition is not always obeyed by the maximally certain states. While it appears at present an intractable problem to characterize the set of all games where the UPQGV correspondence breaks down, we can nevertheless show that the game G^{(7)} is not singular in this respect. Indeed, every twoparty nonmaximally entangled state \(\left \psi \right\rangle\) (i.e., a state not of the form \(\frac{1}{{\sqrt d }}\mathop {\sum}\nolimits_{i = 1}^d \left {i,i} \right\rangle\) for some d > 1) is the optimal state for a game G_{ψ} for which the correspondence does not hold. This is captured in the following proposition (whose proof is given in the Supplementary Note 4).
Proposition 3: For any twoparty entangled, but nonmaximally entangled, state \(\left \psi \right\rangle \in {\Bbb C}^d \otimes {\Bbb C}^d\) for arbitrary Hilbert space dimension d, there exists a game G_{ψ}for which the optimal quantum strategy is given by suitable measurements on \(\left \psi \right\rangle\), and such that the correspondence between the uncertainty principle and the quantum game value does not hold for G_{ψ}.
An interesting open question is whether the conditions in Observation 2 are met for all unique games^{18} which are a natural generalization of XOR games to a larger output alphabet. Also interesting is to find whether the correspondence holds for all games where the optimal strategy involves a maximally entangled state, which would highlight that in the foundational program of seeking an informationtheoretic principle behind the strength of quantum nonlocal correlations, one must go further than the correlations exhibited by the maximally entangled states alone.
Experimental implementation
In our experiment, the physical qubits are singlephoton polarization states and the computational basis corresponds to the horizontal (H) and vertical (V) polarization, i.e., \(\left H \right\rangle \equiv \left 0 \right\rangle\) and \(\left V \right\rangle \equiv \left 1 \right\rangle\). To achieve the maximal violation of the Bell inequality given in (5), we used the following nonmaximally polarizationentangled twophoton state,
This state is produced in two steps. First, we generate entangled photon pairs via spontaneous parametric downconversion (SPDC)^{19}. Then at the second step, these entangled pairs are transformed to the required state Eq. (10) by local rotations^{20}. With this state, one can use simple linear polarization settings \(\left {\phi _x^ \pm } \right\rangle = {\mathrm{cos}}{\kern 1pt} \gamma _x\left H \right\rangle \pm {\mathrm{sin}}{\kern 1pt} \gamma _x\left V \right\rangle\) on both sides, where γ_{0} = π/4 and γ_{1} = 4.7948. The polarization measurement on Alice and Bob’s sides are performed by analyzers consisting of waveplates, polarizing beam splitters (PBS), and singlephoton detectors. An FPGAbased timing system is used to collect data. The experimental setup is outlined in Fig. 3 and its detailed description can be found in the Supplementary Note 5.
The Fidelity, \(F = \left\langle {\mathrm{\Psi }} \right\rho _{exp}\left {\mathrm{\Psi }} \right\rangle\), of the experimentally prepared state ρ_{exp} with respect to Eq. (10) was 0.9933 ± 0.0009. We obtained the experimental Bell inequality violation ω_{q}(G^{(7)}) = 0.7770 ± 0.0002. Note that the theoretical quantum and classical bounds are 0.7822 and 0.7500, respectively. The fidelity of the four maximally certain states v_{0+}, v_{0−}, v_{1+}, and v_{1−} are given by F_{0+} = 0.9990 ± 0.0003, F_{0−} = 0.9888 ± 0.0008, F_{1+} = 0.9899 ± 0.0009, and F_{1−} = 0.9957 ± 0.0004, respectively. Here, v_{ij} is the least uncertain state associated to Alice measurement i having outcome j. In Fig. 4 we represent the least uncertain states (blue) and the states m_{ij} that Alice is able to steer (red) (see Supplementary Notes 3 and 5 for details related to theoretical and experimental results, respectively). Experimental errors determine eight cones in Bob’s Bloch sphere, whose apertures are the largest possible, according to the experimentally obtained errors.
For error estimation, we have considered the error originated from the measurement side only, as the error on the preparation side will just shift the experimentally prepared state away from the desired state and therefore will be apparent from the reported state fidelity or the value of Bell violation. Further details are given in Supplementary Note 8.
We note that the experimental realization is not strictly required for the case of the paper. However, it is fundamental to note that the breakdown in the correspondence between the two major aspects of quantum theory is not a trivial one that would be washed out under inevitable experimental error, since the correspondence was only considered for the optimal quantum value. As such, it is of interest to find that even with current experimental technology, one can achieve sufficient experimental fidelities to make the case of the paper, apart from serving as one of the first experiments to selftest a nonmaximally entangled state. Finally, we remark that the experiment was not performed in a loopholefree manner, as such it would be interesting to check the expectation that the same conclusions also hold in a loopholefree Bell test such as recently done in refs. ^{21,22,23}.
Discussion
In this paper, we have shown that the intriguing correspondence between the uncertainty principle and the quantum game value, proven for the very important class of twoplayer XOR games in^{2}, does not hold for general nonlocal games. In order to prove this result we have put forth an intuitive argument to identify when the correspondence holds in terms of the SHJW theorem.
Many interesting questions remain open. First, note that the CHSH inequality is the only facetdefining inequality in the Bell scenario B(2, 2, 2) and the nonlocal game we consider constitutes a lowerdimensional face of the classical polytope. It is of interest to find whether the correspondence holds for nonlocal games that are tight Bell inequalities (facets of the classical polytope), or for games where the optimal strategy involves a maximally entangled state. Second, while the uncertainty relations always provide a bound on the quantum value, it is now an open question to characterize the class of games for which this bound is saturated and more interestingly those for which the gap is extremal.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
R.R. initiated the research, proposed the Proposition 1 and provided the analytical proof of the Observation 2 and Proposition 3. The paper is supported by ERC AdG grant QOLAPS, by Foundation for Polish Science TEAM project cofinanced by the EU European Regional Development Fund, by the project “Causality in quantum theory: foundations and applications” of the Fondation WienerAnspach and the Interuniversity Attraction Poles 5 program of the Belgian Science Policy Office under the grant IAP P735 photonics@be, by the National Science Centre (NCN) grant 2014/14/E/ST2/00020, by DS Programs of the Faculty of Electronics, Telecommunications and Informatics, Gdańsk University of Technology, by the Swedish Research Council, by Knut and Alice Wallenberg foundation. This work was also made possible through the support of grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.
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R.R. initiated the research, proposed the Proposition 1 and provided the analytical proof of the Observation 2 and Proposition 3. D.G. provided the analytic proof of the Proposition 1, and studied the feasibility of the experiment under presence of realistic errors. P.M. implemented numerical simulations. P.H. contributed to the development of the main idea and theoretical results. S.M. and M.G. designed and carried out the experiment. S.M. performed the data analysis. M.B. supervised the experimental part. R.R. wrote the main text and S.M. and M.B. the experimental part. All the authors discussed the results and contributed to the final version of the manuscript.
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Ramanathan, R., Goyeneche, D., Muhammad, S. et al. Steering is an essential feature of nonlocality in quantum theory. Nat Commun 9, 4244 (2018). https://doi.org/10.1038/s41467018062555
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DOI: https://doi.org/10.1038/s41467018062555
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