3 Bananaworld. 3.1 Einstein Podolsky Rosen bananas

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1 3 Bananaworld The really remarkable thing about our quantum world is the existence of nonlocal correlations correlations between events at separate locations that can t be explained by either of the two sorts of explanation we are familiar with in classical physics or in everyday life: a direct causal connection in which information is transmitted from one event to the other by some physical system moving continuously at finite speed between the correlated events, or a common cause that is the source of the same information transmitted to the correlated events. The plan in this chapter is to consider correlations in Bananaworld, an imaginary island covered with banana trees, with a variety of peculiar correlations between the tastes of the bananas and how they are peeled. To remind you: a Bananaworld banana can only be peeled in one of two ways, either by peeling the stem end (S) or the top end (T ) and, once peeled, a Bananaworld banana tastes just like an ordinary banana (0), or it tastes intense, incredible, indescribably delicious (1). Whether the taste is 0 or 1 is an objective fact, not a subjective matter of opinion. So there are two things you can do to a banana in Bananaworld (corresponding to a binary choice of quantum measurements), and two possible outcomes (corresponding to two possible measurement outcomes). I ll begin with the Einstein Podolsky Rosen correlation and show how to simulate this correlation with a shared random local resource, which is equivalent to showing that the correlation has a common cause explanation. Then I ll consider a superquantum correlation, the Popescu Rohrlich correlation, and I ll show that this correlation can t be perfectly simulated with local resources, but a quantum simulation with entangled quantum states does better than the optimal simulation with local resources. So there s no common cause explanation of the Popescu Rohrlich correlation. Since the correlated events can be arbitrarily far apart, there can t be a direct causal explanation either, because information would have to be transmitted faster than light between the correlated events. As I ll show below, this is a version of a nonlocality result first proved by John Bell for the correlations of entangled quantum states. 3.1 Einstein Podolsky Rosen bananas In May, 1935, Albert Einstein, Boris Podolsky, and Nathan Rosen published an article in the Physical Review with the title Can quantum-mechanical description of physical reality be considered complete? 1 They argued that the only way to make sense of quantum mechanics is to suppose that the state descriptions of the theory are incomplete: something has been left out of the story. The article created quite a stir at the time, with pre-publication headlines in the Science section of the The New York Times of May 4, 1935, based on an interview with Boris Podolsky, announcing: Einstein Attacks Quantum Theory. Scientist and Two Colleagues Find It Is Not Com- 48

2 3.1 Einstein Podolsky Rosen bananas plete Even Though Correct. Einstein was upset by the interview. In a terse statement in the May 7 issue of the New York Times he pointed out that the interview was given without his authority, adding: It is my invariable practice to discuss scientific matters only in the appropriate forum and I deprecate advance publication of any announcement in regard to such matters in the secular press. The response came in an article with the same title by Niels Bohr in the October issue of the same year in the Physical Review. 2 At the time, Bohr s analysis was regarded as sufficiently authoritative that the issue was treated as more or less settled in favor of the orthodox position. Most physicists took the view if it s good enough for Bohr, it s good enough for me or, if there was some lingering worry, would have endorsed Feynman s remark about quantum mechanics: 3 It has not yet become obvious to me that there s no real problem. I cannot define the real problem, therefore I suspect there s no real problem, but I m not sure there s no real problem. In 1952, David Bohm published a hidden variable extension of quantum mechanics in a twopart article in the Physical Review that reproduced all the empirical predictions of quantum mechanics. 4 In the previous year, Bohm had published a widely acclaimed textbook on orthodox quantum mechanics but was dissatisfied with his own discussion of the conceptual problems of the theory. The hidden variable theory is a deterministic theory, in which quantum phenomena arise from the behavior of particles that move on definite trajectories guided by Schrödinger s wave function as a field pervading all of space. Bohm s theory is empirically equivalent to quantum mechanics if you assume that a quantum state at a particular time describes a whole ensemble of Bohmian particles distributed in space in a way that precisely fits the probabilities defined by the state according to the Born rule. It s an ingenious feature of the theory that if the Bohmian particles are distributed in this way at any time, Bohm s equation of motion for the particle trajectories guarantees that this continues to be the case as the quantum state evolves in time. One could say that even if the positions of the Bohmian particles were distributed in some wildly non-quantum way in the early history of the universe, the distribution must have become quantum at some point in time, because that s the way they are distributed now. So if you assume that the Bohmian particles have settled down to an equilibrium distribution in space that fits the probabilities defined by the quantum state of the universe, then the particle positions are hidden variables that remain hidden because they can t be pinned down more precisely than quantum mechanics allows. I first came across the Einstein Podolsky Rosen paper and Bohr s reply as an undergraduate at the University of Cape Town and was quite shocked to discover that physicists like Einstein and Bohr could disagree about the foundations of physics. I thought reasonable people could disagree about art or literature or philosophy, but only cranks disagreed about the basic theories of physics. At the same time, I was introduced to Bohm s hidden variable theory, and I eventually became Bohm s graduate student. I worked on a different sort of hidden variable theory for my dissertation, where the linear quantum dynamics is modified by an additional nonlinear term involving 49

