A system s wave function is uniquely determined by its underlying physical state

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1 A system s wave function is uniquely etermine by its unerlying physical state Roger Colbeck 1, an Renato Renner 2, 1 Department of Mathematics, University of York, YO10 5DD, UK 2 Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerlan (Date: January 13, 2017) We aress the question of whether the quantum-mechanical wave function Ψ of a system is uniquely etermine by any complete escription Λ of the system s physical state. We show that this is the case if the latter satisfies a notion of free choice. This notion requires that certain experimental parameters those that accoring to quantum theory can be chosen inepenently of other variables retain this property in the presence of Λ. An implication of this result is that, among all possible escriptions Λ of a system s state compatible with free choice, the wave function Ψ is as objective as Λ. I. INTRODUCTION The quantum-mechanical wave function, Ψ, has a clear operational meaning, specifie by the Born rule [1]. It asserts that the outcome X of a measurement, efine by a family of projectors {Π x }, follows a istribution P X given by P X (x) = Ψ Π x Ψ, an hence links the wave function Ψ to observations. However, the link is probabilistic: even if Ψ is known to arbitrary precision, we cannot in general preict X with certainty. In classical physics, such ineterministic preictions are always a sign of incomplete knowlege. 1 This raises the question of whether the wave function Ψ associate to a system correspons to an objective property of the system, or whether it shoul instea be interprete subjectively, i.e., as a representation of our (incomplete) knowlege about certain unerlying objective attributes. Another alternative is to eny the existence of the latter, i.e., to give up the iea of an unerlying reality completely. Despite its long history, no consensus about the interpretation of the wave function has been reache. A subjective interpretation was, for instance, supporte by the famous argument of Einstein, Poolsky an Rosen [2] (see also [3]) an, more recently, by information-theoretic consierations [4 6]. The opposite (objective) point of view was taken, for instance, by Schröinger (at least initially), von Neumann, Dirac, an Popper [7 9]. To turn this ebate into a more technical question, one may consier the following geankenexperiment: Assume you are provie with a set of variables Λ that are intene to escribe the physical state of a system. Suppose, furthermore, that the set Λ is complete, i.e., there is nothing that can be ae to Λ to increase the accuracy of any preictions about the outcomes of measurements is complete is not complete P i ( ) P i ( ) P i ( ) P i ( ) ontic -epsistemic FIG. 1: The ifferent possible roles of the wave function Ψ. A moel that uses a variable Λ to escribe a system s physical state can be either Ψ-ontic or Ψ-epistemic, epening on whether or not the wave function Ψ is uniquely etermine by Λ (which takes values enote by λ). Conversely, the relevant parts of Λ may be etermine by Ψ, in which case Ψ is complete. Using free choice (with respect to an appropriate causal orer), [17] rules out the right column, [16] rules out the bottom left case, an the present paper (as well as [14], base on ifferent assumptions) rules out the bottom row. on the system. If you were now aske to specify the wave function Ψ of the system, woul your answer be unique? If so then Ψ is a function of the variables Λ an hence as objective as Λ. The moel efine by Λ woul then be calle Ψ-ontic [10]. Conversely, the existence of a complete set of variables Λ that oes not etermine the wave function Ψ woul mean that Ψ cannot be interprete as an objective property. Λ woul then be calle Ψ-epistemic (see Fig. 1). 2 roger.colbeck@york.ac.uk renner@phys.ethz.ch 1 For example, when we assign a probability istribution P to the outcomes of a ie roll, P is not an objective property but rather a representation of our incomplete knowlege. Inee, if we ha complete knowlege, incluing for instance the precise movement of the thrower s han, the outcome woul be eterministic. 2 Note that the existence or non-existence of Ψ-epistemic theories is also relevant in the context of simulating quantum systems. Here Λ can be thought of as the internal state of a computer performing the simulation, an one woul ieally like that storing Λ requires significantly fewer resources than woul be require to store Ψ. However, a number of existing results alreay cast

