Steering is an essential feature of non-locality in quantum theory
Ravishankar Ramanathan, Dardo Goyeneche, Sadiq Muhammad, Piotr Mironowicz, Marcus Grünfeld, Mohamed Bourennane, Paweł Horodecki
SSteering is an essential feature of non-locality in quantum theory
Ravishankar Ramanathan, ∗ Dardo Goyeneche,
2, 3, 4
Sadiq Muhammad, PiotrMironowicz,
2, 6
Marcus Gr ¨unfeld, Mohamed Bourennane, and Paweł Horodecki
2, 4 Laboratoire d’Information Quantique, Universit´e Libre de Bruxelles, Belgium Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre,Faculty of Mathematics, Physics and Informatics, University of Gda´nsk, Wita Stwosza 57, 80-308 Gda ´nsk, Poland Institute of Physics, Jagiellonian University, Krak´ow, Poland Faculty of Applied Physics and Mathematics, Gda ´nsk University of Technology, 80-233 Gda ´nsk, Poland Department of Physics, Stockholm University, S-10691, Stockholm, Sweden Department of Algorithms and System Modelling, Faculty of Electronics,Telecommunications and Informatics, Gda´nsk University of Technology, 80-233 Gda ´nsk, Poland (Dated: October 15, 2018)
Abstract.
A physical theory is called non-local when observers can produce instantaneous effectsover distant systems. Non-local theories rely on two fundamental effects: local uncertainty relationsand steering of physical states at a distance. In quantum mechanics, the former one dominates theother in a well-known class of non-local games known as XOR games. In particular, optimal quan-tum strategies for XOR games are completely determined by the uncertainty principle alone. Thisbreakthrough result has yielded the fundamental open question whether optimal quantum strategiesare always restricted by local uncertainty principles, with entanglement based steering playing norole. In this work, we provide a negative answer to the question, showing that both steering and un-certainty relations play a fundamental role in determining optimal quantum strategies for non-localgames. Our theoretical findings are supported by an experimental implementation with entangledphotons.
Introduction.
The uncertainty principle is a funda-mental feature of quantum theory, which postulatesthe existence of incompatible observables, the resultsof whose measurements on identically prepared sys-tems cannot be predicted simultaneously with certainty.Recently, the traditional formulation of uncertainty re-lations in terms of standard deviations and commuta-tors has been eschewed in favor of the entropic uncer-tainty relations [1] and the even more fundamental fine-grained uncertainty relations [2]. These fine-grained un-certainty relations are formulated in terms of the basicentities of the theory, namely the probabilities of particu-lar sets of outcomes for given sets of measurements, andare thus able to capture the uncertainty of these mea-surements in a more general manner than the entropicmeasures or the statistical standard deviations. More-over, the uncertainty bounds are expressed in a mannerindependent of the specific underlying quantum state,an advantage over the traditional formulation in termsof average values of commutators on fixed states. An-other fundamental feature of quantum theory is steer-ing, identified by Schr ¨odinger in [3]. This property de-termines, for two systems in a shared (entangled) state,which states can be prepared on one system by a mea-surement on the other. Quantum steering can be usedas a resource to generate ensambles of quantum systemsincompatible with a local hidden variable (LHV) model[4]. For two-qubit states, all states that Alice can steerare restricted to an ellipsoid within the Bloch sphere ofBob [5].The results of measurements on distant quantum sys-tems can be correlated in a way that defies classical local realistic description. This non-locality of quantum the-ory is evidenced in the violation of Bell inequalities byspatially separated quantum systems. Quantum corre-lations are restricted to some extent by the no-signalingprinciple, i.e., the measurement results cannot allow forsignaling between the distant locations. Nevertheless,there exist non-local correlations allowed by the no-signaling principle that cannot be realized in quantumtheory [6, 7].The fundamental question why quantum correlationsare non-local yet not as strong as allowed by the no-signaling principle is an intriguing one that has stimu-lated the formulation of many striking new information-theoretic principles. So far none of the known princi-ples has been able to capture the set of quantum cor-relations in its entirety [8], thus a comprehensive an-swer to this question is still lacking. The test-beds forthese principles are a special class of Bell inequalitiesbased on so-called quantum non-local games which ex-tract purely probabilistic aspects of the non-locality test,independent of the physical realization. Considerationof non-local games lead to a significant breakthrough in[2] where two fundamental concepts of quantum the-ory, the strength of non-local correlations and the uncer-tainty principle, were shown to be inextricably quanti-tatively linked with each other.Moreover it was shown that in a large class of non-local games for which optimal quantum strategies wereexplicitly known (the class of XOR games for which anexplicit characterization of the optimal quantum strat-egy was provided by Tsirelson [9]) these are not onlyjust linked, but one of them - uncertainty - fully deter- a r X i v : . [ qu a n t - ph ] O c t mines the non-locality of quantum theory with steeringplaying no role. An important question left open in [2]was whether such a phenomenon holds in general. If itdid, this would constitute a defining property of quan-tum mechanics: that something fully local (the uncer-tainty principle for a single party’s measurements) gov-erns something non-local (the Bell violation on a sharedsystem).The intriguing result of [2] is that while the degree ofnon-locality in any theory is generally determined by acombination of two factors - the strength of the uncer-tainty principle and the degree of steering allowed inthe theory, in quantum theory the degree of non-localityfor the well-known class of two-player XOR games ispurely determined by the strength of the uncertaintyprinciple alone. More precisely, [2] shows that in a two-party Bell scenario, the strength of non-locality in anytheory is determined by the uncertainty relations forBob’s measurements acting on the states that Alice cansteer to. On the other hand in quantum theory, for allXOR games (aka bipartite correlation Bell inequalities)[9], the states which Alice can steer to are identical to themost certain states, so that only the uncertainty relationsof Bob’s local measurements determine the outcome.In this paper, we show that the one-to-one correspon-dence between the uncertainty principle and the degreeof non-locality in quantum theory (referred hereafter asthe Uncertainty Principle - Quantum Game Value cor-respondence, or UP-QGV correspondence) observed forXOR games in [2] does not hold in general, by present-ing an explicit counter-example of a non-local game vi-olating the correspondence. We provide an intuitive ex-planation in terms of the Schrodinger-Hughston-Jozsa-Wootters theorem [10] for when the UP-QGV correspon-dence breaks down. To show that the game does nothave other optimal strategies that could obey the corre-spondence and to facilitate experimental testing of ourresult, we prove a self-testing 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 toshow that every two-party non-maximally entangledstate | ψ (cid:105) is the optimal state for a game G ψ for whichthe correspondence does not hold. The tradeoff existingbetween steering and uncertainty is conclusively shownby means of an experimental implementation, in whichthe steered states manifestly are seen to be distant fromthe maximally certain state even after the experimentalerrors are taken into account. ResultsUncertainty Principle - Quantum Game Value cor-respondence.