3 3 Bananaworld hidden variables that kicks in during a measurement process and produces the notorious collapse of the wave function (see the section 10.2, The Measurement Problem, in Chapter 10, Making Sense of It All). The theory is empirically distinguishable, in principle, from quantum mechanics and more or less met its demise in a 1967 experiment that failed to confirm a conjecture that the statistics of measurement outcomes in sufficiently rapid sequences of measurements would differ from quantum probabilities. 5 Today there are much more sophisticated dynamical collapse theories, notably the Ghirardi Rimini Weber theory and its variants. 6 These theories also conflict with quantum mechanics, but under conditions that are very difficult to produce with current technology. One might have thought that Bohm s theory was at least a first step to the sort of completion of quantum mechanics that the Einstein Podolsky Rosen paper called for, but in correspondence with Max Born dated May 12, 1952, Einstein dismissed Bohm s theory as too cheap for me. 7 Quite possibly, what Einstein objected to in Bohm s theory was its nonlocality, roughly that what happens over here instantaneously affects what happens over there (which conflicts with special relativity, even though the conflict would be unobservable if the particle positions, the hidden variables in Bohm s theory, are in the equilibrium Born distribution). The behavior of a Bohmian particle where it s trajectory will go at a particular time depends on where all the other Bohmian particles are located. So a change in the position of a particle is instantaneously transmitted nonlocally to remote particles via the quantum wave function as a guiding field, which means that whether a photon goes one way or the other in a beamsplitter instantaneously affects the outcome of a measurement on an entangled remote photon, no matter where it is in the universe. In 1966, John Bell published a critical review of several results about the impossibility of reconstructing quantum mechanics as a hidden variable theory, given various assumptions about the hidden variables. The article concluded with a discussion of Bohm s theory in which Bell pointed out that the theory evades these no go results because the equations of motion are nonlocal, so that an explicit causal mechanism exists whereby the disposition of one piece of apparatus affects the results obtained with a distant piece. He added that the Einstein Podolsky Rosen paradox is resolved in the way in which Einstein would have liked least, and asked whether one could prove that any hidden variable account of quantum mechanics must have this extraordinary character. 8 Bell answered this question by showing that adding hidden variables to quantum mechanics to complete the story is a non-starter if the hidden variables are required to be local, in the sense that they provide a common cause explanation for quantum correlations when a direct causal connection isn t possible. (See subsection 3.5.2, Correlations in the More section at the end of the chapter for the relation between a common cause and Bell s locality condition.) Bell showed that any such theory would have to satisfy a certain inequality, which is violated by the correlations of entangled quantum states for certain combinations of measurements that Einstein, Podolsky, and Rosen didn t consider. He concluded: 9 In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant [so consistent 50