2 2 In a seminal paper [14], Pusey, Barrett an Ruolph showe that any complete moel Λ is Ψ-ontic if it satisfies an assumption, terme preparation inepenence. It emans that Λ consists of separate variables for each subsystem, e.g., Λ = (Λ A, Λ B ) for two subsystems S A an S B, an that these are statistically inepenent, i.e., P ΛA Λ B = P ΛA P ΛB, whenever the joint wave function Ψ of the total system has prouct form, i.e., Ψ = Ψ A Ψ B. Here we show that the same conclusion can be reache without imposing any internal structure on Λ. In more etail, our argument relies on the concept of free choice, which can only be efine with reference to an orering, calle here a causal orer 3. More precisely, we prove that Ψ is a function of any complete set of variables that are compatible with free choice with respect to the causal orer of Figure 3 (see later for more etails). This is state as Corollary 1. The free choice assumption use captures the iea that experimental parameters, e.g., which state to prepare or which measurement to carry out, can be chosen inepenently of all other information (relevant to the experiment), except for information that is create after the choice is mae, e.g., measurement outcomes. While this notion is implicit in quantum theory, we eman that it also hols in the presence of Λ. 4 The proof of our result is inspire by our earlier work [16] in which we observe that the wave function Ψ is uniquely etermine by any complete set of variables Λ, provie that Ψ is itself complete (in the sense escribe above). Together with the result of [17], in which we showe that Ψ is complete, we can conclue that the wave function Ψ is uniquely etermine by Λ. The ifference in the present work is that we can circumvent one of the aspects of quantum theory require by the argument in [17]. In particular, here we prove that Ψ is etermine by Λ without requiring that any quantum measurement on a system correspons to a unitary evolution of an extene system. Being base on weaker assumptions, the resulting no-go theorem is stronger. Furthermore, the argument that the wave function Ψ is complete is quite involve an a beneficial feature of the present work is that we circumvent it 5. II. THE UNIQUENESS THEOREM Our argument refers to an experimental setup where a particle emitte by a source ecays into two, each of which is irecte towars one of two measurement evices (see Fig. 2). The measurements that are performe oubt on this possibility (see, for example, [11 13]). 3 This shoul not be confuse with a causal structure as use in e.g. [15]. 4 Free choice of certain variables is also implie by the preparation inepenence assumption use in [14], as iscusse below. 5 Note, however, that the assumptions use in this work o not allow us to conclue that Ψ is complete. A X measurement measurement { a x} { b y} ecay U source FIG. 2: The experimental setup. The proof of the uniqueness theorem relies on a thought experiment where a source takes as input a escription of a wave function Ψ an prepares a particle in a corresponing state (which, in a general moel, is escribe by a variable Λ). The particle then ecays into two parts, which are measure at separate locations. A an B etermine the measurements that are applie to the two parts, an X an Y are the respective outcomes. epen on parameters A an B, an their respective outcomes are enote X an Y. Quantum theory allows us to make preictions about these outcomes base on a escription of the initial state of the system, the evolution it unergoes an the measurement settings. For our purposes, we assume that the quantum state of each particle emitte by the source is pure, an hence specifie by a wave function 6. As we will consier ifferent choices for this wave function, we moel it as a ranom variable Ψ that takes as values unit vectors ψ in a complex Hilbert space H. Furthermore, we take the ecay to act like an isometry, enote U, from H to a prouct space H A H B. Finally, for any choices a an b of the parameters A an B, the measurements are given by families of projectors {Π a x} x X an {Π b y} y Y on H A an H B, respectively. The Born rule, applie to this setting, now asserts that the joint probability istribution of X an Y, conitione on the 6 We consier it uncontroversial that a mixe state can be thought of as a state of knowlege. Y B