Let us first recall the precise correspon-dence between the fine-grained uncertainty relationsand the strength of non-locality established in [2]. Con-sider a two-player non-local game G , in which Alice and Bob receive questions x, y from respective input sets X , Yaccording to some input distribution π X,Y ( x, y ) . Theyreturn answers a, b from some output sets A , B, respec-tively. The winning constraint is specified by a predicate V ( a, b | x, y ) ∈ { , } . The success probability in the game ω s ( G ) is thus written as ω s ( G ) = max P A,B | X,Y ∈S (cid:88) x ∈ X y ∈ Y π X,Y ( x, y ) (cid:88) a ∈ A b ∈ B V ( a, b | x, y ) P A,B | X,Y ( a, b | x, y ) , (1)where S refers to a set of conditional probability dis-tributions (boxes) P A,B | X,Y . One considers boxes takenfrom sets C , Q , NS corresponding to the set of classical,boxes and general no-signaling boxes, with correspond-ing values ω c ( G ) , ω q ( G ) and ω ns ( G ) respectively. Onemay also restrict attention to the free games for whichthe 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., thevalue obtained from those boxes for which there ex-ists a state ρ on a Hilbert space H d and sets of mea-surement operators (POVMs) { M xa } , { M yb } such that P A,B | X,Y ( a, b | x, y ) = Tr ( ρM xa ⊗ M yb ) . The idea in [2] isto rewrite the game expression in Eq.(1) as (cid:88) x,a π X ( x ) P A | X ( a | x ) (cid:88) y,b π Y | X ( y | x ) V ( a, b | x, y ) P B | Y,X,A ( b | y, x, a ) . (2)Let P B | Y,X,A ( b | y, x, a ) ˆ σ Ba | x be Bob’s marginal probabilitydistribution when his state is steered by Alice to ˆ σ Ba | x .Now, observe that for each ( x, a ) , the expression (cid:88) y,b π Y ( y ) V ( a, b | x, y ) P B | Y,X,A ( b | y, x, a ) ˆ σ Ba | x ≤ ξ ( x,a ) B , (3)constitutes a fine-grained uncertainty relation on Bob’ssystem with ξ ( x,a ) B denoting the maximum over all possi-ble states ˆ σ Ba | x of Bob’s system. When the optimal value ξ ( x,a ) B equals unity, we refer to the corresponding un-certainty relation as trivial, i.e., while the probabilitiesare bounded below unity for some states, there existstates for which the outcomes (for each of Bob’s inputs y ) can be fixed with certainty. On the other hand, when ξ ( x,a ) B < , we infer that one cannot obtain a measure-ment outcome with certainty for all measurements si-multaneously.An example situation of the uncertainty relation isshown in Fig. (1) and steering to the maximally certainstates is exemplified in Fig. (2).Let { ˜ σ Ba | x } denote the set of states of Bob’s system thatachieve the maximum value ξ ( x,a ) B of the uncertainty ex-pressions for each ( x, a ) for given optimal measurement FIG. 1. The uncertainty principle illustrated by randomlyoriented polarizers. Input state | ψ (cid:105) is prepared via a polarizer(Pol) oriented at φ/
2, (which corresponds to orientation φ on the Bloch sphere). A reflecting mirror M is randomlyinserted with probability 0 < p < Q (0), and another onerotated by θ (0 < θ < π ) measures Q ( θ ), such that probabilitythat a photon is transmitted, is P ( transmission ) = (1 − p ) Q (0) | ψ (cid:105) + pQ ( θ ) | ψ (cid:105) and it is upper bounded by ξ ( θ, p ).FIG. 2. (a) The Bloch sphere representation of the mea-surement situation. The state | ψ (cid:105) of the polarized photonis represented by ˆ v , while the projectors Q (0) and Q ( θ ) corre-spond to unit vectors ˆ n and ˆ n θ respectively, and m is give by m = (1 − p )ˆ n + p ˆ n θ . The bound on the probability of trans-mission ξ ( θ, p ) is obtained from the vector m , ξ ( θ, p ) = | m | .The uncertainty relation defined by the probability of trans-mission ( P (transmission) ≤ ξ ( θ, p ) < | ψ (cid:105) with Bloch vector ˆ v parallel to m . (b) The situation whenAlice tries to steer to the least uncertain state. It is achievedonly when ˆ v (cid:107) m . operators { M yb } . The question then arises whether Aliceis able to steer Bob’s system to these maximally certainstates and thus achieve the bound set by the uncertaintyprinciple for the game G . We are thus lead to considerthe effect of steering. For any bipartite no-signalingbox shared by Alice and Bob, any measurement on Al-ice’s system creates a set of single-party boxes on Bob’sside { P B | Y ( b | y ) x,a } = { P B | Y,X,A ( b | y, x, a ) } . We say thatwith this particular input-output pair ( x, a ) , Alice hassteered the state of Bob’s system to the set of boxes { P B | Y ( b | y ) x,a } with probability P A | X ( a | x ) .We see therefore in Eq.(2) the separation of the gameexpression into two components, one where Bob’s (opti-mal) measurements define a set of uncertainty relationsone for each ( x, a ) and a second component wherein Al-ice tries to steer Bob’s system to the maximally certainstates for these relations. The strength of non-localityin any theory is thus seen as a trade-off between thestrength of the uncertainty relations and the amount ofsteering allowed in the theory.In [2], it was shown that for the well-known class oftwo-player XOR games for which the optimal measure-ments are known, the strength of non-locality is purelydetermined by the uncertainty relation with steering notconstraining the value in any way. In other words, theoptimal measurements and the state share the propertythat in all these known instances, Alice is able to steerBob’s system to the most certain states correspondingto the set of uncertainty relations of his system for eachinput-output pair ( x, a ) .Note that the restriction to non-local games ratherthan all Bell inequalities is crucial for the correspon-dence to be meaningful. Indeed, for general Bell in-equalities, where one is allowed to scale the Bell expres-sion with arbitrary multiples of the normalization andno-signaling equalities, it is possible to show that thecorrespondence can always be made to hold up to ar-bitrary high accuracy. This general observation inspiredby recent results in [11] is explained in detail in the Sup-plementary Note 3.Two-player XOR games are non-local games with anarbitrary number of inputs and binary outputs, wherethe winning constraint of the game only depends on the XOR of the parties’ outputs. Building on a breakthroughtheorem by Tsirelson [9], it was shown in [12, 13] thatthe quantum value of two-party
XOR games can be cal-culated precisely by means of a semi-definite program,and the Tsirelson theorem allows to recover the optimalstate and measurement operators for any such game. Ineffect, apart from the pseudo-telepathy games [14] and afew other isolated instances, these are the games wherethe optimal measurements are known and for whichthe relation between the uncertainty principle and non-locality was established in [2]. The difficulty in estab-lishing the relationship for general non-local games isdue to the fact that the problem of finding the quan-tum strategy of arbitrary non-local games is hard [15];one usually uses a hierarchy of semi-definite programs[16, 17] which converge to the true quantum value.Note that it is natural to ask about the relation of thesteering-type representation of the formula (2) to thewell-known Schrodinger-Hughston-Jozsa-Wootters the-orem which defines all the ensembles Alice may steer to.It is tempting to expect that the result [2] is due to ap-plication of that theorem. This is however not the casebecause of the crucial fact it is not guaranteed that themaximally certain states together with the optimal lo-cal probabilities P A | X ( a | x ) obey the no-signaling condi-tion of the SHJW theorem (see Supplementary Note 2for more discussion), viz. (cid:88) a P A | X ( a | x )ˆ σ B a | x = (cid:88) a P A | X ( a | x (cid:48) )ˆ σ B a | x (cid:48) = ˆ σ B = tr A ˆ σ AB . (4) Counter-examples to the correspondence.