4 3.1 Einstein Podolsky Rosen bananas with special relativity]. Bell s theorem, as it is now called, appeared in 1964 two years before the long-delayed publication of the article in which Bell posed the question about Bohm s theory. The original 1935 formulation of the Einstein Podolsky Rosen argument in the Physical Review article was apparently mostly due to Podolsky, and Einstein thought it was rather convoluted. As he put it in a 1948 paper in the journal Dialectica, 10 the argument rests on two assumptions: a separability assumption, that physical objects in different parts of space have their own independent state of existence or being thus ( So-sein in the German original), and a locality assumption, that if A and B are separated, then you can t directly, or instantaneously, affect the being thus of B by doing something to A, and conversely in particular, as he wrote in a letter to Max Born, you can t directly affect B by measuring A: 11 That which really exists in B should therefore not depend on what kind of measurement is carried out in part of space A; it should also be independent of whether or not any measurement at all is carried on in space A. Einstein, Podolsky, and Rosen formulated their argument for correlations between position and momentum measurements on two separated quantum particles in an entangled state. Position and momentum are observables with continuous ranges of values, and a rigorous version of the argument would involve a rather sophisticated mathematical treatment. Instead of position and momentum, most modern discussions follow a version proposed by Bohm and consider correlations between the X and Z observables of a qubit (defined in the previous chapter) for a particular entangled state. Suppose that two separated photons, A and B, are in the entangled state + i = 0i 0i+ 1i 1i = +i +i + i i considered at the end of the previous chapter. (Remember: the equal coefficients 1/ p 2 for each of the product terms in the linear superposition are left out for ease of reading.) If Z stands for polarization in the z direction, and X for polarization in a diagonal direction at a 45 angle to the z direction, then 0i and 1i represent states of horizontal and vertical polarization in the z direction, and +i and i represent states of horizontal and vertical polarization in the diagonal direction. If you measure either Z or X on photon A, you can predict photon B s polarization with certainty, because the outcomes of a Z-measurement on A and B are perfectly correlated for the two photons in this entangled quantum state, and similarly the outcomes of an X-measurement on the two photons are perfectly correlated. On Einstein s locality assumption, a Z-measurement of photon A can t change what really exists in the remote region where photon B is situated. So it seems that you would have to say that photon B has a particular polarization in the z direction, either horizontal or vertical, even before B s polarization is measured. Or you would have to say that B has a definite value of some variable that determines a particular value for polarization in the z direction if one were to measure the polarization, in the sense that the values of this variable provide the photon with an instruction set on the basis of which the photon makes a transition to a state of horizontal or vertical polarization when the polarization is measured in a particular direction. The particular polarization values, or 51

5 3 Bananaworld the instruction sets, correspond to a common cause of the correlation. Similarly, since a measurement of diagonal polarization corresponding to the observable X on photon A reveals B s diagonal polarization with certainty, one would have to say that B has a particular polarization, either horizontal or vertical, in a direction 45 to z as well, or an instruction set for this polarization. But the quantum state + i is inconsistent with the assumption that photon B has a definite polarization in both polarization directions before the measurements. The observables Z and X don t commute, so there is no quantum state in which they both have definite values. (See the section Noncommutativity and Uncertainty in the Supplement Some Mathematical Machinery at the end of the book for more on this.) Einstein concluded that the state descriptions of quantum mechanics are incomplete: they leave something out of the full description. The Einstein Podolsky Rosen argument for photons If you have two separated photons, A and B, in the entangled state + i = 0i 0i + 1i 1i = +i +i + i i, then the outcomes of a Z-measurement (linear polarization in some direction z) on the two photons are perfectly correlated, and the outcomes of an X-measurement (diagonal polarization at an angle 45 to z) are perfectly correlated. So you can predict photon B s polarization, either Z or X, with certainty by measuring the corresponding polarization of photon A. On Einstein s locality assumption, a measurement of photon A can t change what really exists in the remote region where photon B is situated. It seems that you would have to say that photon B has a particular polarization even before B s polarization is measured. Or you would have to say that B has a definite value of some variable that provides the photon with an instruction set on the basis of which the photon makes a transition to a state of horizontal or vertical polarization when the polarization in a particular direction is measured. The instruction set, with a corresponding instruction set for A s correlated measurement outcome, corresponds to a common cause for the correlation. The observables Z and X don t commute, so there is no quantum state in which they both have definite values. Einstein, Podolsky, and Rosen concluded that the state descriptions of quantum mechanics are incomplete: they leave something out of the full description. Actually, as Einstein pointed out, you don t need to consider noncommuting observables, and you could make the argument with just one observable, say Z. The conclusion that B s polarization in the z direction, either horizontal or vertical, is pre-determined by an instruction set, even before it is measured, already shows that the state description by the quantum state + i is incomplete, because the quantum description doesn t include information about this pre-determined polarization value. The Einstein Podolsky Rosen argument starts with an entangled quantum state, and it was this 52