3 3 X A Λ Ψ FIG. 3: The causal orer. Free choice is only well efine if one specifies a causal orer, i.e., a preorer relation on the set of variables relevant to the experiment. The causal orer we use is motivate by the arrangement of variables in the experiment epicte by Fig. 2 in relativistic space time. relevant parameters, is given by P XY ABΨ (x, y a, b, ψ) = ψ U (Π a x Π b y)u ψ. (1) To moel the system s physical state, we introuce an aitional ranom variable Λ. We o not impose any structure on Λ (in particular, Λ coul be a list of values). We will consier preictions P XY ABΛ (x, y a, b, λ) conitione on any particular value λ of Λ, analogously to the preictions base on Ψ accoring to the Born rule (1). To efine the notions of free choice an completeness, as introuce informally in the introuction, we take as motivation that any experiment takes place in spacetime an therefore has a causal orer 7. For example, the measurement setting A is chosen before the measurement outcome X is obtaine. This may be moelle mathematically by a preorer relation 8, enote, on the relevant set of ranom variables. While our technical claim oes not epen on how the causal orer is interprete physically, it is intuitive to imagine it being compatible with relativistic spacetime. In this case, A X woul mean that the spacetime point where X is accessible lies in the future light cone of the spacetime point where the choice A is mae. For our argument we consier the causal orer efine by the transitive completion of the relations Ψ Λ, Λ A, Λ B, A X, B Y (2) (cf. Fig. 3). This reflects, for instance, that Ψ is chosen at the very beginning of the experiment, an that A an B are chosen later, right before the two measurements are carrie out. Note, furthermore, that A Y an B X. With the aforementione interpretation of the relation 7 In previous work we sometimes calle this a chronological structure [18]. 8 A preorer relation is a binary relation that is reflexive an transitive. Y B in relativistic spacetime, this woul mean that the two measurements are carrie out at spacelike separation. Using the notion of a causal orer, we can now specify mathematically what we mean by free choices an by completeness. We note that the two efinitions below shoul be unerstoo as necessary (but not necessarily sufficient) conitions characterising these concepts. Since they appear in the assumptions of our main theorem, our result also applies to any more restrictive efinitions. We remark furthermore that the efinitions are generic, i.e., they can be applie to any set of variables equippe with a preorer relation. 9 Definition 1. When we say that a variable A is a free choice from a set A (w.r.t. a causal orer) this means that the support of P A contains A an that P A A = P A where A is the set of all ranom variables Z (within the causal orer) such that A Z. In other wors, a choice A is free if it is uncorrelate with any other variables, except those that lie in the future of A in the causal orer. For a further iscussion an motivation of this notion we refer to Bell s work [19] as well as to [20]. Crucially, we note that Definition 1 is compatible with the usual unerstaning of free choices within quantum theory. For example, if we consier our experimental setup (cf. Fig. 2) in orinary quantum theory (i.e., where there is no Λ), the initial state Ψ as well as the measurement settings A an B can be taken to be free choices w.r.t. Ψ A, Ψ B, A X, B Y (which is the causal orer efine by Eq. 2 with Λ remove). Definition 2. When we say that a variable Λ is complete (w.r.t. a causal orer) this means that 10 P Λ Λ = P Λ ΛΛ where Λ an Λ enote the sets of ranom variables Z (within the causal orer) such that Λ Z an Z Λ, respectively. Completeness of Λ thus implies that preictions base on Λ about future values Λ cannot be improve by taking into account aitional information Λ available in the past. 11 Recall that this is meant as a necessary criterion for completeness an that our conclusions hol for any more restrictive efinition. For example, one may replace the set Λ by the set of all values that are not in the past of Λ. 9 They are therefore ifferent from notions use commonly in the context of Bell-type experiments, such as parameter inepenence an outcome inepenence. These refer explicitly to measurement choices an outcomes, whereas no such istinction is necessary for the efinitions use here. 10 In other wors, Λ Λ Λ is a Markov chain. 11 Using statistics terminology, one may also say that Λ is sufficient for Λ given ata Λ.