Let usnow exhibit an example of a non-local game for whichthe UP-QGV correspondence does not hold, i.e., onewhere the optimal quantum state and measurements aresuch that Alice is unable to steer Bob’s system to themaximally certain state for each ( x, a ) . Before we pro-ceed to the counter-example, let us mention that it ispossible that the optimal quantum value of a non-localgame can be achieved with different sets of states andmeasurement operators (even going beyond a trivialunitary equivalence), therefore one must check whetherthe relation could hold for at least one optimal quantumstrategy. Thus, in order to give a counter-example to theUP-QGV correspondence, it is necessary to prove thatthe relationship does not hold for all optimal quantumstrategies for the game. We achieve this requirement byproving a self-testing property of the counter-example,i.e., that up to unitary equivalences there is a uniquestate and sets of measurements that achieves the opti-mal value of the game.We consider the Bell scenario B (2 , , of two parties,each performing one of two measurements and obtain-ing one of two outputs. The Bell inequality correspond-ing to the game denoted G (7) is explicitly given by
14 [ P (0 , | ,
0) + P (1 , | ,
0) + P (0 , | ,
1) + P (1 , | , P (0 , | ,
0) + P (1 , | ,
0) + P (0 , | , ≤ , (5)where we have assumed that each party chooses theirinputs uniformly, i.e., π X ( x ) = π Y ( y ) = for x, y ∈{ , } so that π X,Y ( x, y ) = and the classical bound is ω c ( G (7) ) = . The optimal strategy for the game G (7) violates the UP-QGV correspondence (the proof of thefollowing Proposition 1 is given in the SupplementaryNote 1). Proposition . The optimal quantum strategy for thegame G (7) (achieving ω q ( G (7) ) ≈ . ) violates the un-certainty principle - quantum game value correspon-dence, i.e., Alice is unable to steer Bob’s system to themaximally 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 ( x = 0 , a = 0) → P B | Y ( b = 0 | y = 0) + P B | Y ( b = 1 | y = 1) ≤ ξ (0 , B ( x = 0 , a = 1) → P B | Y ( b = 1 | y = 0) + P B | Y ( b = 0 | y = 1) ≤ ξ (0 , B ( x = 1 , a = 0) → P B | Y ( b = 1 | y = 0) + P B | Y ( b = 1 | y = 1) ≤ ξ (1 , B ( x = 1 , a = 1) → P B | Y ( b = 0 | y = 0) ≤ , (6)where the uncertainty bounds are ξ (0 , B = ξ (0 , B ≈ . ,and ξ (1 , B ≈ . . The optimal state and measurementsachieving ω q ( G (7) ) ≈ . are given in the Supplemen-tary Note 4, where it is shown explicitly that while for ( x = 1 , a = 0) Alice steers Bob’s system to the maxi-mally certain state, for ( x = 0 , a = 0) and ( x = 0 , a = 1) Alice is unable to steer Bob’s system to the maximallycertain states of the corresponding (non-trivial) uncer-tainty relations. Further, the trivial uncertainty relationfor ( x = 1 , a = 1) also fails to be saturated. The value ω q ( G (7) ) achievable in quantum theory is thus strictlylower than what is allowed by the uncertainty princi-ple, and therefore the game G (7) violates the UP-QGVcorrespondence.Let us now see why the UP-QGV correspondencebreaks down for the particular game G (7) , and estab-lish conditions for the correspondence to hold. To doso, we examine the assemblage { P A | X ( a | x ) , ˜ σ a | x } of max-imally certain states. For the game G (7) it can be read-ily verified that the corresponding assemblage of max-imally certain states does not obey the no-signaling re-lation Eq.(31), so the SHJW theorem does not guaran-tee the existence of a shared entangled state and mea-surements on Alice’s side that would prepare the cor-responding maximally certain states on Bob’s system.Formally, we may make the observation (which followsfrom well-known demands on steerability [4]) that theUP-QGV correspondence holds when the probabilities P A | X ( a | x ) together with the maximally certain states ˆ σ B a | x obey the no-signaling constraint in Eq.(31). Observation . The uncertainty principle determines thenon-locality of quantum theory whenever the maxi-mally certain states ˆ σ B a | x of one party’s measurementstogether with the optimal local probabilities { P ( a | x ) } of the other party, forms a no-signaling assemblage, i.e.,when { P ( a | x ) , ˆ σ B a | x } obeys Eq.(31).The game G (7) shows that this condition is not alwaysobeyed by the maximally certain states. While it ap-pears at present an intractable problem to characterizethe set of all games where the UP-QGV correspondencebreaks down, we can nevertheless show that the game G (7) is not singular in this respect. Indeed, every two-party non-maximally entangled state | ψ (cid:105) (i.e. a state notof the form √ d (cid:80) di =1 | i, i (cid:105) for some d > ) is the optimalstate for a game G ψ for which the correspondence doesnot hold. This is captured in the following proposition(whose proof is given in the Supplementary Note 4). Proposition . For any two-party entangled, but non-maximally entangled, state | ψ (cid:105) ∈ C d ⊗ C d for arbitraryHilbert space dimension d , there exists a game G ψ forwhich the optimal quantum strategy is given by suit-able measurements on | ψ (cid:105) , and such that the correspon-dence between the uncertainty principle and the quan-tum game value does not hold for G ψ .An interesting open question is whether the condi-tions in Observation 2 are met for all unique games[18] which are a natural generalization of XOR gamesto a larger output alphabet. Also interesting is to findwhether the correspondence holds for all games wherethe optimal strategy involves a maximally entangledstate, which would highlight that in the foundationalprogram of seeking an information-theoretic principlebehind the strength of quantum non-local correlations,one must go further than the correlations exhibited bythe maximally entangled states alone. Experimental Implementation.
In our experiment,the physical qubits are single photon polarization statesand the computational basis corresponds to the horizon-tal (H) and vertical (V) polarization i.e. | H (cid:105) ≡ | (cid:105) and | V (cid:105) ≡ | (cid:105) . To achieve the maximal violation of the Bellinequality given in (5), we used the following polariza-tion entangled two-photon state, | Ψ (cid:105) = 0 . | HH (cid:105) + 0 . | HV (cid:105) (7) +0 . | V H (cid:105) − . | V V (cid:105) . This state is produced in two steps. First, we gener-ate entangled photon pairs via spontaneous parametricdown-conversion (SPDC) [19]. Then at the second step,these entangled pairs are transformed to the requiredstate (45) by local rotations [20]. The polarization mea-surement on Alice and Bob’s sides are performed by an-alyzers consisting of wave-plates, polarizing beam split-ters (PBS) and single photon detectors. An FPGA basedtiming system is used to collect data. The experimentalsetup is outlined in Fig. 3 and its detailed descriptioncan be found in the Supplementary Note 5.The Fidelity, F = (cid:104) Ψ | ρ exp | Ψ (cid:105) , of the experimentallyprepared state ρ exp with respect to (45) was . ± . . With this state and using the settings | φ ± x (cid:105) =cos γ x | H (cid:105) ± sin γ x | V (cid:105) , where γ = π/ and γ = 4 . ,we obtained the experimental Bell inequality violation ω q ( G (7) ) = 0 . ± . . Note that the theoreticalquantum and classical bounds are . and . re-spectively. The fidelity of the four maximally certainstates v , v − , v and v − are given by F = 0 . ± . , F − = 0 . ± . , F = 0 . ± . and F − = 0 . ± . , respectively. Here, v ij is theleast uncertain state associated to Alice measurement i FIG. 3. (Color online) Preparation and measurement stagesfor the state (45) . A UV pump laser at 390 nm was focusedonto two β -barium borate (BBO) crystals placed in cross-configuration to produce photon pairs emitted into two spa-tial modes “ a ” and “ b ” through type-I SPDC process. Anyspatial, temporal or spectral distinguishability between thephotons is removed via a pair of Y V O crystals, narrow-bandwidth filters (F) and coupling into single mode fibers(SMF). Then, the photons in each mode are rotated througha half wave-plate to get the desired state (45) . For measure-ment, Alice and Bob uses polarization analyzers consisting ofa half wave-plate (HWP), a quarter wave-plates (QWP), apolarizing beam splitter (PBS) and D i ( i = { , , , } ) singlephoton avalanche photo-diodes. having outcome j . In Fig. 4 we represent the least un-certain states (blue) and the states m ij that Alice is ableto steer (red) (see Supplementary Notes 3 and 5 for de-tails related to theoretical and experimental results, re-spectively). Experimental errors determine eight conesin Bob’s Bloch sphere, whose apertures are the largestpossible, according to the experimentally obtained er-rors.For error estimation, we have considered the errororiginated from the measurement side only, as the erroron the preparation side will just shift the experimentallyprepared state away from the desired state and there-fore will be apparent from the reported state fidelity orthe value of Bell violation. Further details are given inSupplementary Note 8.We note that the experimental realization is notstrictly required for the case of the paper. However, itis fundamental to note that the breakdown in the cor-respondence between the two major aspects of quan-tum theory is not a trivial one that would be washedout under inevitable experimental error, since the cor-respondence was only considered for the optimal quan-tum value. As such, it is of interest to find that evenwith current experimental technology, one can achievesufficient experimental fidelities to make the case of thepaper, apart from serving as one of the first experiments FIG. 4. (color online). Experimental results. Least uncertainstates v (red) and states m that Alice is able to steer (blue).Cones show experimental errors originating from statistics(Poissonian) and systematic due to limited precision of thesettings and non-ideal components. The experimental resultsillustrate that steering to the maximally certain state is notpossible, as cones associated to v and m do not intersect. to self-test a non-maximally entangled state. Finally,we remark that the experiment was not performed ina loophole-free manner, as such it would be interestingto check the expectation that the same conclusions alsohold in a loophole-free Bell test such as recently done in[21–23]. Discussion
In this paper, we have shown that the intriguing cor-respondence between the uncertainty principle and thequantum game value, proven for the very importantclass of two-player XOR games in [2], does not hold forgeneral non-local games. In order to prove this result wehave put forth an intuitive argument to identify whenthe correspondence holds in terms of the SHJW theo-rem.Many interesting questions remain open. Firstly, notethat the CHSH inequality is the only facet-defining in-equality in the Bell scenario B (2 , , and the non-localgame we consider constitutes a lower-dimensional faceof the classical polytope. It is of interest to find whetherthe correspondence holds for non-local games that aretight Bell inequalities (facets of the classical polytope),or for games where the optimal strategy involves a max-imally entangled state. Secondly, while the uncertaintyrelations always provide a bound on the quantum value,it is now an open question to characterize the class ofgames for which this bound is saturated and more inter-estingly those for which the gap is extremal. Data availability statement.