6 3.1 Einstein Podolsky Rosen bananas argument that brought entanglement to the attention of the physics community as a surprising and possibly problematic quantum phenomenon. But the argument itself does not really depend on anything particularly quantum about the Z and X correlations they consider. What Einstein, Podolsky, and Rosen do with an entangled state amounts to saying: Look, the correlations between measurement outcomes for two particular noncommuting observables are exactly like the classical correlations you get when you have two systems that start out correlated from a common source, like two billiard balls moving in different directions after a collision, where the positions and momenta are correlated. You can predict with certainty the outcome of a measurement on one billiard ball from a measurement on the other, for either position or momentum as you choose. For two entangled photons in the state + i, the outcome of a Z measurement on one photon is uncorrelated with the outcome of an X measurement on the second photon, but the outcomes of Z measurements on the two photons are perfectly correlated, as well as the outcomes of X measurements. That s like the relation between position and momentum in the original Einstein Podolsky Rosen paper. So if Alice and Bob are separated and share two photons in the entangled quantum state + i, and they each have a choice between Z and X measurements, then the outcomes are perfectly correlated if they measure the same observables, but uncorrelated if they measure different observables. This is a classical correlation that can be explained quite simply by a common cause, as I ll show below. That s the point of the Einstein Podolsky Rosen argument: there must be elements of reality for such a correlation, to use their terminology (Einstein s being thus ), which are the common cause of the correlations, and quantum mechanics is incomplete because it doesn t include these elements of reality in the quantum state description. Putting it a little differently, the Einstein Podolsky Rosen argument is that entangled states in quantum mechanics involve classical correlations that can be simulated with local resources, and this means that something has been left out of the quantum description the theory is incomplete. For two photons A and B in the entangled state + i, the sort of correlation Einstein, Podolsky, and Rosen are talking about is between an A-observable, either A 1 or A 2, and a B-observable, either B 1 or B 2, where A 1,A 2 are the same observables for A as B 1,B 2 are for B. The possible outcomes of measuring the observables can be labeled 0 and 1, and the quantum state + i encodes correlations between A and B measurement outcomes which persist as the photons separate of the following sort: if the same observable is measured on A and on B, the outcomes are the same, with equal probability for 00 and 11 if different observables are measured on A and on B, the outcomes are uncorrelated, with equal probability for 00, 01, 10, 11 the marginal probabilities the probabilities for each possible outcome, 0 or 1, when either observable is measured on A, or when either observable is measured on B are 1/2, irrespective of what observable is measured on the paired photon, or whether any observable is measured at all These are just the probabilities for the possible outcomes of measurements of the observables Z (A 1 or B 1 ) or X (A 2 or B 2 ) on two photons in the entangled state + i. 53

7 3 Bananaworld The Einstein Podolsky Rosen correlation For two photons A and B in the entangled state + i and measurements of observables Z and X, the Einstein Podolsky Rosen corelation is (i) if the same observable is measured on A and on B, the outcomes are the same, with equal probability of 1/2 for 00 and 11, and if different observables are measured, the outcomes are uncorrelated, with equal probability of 1/4 for 00, 01, 10, 11 (ii) the marginal probabilities the probabilities for each possible outcome, 0 or 1, when either observable is measured on A, or when either observable is measured on B are 1/2, irrespective of what observable is measured on the paired photon, or whether any observable is measured at all The analogous correlation in Bananaworld is (i) if the peelings are the same (SS or TT), the tastes are the same, with equal probability of 1/2 for 00 and 11, and if the peelings are different (ST or TS), the tastes are uncorrelated, with equal probability of 1/4 for 00, 01, 10, 11 (ii) the marginal probabilities for the tastes 0 or 1 if a banana is peeled S or T are 1/2, irrespective of how the paired banana is peeled, or whether or not the paired banana is peeled The term marginal probability is used when probabilities are defined for two or more random variables, A, B,..., and you want to consider just one random variable, say A, or, more generally, some subset of all the random variables. (A random variable is just a variable whose values are associated with certain probabilities.) The marginal probabilities of A are the probabilities that A has certain values, irrespective of the values of the other random variables. The joint probabilities of A, B,... are the probabilities of sequences of values for A, B,.... These notions apply to the probabilities of measurement outcomes of quantum observables. In Bohm s version of the Einstein Podolsky Rosen argument, the entangled state is i = 0i 1i 1i 0i and the observables are oppositely correlated rather than correlated, so the outcomes are different if the same observable is measured on the two qubits, rather than the same. The Einstein Podolsky Rosen argument is the same for the two versions: you could convert one version to the other just by having Alice (but not Bob) flip her outcomes, exchanging 0 for 1 and 1 for 0. In Bananaworld, there are Einstein Podolsky Rosen banana trees with bunches of just two bananas, where peelings and tastes are correlated as in the Einstein Podolsky Rosen argument, with correlations that persist when the bananas are separated by an arbitrary distance, as follows (see Figure 3.1): if the peelings are the same (SS or TT), the tastes are the same, with equal probability for 00 and 11 if the peelings are different (ST or TS), the tastes are uncorrelated, with equal probability 54