4 4 We are now reay to formulate our main result as a theorem. Note that, the assumptions of the theorem as well as its claim correspon to properties of the joint probability istribution of X, Y, A, B, Ψ an Λ. Theorem 1. Let Λ an Ψ be ranom variables an assume that the support of Ψ contains two wave functions, ψ an ψ, with ψ ψ < 1. If for any isometry U an measurements {Π a x} x an {Π b y} y, parameterise by a A an b B, there exist ranom variables A, B, X an Y such that 1. P XY ABΨ satisfies the Born rule (1); 2. A an B are free choices from A an B, w.r.t. (2); 3. Λ is complete w.r.t. (2) then there exists a subset L of the range of Λ such that P Λ Ψ (L ψ) = 1 an P Λ Ψ (L ψ ) = 0. The theorem asserts that, assuming valiity of the Born rule an freeom of choice, the values taken by any complete variable Λ are ifferent for ifferent choices of the wave function Ψ. This implies that Ψ is inee a function of Λ. To formulate this implication as a technical statement, we consier an arbitrary countable 12 set S of wave functions such that ψ ψ < 1 for any istinct elements ψ, ψ S. Corollary 1. Let Λ an Ψ be ranom variables with Ψ taking values from the set S of wave functions. If the conitions of Theorem 1 are satisfie then there exists a function f such that Ψ = f(λ) hols almost surely. III. The proof of this corollary is given in Appenix A. PROOF OF THE UNIQUENESS THEOREM The argument relies on specific wave functions, which epen on parameters, k N an ξ [0, 1], with k <. They are efine as unit vectors on a prouct space H A H B, where H A an H B are ( + 1)- imensional Hilbert spaces equippe with an orthonormal basis { j } j=0,13 φ = 1 j j (3) j=0 φ = 1 k 1 (ξ j j + ) 1 ξ 2. (4) k j=1 Lemma 1. For any 0 α < 1 there exist k, N with k < an ξ [0, 1] such that the vectors φ an φ efine by (3) an (4) have overlap φ φ = α. Proof. If α = 0, set k = 1, = 2 an ξ = 0. Otherwise, set 1/(1 α 2 ), k = α 2 an ξ = α k k + 1, so that ξ [0, 1] an φ φ = α. Furthermore, the choice of ensures that α 2 + 1, which implies k <. Furthermore, for any n N, we consier projective measurements {Π a x} x X an {Π b y} y X on H A an H B, parameterise by a A n {0, 2, 4,..., 2} an b B n {1, 3, 5,..., 1}, an with outcomes in X {0,..., }. For x, y {0,..., 1}, the projectors are efine in terms of the generalise Pauli operator, ˆX 1 l=0 l l 1 (where enotes aition moulo ) by Π a x ( ˆX ) a x x ( ˆX ) a (5) Π b y ( ˆX ) b y y ( ˆX ) b. (6) We also set Π a = Πb =. The outcomes X an Y will generally be correlate. To quantify these correlations, we efine 14 I n, (P XY AB ) P XY AB (x, x 1 0, 1) x=0 a,b x=0 a b =1 P XY AB (x, x a, b). For the correlations preicte by the Born rule for the measurements {Π a x} x X an {Π b y} y X applie to the state φ efine by (3), i.e., P XY AB (x, y a, b) = φ Π a x Π b y φ, we fin (see Appenix B) I n, (P XY AB ) π2 6n. (7) The next lemma shows that I n, gives an upper boun on the istance of the istribution P X AΛ from a uniform istribution over {0,..., 1}. The boun hols for any ranom variable Λ, provie the joint istribution P XY Λ AB satisfies certain conitions. Lemma 2. Let P XY ABΛ be a istribution that satisfies P XΛ AB = P XΛ A, P Y Λ AB = P Y Λ B an P ABΛ = P A P B P Λ with supp(p A ) A n an supp(p B ) B n. Then P Λ (λ) PX AΛ (x 0, λ) 1 2 I n,(p XY AB ). x=0 12 The restriction to a countable set is ue to our proof technique. We leave it as an open problem to etermine whether this restriction is necessary. 13 We use here the abbreviation j j for j j. 14 Note that the first sum correspons to the probability that X 1 = Y, conitione on A = 0 an B = 1. The terms in the secon sum can be interprete analogously.