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R.R. thanks Nicolas Brunnerand Antonio Ac´ın for useful discussions. The paper issupported by ERC AdG grant QOLAPS, by Foundationfor Polish Science TEAM project co-financed by theEU European Regional Development Fund, by theproject ”Causality in quantum theory: foundationsand applications” of the Fondation Wiener-Anspachand the Interuniversity Attraction Poles 5 program ofthe Belgian Science Policy Office under the grant IAPP7-35 photonics@be, by the National Science Centre(NCN) grant 2014/14/E/ST2/00020, by DS Programsof the Faculty of Electronics, Telecommunications andInformatics, Gda ´nsk University of Technology, by theSwedish Research Council, Knut and by the Alice Wal-lenberg foundation. This work was also made possiblethrough the support of grant from the John TempletonFoundation. The opinions expressed in this publicationare those of the authors and do not necessarily reflectthe views of the John Templeton Foundation.
Author Contributions.
R.R. initiated the research,proposed the Proposition 1 and provided the analyticalproof of the Propositions 2 and 3. D.G. provided theanalytic proof of the Proposition 1, and studied thefeasibility of the experiment under presence of realisticerrors. P.M. implemented numerical simulations. P. H.contributed to the development of the main idea andtheoretical results. S. M and M. G designed and carriedout the experiment. S.M. performed the data analysis.M. B. supervised the experimental part. R.R. wrote themain text and S.M. and M.B. the experimental part. Allthe authors discussed the results and contributed to the final version of the manuscript.
Competing interests.
The authors declare no compet-ing interests.
Supplementary Information:Steering is an essential feature ofnon-locality in quantum theory
Supplementary Note 1 - Formal Proofs
Proposition 1.
The optimal quantum strategy for thegame G (7) (achieving ω q ( G (7) ) ≈ . ) violates the un-certainty principle - quantum game value correspon-dence, i.e., Alice is unable to steer Bob’s system to themaximally certain states and vice versa.In the proof, we will use the Lemmas 1 and 2 statedbelow. Lemma . In the Bell scenario B ( n, , for anynumber of parties n , all extreme boxes P ( a, b | x, y ) of thequantum set Q ( n, , can be realized by measuring n qubit pure states with projective observables. Lemma
ORDAN ’ S LEMMA [2]) . Any two binary ob-servables A and A acting on a finite-dimensionalHilbert space C n with n ≥ can be simultaneouslyblock-diagonalized into × and × blocks.Lemma 1 allows us to find ω q ( G (7) ) by optimizingover projective measurements on pure two qubit states.Let ( | ψ (cid:105) , { M x } , { M y } ) be a qubit strategy for the game G (7) with M x = (cid:88) a =0 ( − a Π xa = (cid:18) e iα x e − iα x (cid:19) (8) M y = (cid:88) b =0 ( − b Π yb = (cid:18) e iβ y e − iβ y (cid:19) , (9)over the computational basis {| (cid:105) , | (cid:105)} , where withoutloss of generality α = β = 0 and α , β ∈ [ − π, π ] . Here Π xa and Π yb are the projectors associated to the values , obtained by diagonalizing M x , M y respectively. We canwrite the Bell operator in terms of the Π xa and Π yb as B ( G (7) ) = (cid:88) x,y ∈{ , } π AB ( x, y ) (cid:88) a,b ∈{ , } V G (7) ( a, b | x, y )Π xa ⊗ Π yb , (10)with π AB ( x, y ) = π A ( x ) π B ( y ) and π A ( x ) = π B ( y ) = for each x, y ∈ { , } . Our task is to find the op-timal α , β that lead to the maximum eigenvalue ofthe operator ˜B ( G (7) ) = 4 B ( G (7) ) . The eigen-equation det[ ˜B ( G (7) ) − λ ] = 0 simplifies to λ [ −
30 + λ (33 + 2 λ ( λ − λ − λ −
1) [cos ( α ) − cos ( α ) cos ( β ) + cos ( β )]+ 9 − sin ( α ) sin ( β ) = 0 . (11) To find the optimum quantum value λ max , we use theKKT conditions, i.e., we investigate the expression for λ in terms of α and β in four sectors.Case I: α = β = π . In this case, the eigen-equation(11) directly solves to λ ( I ) = 3 corresponding to the(classical) game value of .Case II: β = π , optimize over α . In this case theeigen-equation simplifies to ( λ − λ + 3)(1 − λ + λ + cos ( α )) = 0 , (12)which has the four solutions λ = 1 , , (cid:0) ± √ − α (cid:1) , (13)so that we obtain the maximum value of λ in this sectorto be , corresponding to a game value of .Case III: α = π , optimize over β . As in the previouscase II, the eigen-equation simplifies to SupplementaryEquation (12) giving the maximum value of λ = 3 or agame value of .Case IV: optimize over α , β . Here we have ∂λ∂α =0 and ∂λ∂β = 0 . We implicitly differentiate the eigen-equation (11) with respect to α and set ∂λ∂α = 0 to get ( λ − λ −
1) sin ( α )( − α )) = sin (2 α ) sin ( β ) . (14)Similarly implicit differentiation of SupplementaryEquation (11) with respect to β and setting ∂λ∂β = 0 gives (3 − λ )( λ − − cos ( α )) = sin ( α ) sin (2 β ) . (15)Solving Eqs. (14) and (15) yields that either α = 0 giv-ing λ = 3 or that the following relation holds betweenthe optimal α and β cos ( α ) = cos ( β ) = ⇒ α = β , (16)since α , β ∈ [ − π, π ] . We can now simplify the eigen-equation (11) setting α = β . Implicit differentiationof the resulting expression with respect to α and againsetting ∂λ∂α = 0 gives the λ in terms of the α as (cid:2) ( − λ ) + 2 cos ( α ) + cos (2 α ) (cid:3) ×× cos (cid:16) α (cid:17) sin (cid:16) α (cid:17) = 0 , ⇒ λ = 2 ± (cid:112) − α ) − cos (2 α ) . (17)Substituting Supplementary Equation (17) back into theeigen-equation given by Supplementary Equation (11)and solving gives the optimal α α = 2 arctan (cid:115) η + 2 η − η , (18)with η = (cid:2) (43 + 9 √ (cid:3) . Thus the optimal value λ max is given by Supplementary Equation (17) with α in Sup-plementary Equation (18) as ω q ( G (7) ) = λ max ( ˜B ( G (7) ))4= 1108 (35 + (15740 − √ + 2 (3935 + 243 √ ) ≈ . . (19)We thus have the optimal quantum strategy for game G (7) given by the measurement operators in Supple-mentary Equation (8) with α = β given by Sup-plementary Equation (18). The corresponding optimalquantum (qubit) state is the maximal eigenvector of B ( G (7) ) with these measurements.Applying Jordan’s Lemma 2 to Alice’s observables M x for x = 0 , , we obtain that M x = ⊕ i M x ( i ) = (cid:88) i M x ( i ) ⊗ | i (cid:105)(cid:104) i | , (20)where i labels the block index and each M x ( i ) is a × block (the × blocks can be enhanced by adding addi-tional dimensions without loss of generality). This givesa basis in which the Hilbert space of Alice’s observables H A can be written as H A = C ⊗ H (cid:48) A with H (cid:48) A de-noting the Hilbert space with basis {| i (cid:105)} and C beinga qubit space. A similar structure exists for the Hilbertspace of Bob’s observables H B . The crucial part of theabove structure is that a measurement of the block index i commutes with both Alice’s observables M x , so onecan consider a general strategy in which Alice measures | i (cid:105)(cid:104) i | first, and similarly Bob measures the block indexfor his observables M y . This reduces the whole prob-lem to the case where H