8 3.1 Einstein Podolsky Rosen bananas for 00, 01, 10, 11 the marginal probabilities for the tastes 0 or 1 if a banana is peeled S or T are 1/2, irrespective of whether or not the paired banana is peeled Figure 3.1: The Einstein Podolsky Rosen correlation in Bananaworld. The inputs are the possible choices of how to peel a banana. The outputs are the possible tastes of a peeled banana. The = sign indicates that the tastes are the same. The dice symbol indicates that the tastes are uncorrelated, so that each possible pair of outcomes occurs with the same probability. The symbol ST is short for Alice peels S and Bob peels T, and similarly 01 is short for Alice s banana tastes 0 and Bob s banana tastes 1, with Alice s peeling or taste represented by the first member of the pair, and Bob s peeling or taste represented by the second member of the pair. The stipulation that the marginal probabilities are all 1/2 the probabilities that a banana tastes 0 or 1 when peeled a certain way guarantees that the correlations satisfy a no-signaling principle. Expressed in Bananaworld terms, the principle is that the taste of a banana is independent of how a remote banana is peeled, or whether or not a remote banana is peeled (which doesn t rule out correlations between the tastes). More generally, if you are talking about measurements, no information should be available in region A about alternative choices made by Bob in region B Alice, in region A shouldn t be able to tell what Bob measured in region B, or whether Bob 55

9 3 Bananaworld performed any measurement at all, by looking at the statistics of her measurement outcomes, and conversely. As Nicolas Gisin puts it, the no-signaling principle is no communication without transmission, where transmission involves a physical signal that moves along a continuous trajectory at a certain finite speed through the space-time continuum. 12 Violating the no-signaling principle would mean the possibility of superluminal, or even instantaneous, signaling. But the no-signaling principle is not specifically a relativistic constraint on the motion of a physical entity through space-time there s no reference to the velocity of light, even implicitly. It s a much more fundamental constraint. What s excluded is that something happening here has an effect over there, without delay and without anything physical passing from here to there. You might say that the principle is part of what s meant by treating two physical systems as separate systems. The no-signaling principle The no-signaling principle: no information should be available in region A about alternative choices made by Bob in region B Alice, in region A shouldn t be able to tell what Bob measured in region B, or whether Bob performed any measurement at all, by looking at the statistics of her measurement outcomes, and conversely. Expressed in Bananaworld terms, the principle is that the taste of a banana is independent of how a remote banana is peeled, or whether or not a remote banana is peeled. Before considering how Alice and Bob, limited to local resources, could simulate the correlations of Einstein Podolsky Rosen banana pairs, consider a simpler problem. Suppose the correlation is that if the peelings are the same, the tastes are the same, and if the peelings are different, the tastes are different. if you think about it for a moment, there s a simple way Alice and Bob could win the simulation game for this correlation. (To remind you see section 1.2, Why Bananaworld? in Chapter 1 the moderator, who can communicate with Alice and Bob separately, gives each player one of two prompts, S or T, at the beginning of each round of the game, and the players, who are not allowed to communicate with each other during the game, are supposed to respond to their prompts with one of two responses, 0 or 1. They win a round if the responses and the prompts are correlated in the right way. The game is played over many rounds, and at the end of the game they win a prize, where the value of the prize depends on the number of rounds they win. So the aim is to figure out a strategy that will enable them to win the maximum number of rounds.) For this game, a winning strategy is to respond 0 if the moderator s prompt is S and 1 if the prompt is T. Then Alice and Bob both respond 0 or both respond 1 if the prompts are the same, and their responses will be different if the prompts are different. That s easy, but now suppose the correlation is a little more complicated. Suppose the response for a given prompt has to be random, so for the prompt S the response can t always be 0. Both possible responses, 0 and 1, should come up with equal probability over many rounds of the game, and similarly for the prompt T. Can Alice and Bob win this game? They can if they are allowed to consult the same list of random numbers during the game. You might want to think how they could do this before I give you the solution. 56