5 5 (Although our proof eals with the general case, the main ieas can be seen by working through the analogous argument in the slightly simpler (but less general) case in which Λ is iscrete, so that P Λ (λ) is replace by λ P Λ(λ).) The proof of Lemma 2 is given in Appenix C. It generalises an argument escribe in [17], which is in turn base on work relate to chaine Bell inequalities [21, 22] (see also [23, 24]). We have now everything reay to prove the uniqueness theorem. Proof of Theorem 1. Let α, γ R such that e iγ α = ψ ψ. Furthermore, let k,, ξ be as efine by Lemma 1, so that φ φ = α. Then there exists an isometry U such that Uψ = φ an Uψ = e iγ φ (see Lemma 3 of Appenix D). 15 Now let n N an let A, B, X an Y be ranom variables that satisfy the three conitions of the theorem for the isometry U an for the projective measurements efine by (5) an (6), which are parameterise by a A n an b B n, respectively. Accoring to the Born rule (Conition 1), the istribution P XY ABψ P XY ABΨ (,,, ψ) conitione on the choice of initial state Ψ = ψ correspons to the one consiere in (7), i.e., I n, (P XY ABψ ) π2 6n. (8) Note that P A BΨ P Y Λ ABΨ = P AY Λ BΨ = P A BY ΛΨ P Y Λ BΨ. Freeom of choice (Conition 2) implies that P A BΨ = P A BY ΛΨ. It follows that P Y Λ ABΨ = P Y Λ BΨ. By a similar reasoning, we also have P XΛ ABΨ = P XΛ AΨ. The freeom of choice conition also ensures that P ABΛ Ψ = P A P B P Λ Ψ with supp(p A ) A n an supp(p B ) B n. We can thus apply Lemma 2 to give, with (8), P Λ ψ (λ) PX AΛΨ (x 0, λ, ψ) 1 π2 1. x=0 Consiering only the term x = k (recall that k < ) an noting that the left han sie oes not epen on n, we have P Λ ψ (λ) PX AΛΨ (k 0, λ, ψ) 1 = 0 (otherwise, by taking n sufficiently large, we will get a contraiction with the above). Let L be the set of all elements λ from the range of Λ for which P X AΛΨ (k 0, λ, ψ) is efine an equal to 1. The above implies that P Λ Ψ (L ψ) = 1. Furthermore, completeness of Λ (Conition 3) implies that for any λ L for which P X AΛΨ (k 0, λ, ψ ) is efine P X AΛΨ (k 0, λ, ψ ) = P X AΛΨ (k 0, λ, ψ) = 1. Thus, using P Λ AΨ = P Λ Ψ (which is implie by the freeom of choice assumption, Conition 2) an writing δ L for the inicator function, we have P X AΨ (k 0, ψ ) = P Λ Ψ (λ ψ )P X AΛΨ (k 0, λ, ψ ) (9) δ L (λ)p Λ Ψ (λ ψ )P X AΛΨ (k 0, λ, ψ ) = 1 δ L (λ)p Λ Ψ (λ ψ ) = 1 P Λ Ψ(L ψ ). However, because the vector e iγ φ = Uψ has no overlap with k (because k < ) an because the measurement {Π a x} x X for a = 0 correspons to projectors along the { x } x=0 basis, we have P X AΨ (k 0, ψ ) = 0 by the Born rule (Conition 1). Inserting this in (9) we conclue that P Λ Ψ (L ψ ) = 0. IV. DISCUSSION It is interesting to compare Theorem 1 to the result of [14], which we briefly escribe in the introuction. The latter is base on a ifferent experimental setup, where n particles with wave functions Ψ 1,..., Ψ n, each chosen from a set {ψ, ψ }, are prepare inepenently at n remote locations. The n particles are then irecte to a evice where they unergo a joint measurement with outcome Z. The main result of [14] is that, for any variable Λ that satisfies certain assumptions, the wave functions Ψ 1,..., Ψ n are etermine by Λ. One of these assumptions is that Λ consists of n parts, Λ 1,..., Λ n, one for each particle. To state the other assumptions an compare them to ours, it is useful to consier the causal orer efine by the transitive completion of the relations 16 Ψ i Λ i ( i), (Λ 1,..., Λ n ) Λ, Λ Z. (10) It is then easily verifie that the assumptions of [14] imply the following: 1. P Z Ψ1 Ψ n satisfies the Born rule; 2. Ψ 1,..., Ψ n are free choices from {ψ, ψ } w.r.t. (10); 3. Λ is complete w.r.t. (10). 15 If H has a larger imension than H A H B (e.g., because H is infinite imensional) then we can consier an (infinite imensional) extension of H B, keeping the same notation for convenience. 16 Note that this causal orer captures the aforementione experimental setup. In particular, we have Ψ i Λ j for i j, reflecting the iea that the n particles are prepare in separate isolate evices.