A = C and H B = C . Withinthe qubit subspace, we know by Lemma 1 that projec-tive measurements on pure two qubit states achieve theextremal points, and as we have shown any qubit strat-egy is equivalent, up to local unitaries to the strategygiven by the measurements in Supplementary Equation(8).Let us now examine the uncertainty relations corre-sponding to the game G (7) for each input-output pair ( x, a ) of Alice. In what follows, we simplify notation byexplicitly specifying numerical values to avoid cumber-some analytical expressions, however note that all the results are fully analytical. ( x, a ) = (0 , → P ( b = 0 | y = 0) + P ( b = 1 | y = 1) ≤ ξ (0 , B ( x, a ) = (0 , → P ( b = 1 | y = 0) + P ( b = 0 | y = 1) ≤ ξ (0 , B ( x, a ) = (1 , → P ( b = 1 | y = 0) + P ( b = 1 | y = 1) ≤ ξ (1 , B ( x, a ) = (1 , → P ( b = 0 | y = 0) ≤ , (21)where we have used π B ( y = 0) = π B ( y = 1) = , andthe uncertainty bounds are ξ (0 , B = ξ (0 , B = 14 e − iα (cid:16) e iα − (cid:112) − e iα ( − e iα ) (cid:17) ≈ . , (22)and ξ (1 , B = 14 e − i α (cid:16) e i α (cid:17) ≈ . , (23)with the optimal α given in Supplementary Equation(18).Case I: (x,a) = (0,0). The maximally certain state for ( x, a ) = (0 , is | ψ (0 , (cid:105) A = 1 √ (cid:32) (cid:112) − e iα (1 − e iα ) (1 − e iα ) | (cid:105) + | (cid:105) (cid:33) . (24)Projecting onto the optimal state with the optimal pro-jector Π x =0 a =0 = | + (cid:105)(cid:104) + | , we see that Alice only manages tosteer Bob’s system to the state | ˜ ψ (cid:105) = 1 √ (cid:0) e − . i | (cid:105) + | (cid:105) (cid:1) , (25)which achieves value . < ξ (0 , B for the correspond-ing uncertainty relation. Note that here we have statedthe expression numerically simply to avoid the cumber-some notation associated with the exact analytical ex-pression.Case II: (x,a) = (0,1). The maximally certain state for ( x, a ) = (0 , is | ψ (0 , (cid:105) A = 1 √ (cid:32) (cid:112) − e iα (1 − e iα ) ( − e iα ) | (cid:105) + | (cid:105) (cid:33) . (26)Projecting onto the optimal state with the optimal pro-jector Π x =0 a =1 = |−(cid:105)(cid:104)−| , we see that Alice only manages tosteer Bob’s system to the state | ˜ ψ (cid:105) = 1 √ (cid:0) − e − . i | (cid:105) + | (cid:105) (cid:1) , (27)which achieves value . < ξ (0 , B for the correspond-ing uncertainty relation.0Case III: (x,a) = (1,0). The maximally certain state for ( x, a ) = (1 , is | ψ (1 , (cid:105) A = 1 √ (cid:16) − e i α | (cid:105) + | (cid:105) (cid:17) . (28)Projecting onto the optimal state with the projector Π x =1 a =0 , we see that Alice manages to steer Bob’s systemto the maximally certain state in Supplementary Equa-tion (28) for this uncertainty relation.Case IV: (x,a) = (1,1). The maximally certain statefor ( x, a ) = (1 , corresponds to the projector Π y =0 b =0 = | + (cid:105)(cid:104) + | . Projecting onto the optimal state with the projec-tor Π x =1 a =1 , we see that Alice only manages to steer Bob’ssystem to the state | ˜ ψ (cid:105) = 1 √ (cid:0) e . i | (cid:105) + | (cid:105) (cid:1) , (29)which achieves value . < ξ (1 , B = 1 for this uncer-tainty relation.We thus see that for the game G (7) , Alice is unable tosteer Bob’s system to the maximally certain states evenfor the non-trivial uncertainty relations correspondingto ( x, a ) = (0 , and ( x, a ) = (0 , . In order to for-mally complete the argument for every optimal quan-tum strategy, we note that in the general case, one ob-tains a mixture of uncertainty relations over the out-comes i of the block index measurement by Alice andBob. Since the uncertainty relation fails to be saturatedin each block, this implies that the same holds true alsoin the convex mixture of uncertainty relations.The quantum value of the game ω q ( G (7) ) is thus lowerthan what could have been achieved if the non-localityof the theory were bounded by the uncertainty princi-ple alone. The same holds true from the point of viewof Bob steering Alice’s system. While Bob is able tosteer Alice’s system to the maximally certain state for ( y, b ) = (1 , , he is unable to do so for the non-trivialuncertainty relations corresponding to ( y, b ) = (0 , and ( y, b ) = (0 , as well as for the trivial uncertainty rela-tion for ( y, b ) = (1 , . Thus, this example proves that thenon-locality of quantum theory is not determined by theuncertainty principle alone, and steering plays a definiterole. Supplementary Note 2 - Relation with theSchrodinger-Hughston-Jozsa-Wootters theorem
Let us remark upon a curious feature of the rewritingin Eq.(2) of the main text with regard to steerability ofquantum systems. Consider a set of measurement oper-ators M xa on Alice’s side, i.e., positive operators M xa ≥ satisfying (cid:80) a M xa = . Such a collection represents apositive-operator valued measure (POVM) for each x .For any fixed bipartite quantum state ˆ σ AB , every mea-surement on Alice’s side gives rise to an assemblage { σ Ba | x } a,x = { P A | X ( a | x ) , ˆ σ B a | x } a,x . Here σ B a | x = tr A [( M xa ⊗ )ˆ σ AB ] , (30)are the conditional (unnormalised) states of Bob’s sys-tem prepared by Alice’s measurement. We have that (cid:88) a P A | X ( a | x )ˆ σ B a | x = (cid:88) a P A | X ( a | x (cid:48) )ˆ σ B a | x (cid:48) = ˆ σ B = tr A ˆ σ AB , (31)for every x, x (cid:48) ∈ X , in order to obey the no-signalingprinciple; i.e., without the knowledge of Alice’s out-come a , Bob’s state is independent of the measure-ment choice x . The well-known Schr ¨odinger-Hughston-Jozsa-Wootters (SHJW) theorem [3, 4] shows that everyassemblage { P A | X ( a | x ) , ˆ σ B a | x } a,x satisfying Supplemen-tary Equation (31) has a quantum realization as in Sup-plementary Equation (30) for some quantum state ˆ σ AB and for some set of measurement operators { M xa } . Now,the set of states { ˜ σ B a | x } achieving the maximum value ofthe uncertainty relations together with the optimal prob-abilities { P A | x ( a | x ) } forms an assemblage. One mightthen wonder whether the result of [5] is a direct con-sequence of the SHJW theorem, since Alice might steerto the assemblage corresponding to the maximally cer-tain states. However, maximally certain states togetherwith the optimal local probabilities P ( a | x ) do not guar-antee that the no-signaling principle (31) holds. Thus,the UP-QGV correspondence found in [5] is a non-trivialproperty of the optimal states and measurements. It wasposed as an open question in [5] whether the correspon-dence holds for all non-local games. Supplementary Note 3 - Games vs. General Bellinequalities
As observed in the main text, it is crucial for the cor-respondence between uncertainty relations and optimalquantum strategy to be meaningful that the form of theBell expressions is restricted, for instance to the form ofnon-local games as in the paper [5]. It is readily seenthat if one allows an arbitrary freedom in rewriting theBell expressions up to normalization and no-signalingequality constraints as suggested in [6], then one can al-ways find a form of the Bell expression where the cor-respondence holds approximately, up to an arbitrarilysmall error. For instance, consider the following Bell ex-pression, where the game has been supplemented withmultiples λ x,y of the normalization constraints for inputpairs ( x, y ) ∈ X × Y (cid:88) x,y π A,B ( x, y ) (cid:88) a,b V ( a, b, x, y ) P A,B | X,Y ( a, b | x, y )+ (cid:88) x,y λ x,y (cid:88) a,b P A,B | X,Y ( a, b | x, y ) ≤ β c . (32)When λ x,y factorize as λ x,y = π A ( x ) β y , the resultingfine-grained uncertainty relation for fixed ( x, a ) is given1as (cid:88) y,b π B ( y | x ) V ( a, b, x, y ) P B | Y,X,A ( b | y, x, a ) + (cid:88) y β y ≤ ˜ ξ x,a , (33)where we have used (cid:80) b P B | Y,X,A ( b | y, x, a ) = 1 for all y in the second term. Now clearly, the absolute valueof the multipliers β y need to be bounded at least as | β y | ≤ to be comparable with π B ( y | x ) . Similar consid-erations hold also for the multipliers associated to theno-signaling constraints. Failing such restrictions, onemight always choose appropriately large (in compari-son with π B ( y | x ) ) β y that lead to the saturation of ˜ ξ x,a inthe uncertainty relation, up to an arbitrary small devia-tion. Even otherwise, the artificial addition of normal-ization and no-signaling constraints which are satisfiedby all boxes in the set, leads to the question whether theresulting saturation of the uncertainty relations is intrin-sic to the non-local correlations that maximally violatethe inequality. To avoid such mathematical sleight ofhand (which is also inherent in questions such as that ofunbounded violation of Bell inequalities [7], the uniquegames conjecture [8], etc.) we follow [5] in restricting tonon-local games, i.e., where P A,B | X,Y ( a, b | x, y ) appear inthe Bell expression only with non-negative coefficientsand all non-zero coefficients are equal (to π A,B ( x, y ) ) fora fixed input pair ( x, y ) . Note that this difference be-tween non-local games and general Bell inequalities hasalso been noted previously in [9]. Supplementary Note 4 - Constructingcounter-example games for all non-maximallyentangled states
In the previous sections, we have seen an example ofa non-local game with the optimal state being a non-maximally entangled two-qubit state, for which the UP-QGV correspondence breaks down. In this section, weprove that this is not a one-off instance, indeed for everynon-maximally entangled two-qudit state | ψ (cid:105) ∈ C d ⊗ C d (for an Hilbert space of arbitrary dimension d ), one canconstruct a two-player game G ψ such that | ψ (cid:105) is optimalfor G ψ and such that the UP-QGV correspondence doesnot hold for G ψ . We present the construction in this sec-tion as the proof of the following Proposition from themain text. The construction we use resembles that usedby Coladangelo et al. in [11] to show that all bipartitepure entangled states can be self-tested. Proposition 3.
For any two-party entangled, but non-maximally entangled, state | ψ (cid:105) ∈ C d ⊗ C d for arbitraryHilbert space dimension d , there exists a game G ψ forwhich the optimal quantum strategy is given by suit-able measurements on | ψ (cid:105) , and such that the correspon-dence between the uncertainty principle and the quan-tum game value does not hold for G ψ . Proof.
The first step in the construction is to show thestatement for all entangled, but non-maximally entan-gled, two-qubit states; i.e., states of the form | ψ θ (cid:105) = cos θ | (cid:105) + sin θ | (cid:105) , (34)with θ ∈ (0 , π ) . To do this, we use a slightly differ-ent game than G (7) , namely the tilted CHSH inequal-ity of [10], which we reformulate as a game. In the tiltedCHSH game CHSH tilt , Alice and Bob each receive inputs x ∈ { , } and y ∈ { , , } with probabilities π X (0) = β β , π X (1) = β , and π Y | X (0 |
0) = π Y | X (1 |
0) = β , π Y | X (2 |
0) = β β , π Y | X (0 |
1) = π Y | X (1 |
1) = , π Y | X (2 |
1) =0 , with β = √ θ , and return binary answers a, b ∈ { , } . The winning constraints for the game arethe same as the usual CHSH game for x, y ∈ { , } , i.e., V ( a, b | x, y ) = 1 if a ⊕ b = x · y , while for x = 0 , y = 2 , V ( a, b | x, y ) = 1 if a = 0 . The Bell expression for thegame is thus given by
14 + β (cid:88) x,y =0 , P ( a ⊕ b = x · y | x, y )+ β β P ( a = 0 | , ≤ β β . (35)The classical value of the game is ω c ( CHSH tilt ) = β β .The quantum value of the game is ω q ( CHSH tilt ) = + √ β β and is achieved when the parties perform thefollowing measurements A x , B y on the state | ψ θ (cid:105) : A = σ z , A = σ x ,B = cos µσ z + sin µσ x , B = cos µσ z − sin µσ x , B = σ z , (36)with µ = arctan (sin 2 θ ) .The corresponding uncertainty relations are given as ( x, a ) = (0 , → P B | Y,X,A (0 | , (0 , P B | Y,X,A (0 | , (0 , βP B | Y,X,A (0 | , (0 , βP B | Y,X,A (1 | , (0 , ≤ (2 + β ) ξ (0 , B ( x, a ) = (0 , → P B | Y,X,A (1 | , (0 , P B | Y,X,A (1 | , (0 , ≤ (2 + β ) ξ (0 , B ( x, a ) = (1 , → P B | Y,X,A (0 | , (1 , P B | Y,X,A (1 | , (1 , ≤ ξ (1 , B ( x, a ) = (1 , → P B | Y,X,A (1 | , (1 , P B | Y,X,A (0 | , (1 , ≤ ξ (1 , B , (37)where ξ ( x,a ) B are the bounds on the uncertainty expres-sions. We find that while the first two inequalities aboveare saturated by the optimal quantum strategy, the thirdand fourth inequalities fail to be saturated except when θ = π , i.e., for the maximally entangled state | ψ π (cid:105) . For2the third expression, i.e., when ( x, a ) = (1 , , the boundis ξ (1 , B = √ − cos 4 θ + √ θ √ − cos 4 θ (38)with the maximally certain state being | + (cid:105) = √ ( | (cid:105) + | (cid:105) ) . On the other hand, we see that for ( x, a ) = (1 , using the optimal strategy Alice steers Bob’s state to | ˜ ψ (1 , (cid:105) = cos θ | (cid:105) + sin θ | (cid:105) , so that only for the maxi-mally entangled state ( θ = π , β = 0 ) does Alice manageto steer Bob’s system to the least uncertain state. Simi-larly for the case ( x, a ) = (1 , , the bound is ξ (1 , B = √ − cos 4 θ + √ θ √ − cos 4 θ (39)with the maximally certain state being |−(cid:105) = √ ( | (cid:105) −| (cid:105) ) . On the other hand, for ( x, a ) = (1 , using theoptimal strategy Alice steers Bob’s state to | ˜ ψ (1 , (cid:105) =cos θ | (cid:105) − sin θ | (cid:105) . So that it is again only for the max-imally entangled state that Alice manages to steer Bob’ssystem to the least uncertain state. Thus, the tiltedCHSH inequality of [10] expressed as a game, showsthat every non-maximally entangled two-qubit stateserves as the optimal state for a game in which the un-certainty principle - quantum game value correspon-dence does not hold.It now remains to generalize the result even further, toall two-qudit states that are non-maximally entangled.Consider a general two-qudit non-maximally entangledstate, written as | φ (cid:105) = d (cid:88) i =1 λ i | i, i (cid:105) (40)with the Schmidt coefficients λ i ∈ R obeying < λ i < for all i and (cid:80) i λ i = 1 with not all λ i equal to √ d . Wefirst deal with the case when d is even. The idea is todesign a game with d -outcome