6 6 These conitions are essentially in one-to-one corresponence with the assumptions of Theorem The main ifference thus concerns the moelling of the physical state Λ, which in the approach of [14] is assume to have an internal structure. A main goal of the present work was to avoi using this assumption (see also [25, 26] for alternative arguments). We conclue by noting that the assumptions of Theorem 1 an Corollary 1 may be weakene. For example, the inepenence conition that is implie by free choice may be replace by a partial inepenence conition along the lines consiere in [27]. An analogous weakening was given in [28, 29] regaring the argument of [14]. More generally, recall that all our assumptions are properties of the probability istribution P XY ABΨΛ. One may therefore replace them by relaxe properties that nee only be satisfie for istributions that are ε- close (in total variation istance) to P XY ABΨΛ. (For example, the Born rule may only hol approximately.) It is relatively straightforwar to verify that the proof still goes through, leaing to the claim that Ψ = f(λ) hols with probability at least 1 δ, with δ 0 in the limit where ε 0. Nevertheless, none of the three assumptions of Theorem 1 can be roppe without replacement. Inee, without the Born rule, the wave function Ψ has no meaning an coul be taken to be inepenent of the measurement outcomes X. Furthermore, a recent impossibility result [30] implies that the analogous theorem with the secon assumption omitte oes not hol. It also implies that the statement of Theorem 1 cannot hol for a setting with only one single measurement. This means that there exist Ψ-epistemic theories compatible with the remaining assumptions. However, in this case, it is still possible to exclue a certain subclass of such theories, calle maximally Ψ-epistemic theories [31] (see also [32]). Finally, completeness of Λ is necessary because, without it, Λ coul be set to a constant, in which case it clearly cannot etermine Ψ. Acknowlegments We thank Omar Fawzi, Michael Hush, Matt Leifer, Matthew Pusey an Rob Spekkens for useful iscussions. Research leaing to these results was supporte by the Swiss National Science Founation (through the National Centre of Competence in Research Quantum Science an Technology an grant No ), the CHIST- ERA project DIQIP, an the European Research Council (grant No ). [1] M. Born, Zur Quantenmechanik er Stoßvorgänge, Zeitschrift für Physik 37, (1926). [2] A. Einstein, B. Poolsky an N. Rosen, Can quantummechanical escription of physical reality be consiere complete?, Phys. Rev. 47, (1935). [3] A. Einstein, Letter to Schröinger (1935). Translation from D. Howar, Stu. Hist. Phil. Sci. 16, 171 (1985). [4] E. T. Jaynes, Probability in quantum theory, in Complexity, Entropy an the Physics of Information, e. by W.H. 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7 7 [25] L. Hary, Are quantum states real?, Int. J. Mo. Phys. B 27, (2013). [26] S. Aaronson, A. Boulan, L. Chua an G. Lowther, ψ- epistemic theories: the role of symmetry, Phys. Rev. A 88, (2013). [27] R. Colbeck an R. Renner, Free ranomness can be amplifie, Nat. Phys. 8, (2012). [28] M.J.W. Hall, Generalisations of the recent Pusey- Barrett-Ruolph theorem for statistical moels of quantum phenomena, arxiv: (2011). [29] M. Schlosshauer an A. Fine, Implications