measurements on eachside, such that the correlation tables for some measure-ment settings are block-diagonal with blocks of size × each. The j -th × block will correspond to a tiltedCHSH game that is maximally violated by a two qubitstate, which is a normalized version of λ j − | j − , j − (cid:105) + λ j | j, j (cid:105) , with j = 1 , . . . , d/ . Accordingly, weconstruct a game with two inputs x = 0 , for Alice, and d + 2 inputs y = 0 , . . . , d + 1 for Bob, with d outputseach. Given | φ (cid:105) , the ratios (cid:110) λ j λ j − (cid:111) for j = 1 , . . . , d/ determine the game as follows. For inputs x = 0 , and y = 0 , . . . , d/ , the players play a set of d/ tiltedCHSH games determined by the following procedure.Set θ j = arctan λ j λ j − , with corresponding optimalstates | ψ θ j (cid:105) = cos θ j | (cid:105) + sin θ j | (cid:105) . We obtain the set of d/ tilted CHSH games with tilting parameters β j for j = 1 , . . . , d/ given by β j = 2 (cid:112) θ j . (41)The j -th tilted CHSH game CHSH ( j ) tilt is played on inputs x = 0 , for Alice and y = 0 , , j + 1 for Bob, so that eachof the settings y = 2 , . . . , d + 1 appears in a distinct tiltedCHSH game. With an appropriate choice of observables,the players are able to achieve a value proportional to ( λ j + λ j − ) ω (2 j ) q for the game with ω (2 j ) q :=
12 + (cid:113) β j β j (4 + β j ) (42)for j = 1 , . . . , d/ . Let ω ∗ q := max j ω (2 j ) q , and note that ω ∗ q is determined once the λ i are given.To complete the construction, we now consider an-other set of d/ tilted CHSH games with parameters β j , this time played by Alice and Bob on the inputs x = 0 , and y = d/ j − , d/ j, d/ j + 1 for j = 1 , . . . , d/ . It remains to specify theinput distributions, these are given with τ := 4 + (cid:80) d/ j =1 (cid:18) β j + ω ∗ q − ω (2 j ) q ω (2 j ) q (4 + β j ) (cid:19) as π X (0) = 1 τ (cid:88) j (cid:32) β j + (2 + β j ) ω ∗ q − ω (2 j ) q ω (2 j ) q (cid:33) ,π X (1) = 1 − π X (0) = 2 + 2 (cid:80) d/ j =1 ( ω ∗ q − ω (2 j ) q ) / ( ω (2 j ) q ) τ ,π Y | X (0 |
0) = π Y | X (1 |
0) = 1 π X (0) τ ,π Y | X ( j + 1 |
0) = β j π X (0) τ , j = 1 , . . . , d/ ,π Y | X ( d/ j − |
0) = π Y | X ( d/ j |
0) = ω ∗ q − ω (2 j ) q ω (2 j ) q π X (0) τ ,π Y | X ( d/ j + 1 |
0) = β j ( ω ∗ q − ω (2 j ) q ) ω (2 j ) q π X (0) τ ,π Y | X ( d/ j − |
1) = π Y | X ( d/ j |
1) = ω ∗ q − ω (2 j ) q ω (2 j ) q π X (1) τ ,π Y | X (0 |
1) = π Y | X (1 |
1) = 1 π X (1) τ . (43)With the above input distributions, we can now di-rectly calculate the value achieved by a quantum strat-egy given by the shared state | φ (cid:105) and observables A = ⊕ d/ j =1 σ ( j ) z , A = ⊕ d/ j =1 σ ( j ) x , B = B d/ j − = ⊕ d/ j =1 (cid:16) cos µσ ( j ) z + sin µσ ( j ) x (cid:17) and B = B d/ j = ⊕ d/ j =1 (cid:16) cos µσ ( j ) z − sin µσ ( j ) x (cid:17) , B j +1 = B d/ j +1 = ⊕ d/ j =1 σ ( j ) z . In the j -th × sector, the strategy achievesa value of ( λ j + λ j − ) ω ∗ q τ , so that summing over all j = 1 , . . . , d/ , we obtain the quantum value of our gen-eralized tilted CHSH game to be ω q ( CHSH gen-tilt ) = d/ (cid:88) j =1 ( λ j + λ j − ) ω ∗ q τ = ω ∗ q τ , (44)by virtue of the fact that (cid:80) i λ i = 1 . Let us now ver-ify that this is in fact the optimal quantum value ofthe game CHSH gen-tilt . This is seen by the fact that thegame decomposes into × blocks, and the maximumquantum value within each block is ω ∗ q τ , obtained from ω q ( CHSH tilt ) presented earlier. Moreover, we see thatthe uncertainty relations fail to be saturated within each × block, except those which correspond to λ j = λ j − , from the results for the qubit case. Finally, thecase for odd d works in a very similar manner to that foreven d , we use the generalized tilted CHSH game corre-sponding to d − which is even, and augment the gamewith a × block, i.e., the entries P A,B | X,Y ( d, d | x, y ) for x, y ∈ { , } and x = 0 , , y = 2 d, d + 1 . Similarly, weaugment the observables with the projector | d (cid:105)(cid:104) d | , i.e., A = ⊕ ( d − / j =1 (cid:16) σ ( j ) z ⊕ | d (cid:105)(cid:104) d | (cid:17) . While the uncertainty re-lation corresponding to the × block is saturated, forall non-maximally entangled states, the uncertainty re-lations for the × blocks are not, as we have seen inthe even d case. Thus, we have constructed for every d ≥ , a non-local game with the optimal strategy beinggiven by the state | φ (cid:105) = (cid:80) i λ i | i, i (cid:105) and such that the cor-respondence between the uncertainty principle and thequantum game value is broken. Supplementary Note 5 - Experimentalimplementation
In the experiment, we used single photon’s polariza-tion state as the physical qubit. To maximally violate theBell inequality given in Eq. (5) of the main text, we pre-pare the following polarization entangled two-photonsstate, | Ψ (cid:105) =0 . | HH (cid:105) + 0 . | HV (cid:105) (45) +0 . | V H (cid:105)− . | V V (cid:105) . This state is produced as follows, firstly, we preparemaximally entangled photons pairs (see Fig. (3) ofthe main paper). For this an Ultraviolet light centeredat wavelength of 390 nm was focused onto two 2 mmthick β barium borate (BBO) nonlinear crystals placedin cross-configuration to produce photon pairs emittedinto two spatial modes a and b through the second or-der degenerate type-I spontaneous parametric down-conversion (SPDC) process [12]. Any spatial or tempo-ral distinguishability between the down-converted pho-tons is carefully removed through quartz wedges placedin the pump beam (not shown in the figure) and a pairof Y V O crystals located in each of the down-convertedbeams. The emitted photons were passed through thenarrow-bandwidth interference filters (IF) ( ∆ λ = 3 nm)and coupled into 2 m single mode optical fibers (SMF) tosecure well defined spatial and spectral emission modes.Secondly, to prepare the desired state as outlined in [13],the pump polarization is altered to produce the state cos θ p | HH (cid:105) − sin θ p | V V (cid:105) with θ p = 31 . ◦ . Then, as thefinal step this state is rotated to | Ψ (cid:105) by the use of a halfwave-plate (HWP) placed after the output fiber couplerin each of the mode (a) and (b) at an angle of . ◦ and . ◦ respectively. Supplementary Note 6 - State tomography
To estimate the fidelity of the two-photon preparedstate ( ρ exp ) with respect to the ρ th = | Ψ (cid:105)(cid:104) Ψ | , we car-ried out quantum state tomography as described in Ref.[14], and where we have measured each of the two pho-tons in three mutually unbiased basis (H/V, D/A, L/R).These polarization measurements were performed byusing HWPs, quarter wave plates (QWP) and polarizingbeam splitters (PBS) followed by single photon detec-tors (actively quenched Si-avalanche photodiodes (Si-APD)). An FPGA based timing system is used to recordthe number of coincidence events with a detection timewindow of . ns. For each setting, we have obtainedapproximately 1.6 Million events.The obtained density matrix of the prepared state is ρ exp = .
065 0 . − . i . − . i − . − . i .
124 + 0 . i .
242 0 . − . i − . − . i .
197 + 0 . i .
39 + 0 . i . − . − . i − .
059 + 0 . i − .
116 + 0 . i − .
186 + 0 . i . , (46)where entry–wise error bars are given by ∆ ρ exp = .
002 0 .
004 0 .
007 0 . .
004 0 .
001 0 .
010 0 . .
007 0 .
010 0 .
002 0 . .
003 0 .
004 0 .
007 0 . . (47) The real and imaginary parts of this density matrix areshown in Supplementary figure 1. From experimen-4tally collected data, we reconstructed the density matrix ρ exp by considering maximum likelihood estimation, asdescribed in Ref. [14]. Our figure of merit to quan-tify effectiveness of state reconstruction is the relativefidelity with respect to the theoretical state | Ψ (cid:105) . Errorsfor fidelity reconstruction have been estimated by tak-ing into account the Poissonian statistical distribution ofthe photon number counting (see Section VI, Ref. [14]).We obtained the following state fidelity a b Supplementary Figure 1. (Color online). Experimental den-sity matrix. a Real and, b imaginary part, shown in compu-tational basis (HH, HV,VH and VV), of the experimentallyobtained elements of the two qubit density matrix ρ Exp , con-structed using maximum likelihood quantum state tomogra-phy. F = (cid:104) Ψ | ρ exp | Ψ (cid:105) = 0 . ± . . (48) Supplementary Note 7 - Steering and Bell Violation
Alice and Bob can perform Bell test on experimentallyprepared state with the following settings | φ ± (cid:105) = 1 √ | H (cid:105) ± | V (cid:105) ) | φ ± (cid:105) = cos γ | H (cid:105) ± sin γ | V (cid:105) (49)where γ = 4 . .To perform Bell test and the tomography of thesteered state, Bob randomly chooses if he wants to re-alize tomographic measurement or the Bell test settings.Sequence of measurements performed for each task aregiven in the table I. These choices are randomly madeand executed via computer controlled rotation stagescarrying wave-plates of Bob’s analyzer. To decrease thestatistical counts error further, we have collected ap-proximately 6.48 Million events per setting.For the Bell test we obtained ω q ( G (7) ) = 0 . ± . , when the corresponding theoretical value is . . In the following, we report theoretical and theexperimentally obtained density matrices of Bob stateswhen Alice projected her photon. Supplementary Table I. For a particular setting of Alice (col-umn 1), Bob chooses randomly if he wants to perform se-quence of measurements for the Bell test (column 2, upperrow for the corresponding Alice setting) or tomography of thephoton he possess (column 2, lower row for the correspondingAlice setting).
Alice Settings Bob Settings | φ ± (cid:105) | φ ± (cid:105) , | φ ± (cid:105) H/V, D/A, L/R | φ ± (cid:105) | φ ± (cid:105) , | φ ± (cid:105) H/V, D/A, L/R
1. When Alice’s photon is projected to | φ +0 (cid:105) , Bobstate becomes (cid:32) . . . . (cid:33) and we obtained experimentally (cid:32) .
943 0 . − . i .
231 + 0 . i . (cid:33) ± (cid:32) .
002 0 . .
011 0 . (cid:33) , which corresponds to the fi-delity of . ± . .2. When Alice photon is projected to | φ − (cid:105) , Bob statebecomes (cid:32) . − . − . . (cid:33) and we obtained experimentally (cid:32) . − . − . i − .
471 + 0 . i . (cid:33) ± (cid:32) .
004 0 . .
011 0 . (cid:33) , which corresponds to thefidelity of . ± . .3. When Alice photon is projected to | φ +1 (cid:105) , Bob statebecomes (cid:32) . − . − . . (cid:33) and we obtained experimentally (cid:32) . − . − . i − .
312 + 0 . i . (cid:33) ± (cid:32) .
002 0 . .
01 0 . (cid:33) , which corresponds to thefidelity of . ± . .4. When Alice photon is projected to | φ − (cid:105) , Bob statebecomes (cid:32) . . . . (cid:33) and we obtained experimentally (cid:32) .
329 0 .
465 + 0 . i . − . i . (cid:33) ± (cid:32) .
004 0 . .
011 0 . (cid:33) , which corresponds to thefidelity of . ± . . Supplementary Note 8 - Error estimation
We have considered error originated from the mea-surement side only, as the error on the preparation sidewill just shift the experimentally prepared state awayfrom the desire state and therefore will be apparent fromthe reported state fidelity or the value of Bell inequalityviolation.To estimate the error in our experiment, we have con-sidered errors due to cross-talks–originated from thePBS extinction and absorption–wave-plate setting er-rors, wave-plates offset error, wave-plates retardancetolerance and error due to Poissonian statistics of theincoming photons. Cross-talk is considered here as theused PBSs were not perfect. To calculate the PBS extinc-tion, we have carefully estimated the extinction ratio ofthe PBSs used on both sides (Alice and Bob) with theirtransmission and absorption for each polarizations.The wave-plate setting error is considered as one hasto switch the settings during collecting data for the esti-mation of the state fidelity. In the experiment, we usedmotorized stages to rotate the wave-plates to switchamong the different settings. These mounts have re-peatability of less than . ◦ . Therefore, for error estima-tion, we assumed that the wave-plates setting error hasnormal distribution with standard deviation of . ◦ .A normally distributed offset error of . ◦ in settingthe wave-plates is also assumed as the zero of a givenwave-plate could not be adjusted better than . ◦ .The wave-plates retardance tolerance of λ is alsotaken into account by assuming a normally distributedretardance error in each of the wave plate used. Notethat, among all these errors, wave-plates retardance er-ror is leading and it is over estimated as it is fixed witheach wave-plate chosen for the experiment, moreover,we are carefully characterizing wave plates which wehave not assumed here. Finally, we considered errors arising due to the photon counts following the Poisso-nian statistics. ∗ [email protected] Supplementary References: [1] Masanes, Ll.,
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