of the Pusey- Barrett-Ruolph quantum no-go theorem, Phys. Rev. Lett. 108, (2012). [30] P.G. Lewis, D. Jennings, J. Barrett an T. Ruolph, Distinct quantum states can be compatible with a single state of reality, Phys. Rev. Lett. 109, (2012). [31] O.J.E. Maroney, How statistical are quantum states?, arxiv: (2012). [32] M.S. Leifer an O.J.E. Maroney, Maximally epistemic interpretations of the quantum state an contextuality, Phys. Rev. Lett. 110, (2013). Appenix A: Proof of Corollary 1 For any istinct ψ, ψ S, let L ψ,ψ be the set efine by Theorem 1, i.e., P Λ Ψ (L ψ,ψ ψ) = 1 P Λ Ψ (L ψ,ψ ψ ) = 0, an for any ψ S efine the (countable) intersection L ψ ψ S\{ψ} L ψ,ψ. This satisfies { P Λ Ψ (L ψ ψ 1 if ψ = ψ ) = 0 otherwise. (Here we have use that for any probability istribution P an for any events L, L, P (L) = P (L ) = 1 implies that P (L L ) = 1.) To efine the function f, we specify the inverse sets f 1 (ψ) = L ψ \ ( L ψ ). ψ S\{ψ} The function f is well efine on ψ S f 1 (ψ) because, by construction, the sets f 1 (ψ) are isjoint for ifferent ψ S. Furthermore, it follows from the above that for any ψ S P Λ Ψ (f 1 (ψ) ψ) = 1. This implies that f(λ) = Ψ hols with probability 1 conitione on Ψ = ψ. The assertion of the corollary then follows because this is true for any ψ S. Appenix B: Quantum correlations The aim of this appenix is to erive the boun (7) use in the proof of the uniqueness theorem. Note that the state φ, efine by (3), has support on H H, where H = span{ 0, 1,..., 1 }. Since the projectors Π a x an Π b y, efine by (5) an (6), for a A n an b B n an for x, y {0,..., 1} also act on H, we can restrict to this subspace. For j {0,..., 1} an k {0,..., 1} the projectors Π k j are along the vectors ζ k j = ( ˆX ) k j, where ˆX enotes the generalise Pauli operator (efine in the main text). To write these vectors out more explicitly, we consier the iagonal operator Ẑ 1 j=0 e2πij/ j j an the unitary U 1 jk e2πijk/ j k. These have the property that ˆX = U Ẑ U, an hence it follows that ( ˆX ) k = U (Ẑ) k U. Thus, we can write ζ k j = 1 m=0 1 exp[ 2πi 1 exp[ ikπ n ] (m + k/ j)] m, for k 0. Note that ζj k ζk j = δ j,j, implying that, for each k, {Π k j } j is a projective measurement on H. Recall that the probability istribution in (7) is obtaine from a measurement of φ with respect to these projectors, i.e., P XY AB (x, y a, b) = ( ζx ζ a y ) φ b 2. We are now going to show that P XY AB (x, x a, b) = sin2 π x for a b = 1, an 2 sin 2 π P XY AB (x, x 1 0, 1) = sin2 π x 2 sin 2 π, (B1). (B2) For this it is useful to use the relation that for any operator C, (11 C) φ = (C T 11) φ, where C T enotes the transpose of C in the i basis. Thus, noting that = U, we have U T ( ζx ζ a x ) φ b = 1 x U Ẑ a (U )2 Ẑ b U x. Then, using (U )2 = 1 e 2πij(k+m)/ k m = k k, we fin jkm ( ζx ζ a x ) φ b = 1 3/2 j k=0 e πij n (a b) = 1 We can hence use 1 e iy 2 = 4 sin 2 y 2 x πi 1 e 3/2 1 e πi to obtain P n, XY AB (x, x a, b) = sin2 π(a b), 2 sin 2 π(a b) n (a b). n (a b)

8 8 from which (B1) follows. (B2) can be obtaine by a similar argument. These two expressions immeiately imply I n, (P XY AB ) = (1 sin2 π 2 sin 2 π Using x 2 x 4 /3 sin 2 x x 2 for 0 x 1 leas to the boun (7). Appenix C: Proof of Lemma 2 In the following we use the abbreviations P XY ABλ P XY ABΛ (,,, λ) an P XY abλ = P XY ABλ (, a, b) for the istributions conitione on Λ = λ an (A, B) = (a, b). The inequality in Lemma 2 can be expresse in terms of the total variation istance, efine by D(P X, Q X ) 1 2 x P X(x) Q X (x), as P Λ (λ)d(p X a0λ, 1/) 4 I n,(p XY AB ). where 1/ enotes the uniform istribution over {0,..., 1}, an where a 0 = 0. Furthermore, using P XY AB = P Λ (λ)p XY ABλ (which hols because P Λ AB = P Λ ) an that I n, is a linear function, we have I n, (P XY AB ) = P Λ (λ)i n, (P XY ABλ ). It therefore suffices to show that, for any λ, D(P X a0λ, 1/) 4 I n,(p XY ABλ ). For this, we consier the istribution P X 1 aλ, which correspons to the istribution of X if its values are shifte by one (moulo ). Accoring to Lemma 5 an using we have ). D(P X a0λ, 1/) 4 D(P X 1 a 0λ, P X a0λ). The assertion then follows with I n, (P XY ABλ ) = x P XY a0b 0λ(x, x 1) x,a,b a b =1 D(P X 1 a0b 0λ, P Y a0b 0λ) + a,b a b =1 D(P X 1 a0λ, P X a0λ), P XY abλ (x, x) D(P X abλ, P Y abλ ) where we have set b 0 1; the first inequality follows from Lemma 4; the secon is obtaine with P X abλ = P X aλ an P Y abλ = P Y bλ (which are implie by the conitions state in the lemma) as well as the triangle inequality for D(, ). Appenix D: Aitional Lemmas Lemma 3. For any unit vectors ψ, ψ H 1 an φ, φ H 2, where im(h 1 ) im(h 2 ) an ψ ψ = φ φ, there exists an isometry U : H 1 H 2 such that Uψ = φ an Uψ = φ. Proof. With α = ψ ψ = φ φ an β = 1 α 2 we can write ψ = αψ + βψ an φ = αφ + βφ with unit vectors ψ an φ orthogonal to ψ an φ, respectively. The isometry U can be taken as any that acts as φ ψ + φ ψ on the subspace spanne by ψ an ψ. Lemma 4. For two ranom variables X an Y with joint istribution P XY, the total variation istance between the marginal istributions P X an P Y satisfies D(P X, P Y ) 1 x P XY (x, x). Proof. Consier P XY P XY X Y, the istribution of X an Y conitione on the event that X Y, as well as P XY = P XY X=Y so that P XY = p P XY + (1 p )P = XY where p 1 x P XY (x, x). The marginals also obey this relation, i.e., P X P Y = p P X + (1 p )P X = = p P Y + (1 p )P Y =. Hence, since the total variation istance is convex, D(P X, P Y ) p D(P X, P Y ) + (1 p )D(P = X, P = Y ) p, where we have use the fact that the total variation istance is at most 1, as well as D(P X =, P Y = ) = 0 in the last line. Lemma 5. The total variation istance between any probability istribution with range {0, 1,..., 1} an the uniform istribution over this set, 1/, is boune by Proof. Using 1 D, we fin D(P X, 1/) D(P X 1, P X ). 1 P X i = 1/ an the convexity of D(P X, 1/) = D( 1 P X, 1 P X i ) 1 D(P X, P X i ).

9 9 Because D(P X (i 1), P X i ) = D(P X 1, P X ) for all i we have for i /2 Combining this with the above conclues the proof. D(P X, P X i ) D(P X, P X (i 1) ) + D(P X (i 1), P X i ) = D(P X, P X (i 1) ) + D(P X 1, P X ). Using this multiple times yiels D(P X, P X i ) id(p X 1, P X ). Similarly, for i /2, we use D(P X, P X i ) D(P X, P X (i+1) ) + D(P X (i+1), P X i ) = D(P X, P X (i+1) ) + D(P X 1, P X ) multiple times to yiel D(P X, P X i ) ( i)d(p X 1, P X ). Thus, D(P X, P X i ) /2 i + i= /2 +1 ( i) D(P X 1, P X ) 2 = D(P X 1, P X ). 4 Note that there are istributions that achieve the boun of Lemma 5, as can be seen for even an the istribution P X = (2/, 2/,..., 2/, 0, 0,...), for which D(P X, 1/) = 1/2 an D(P X 1, P X ) = 2/.