10 3.1 Einstein Podolsky Rosen bananas Here s what they could do. Alice and Bob generate a long list of random bits, 0 s and 1 s, during the strategy session before the game starts, perhaps by tossing a fair coin, with 0 for heads and 1 for tails, with at least as many bits in the list as there are rounds in the simulation game. They each make a copy of the list of bits, which they take with them to consult during the game. The strategy is to respond with the bit on the list if the prompt is S, in order down the list for successive rounds, and to flip the bit on the list if the the prompt is T. Then if Alice and Bob receive the same prompt, they both respond 0 or both respond 1, but the response will be independent of the prompt, and random, because the list is random. If they receive different prompts, one of them responds with the bit on the list, and the other with the flipped bit, so the responses are different, but 0 s and 1 s come up with equal probability. Now consider how Alice and Bob, limited to local resources, could simulate the correlations of Einstein Podolsky Rosen banana pairs. They could adopt a modification of the previous strategy and generate two long lists of random 0 s and 1 s by tossing two fair coins in the strategy session before the start of the game. So Alice and Bob each have copies of two random lists of 0 s and 1 s that I ll call the S-list and the T -list, with at least as many bits in each list as there are rounds in the simulation game. At each round, they adopt the following strategy: if the prompt is S, the response (for both Alice and Bob) is a bit from the S-list if the prompt is T, the response (for both Alice and Bob) is a bit from the T -list If the prompts are the same, they respond with the same random bit, 0 or 1, because they consult the same list of shared bits, and they read off the random bits in order for successive rounds. If the prompts are different, they respond with bits from different lists, which are randomly related to each other. The marginal probabilities of 0 and 1 for Alice and Bob separately are 1/2 for either prompt, irrespective of the prompt to the remote player. Figure 3.2 illustrates the idea. What does this have to do with the Einstein Podolsky Rosen argument for the incompleteness of quantum mechanics? The local simulation of Einstein Podolsky Rosen bananas shows that it is possible for two systems to exhibit Einstein Podolsky Rosen correlations if they both have their own being thus, but the separate being thuses originate from a common source as the common cause of the correlation. In the case of the bananas, this will be the case for two bananas on an Einstein Podolsky Rosen bunch that grow as a pair on a single banana tree, each with a shared value of some banana variable, the being thus of the bananas. The banana variable corresponds to the shared simulation random variable, represented by a pair of random bits from the S-list and the T -list. The tastes for all possible pairs of peelings are determined by the shared value of, and the distribution of shared values for many Einstein Podolsky Rosen pairs of bananas provides an explanation of the correlation. The correlations are just like the correlations you would get if you took playing cards from a deck of cards, cut each card in half and mailed the two halves to separate addresses. Opening an envelope containing a half-card would reveal the color of the half-card and, instantaneously, the color of the half-card at the distant address. In this case, the being thus of a half-card would be specified by a shared color variable, with two possible values, red or black. Here each of the subsystems, a banana in an Einstein Podolsky Rosen pair or a half-card, has its own being thus. 57

11 3 Bananaworld Figure 3.2: The Einstein Podolsky Rosen simulation game. Alice and Bob can win the game if they respond on the basis of two shared random lists of bits that they consult in order for each round of the game, an S-list for S prompts and a T -list for T prompts. This is Round 4, the prompts are both S, so according to the shared S-list, Alice and Bob both respond with a 0. This is characteristic of correlations that can be simulated with local classical resources. As I ll show in the following section, this is not the case for Popescu Rohrlich bananas: the Popescu Rohrlich correlations exclude a being thus in this sense. In effect, the Einstein Podolsky Rosen incompleteness argument rests on showing that the shared randomness required for the simulation the common cause of the correlations is missing 58

12 3.1 Einstein Podolsky Rosen bananas from the quantum description. Einstein s being thus, represented by the value of a shared random variable, is the common cause of the correlations, the instruction set that tells the system how to respond to a measurement. What s shared is a random distribution of values of a common cause variable, like the values of the banana variable or the random sequence of 0 s and 1 s from the S-list and the T -list in the simulation game. Each banana in a pair picked from the same bunch on an Einstein Podolsky Rosen tree shares a value of, and the values of are distributed randomly over different bunches on the tree. Similarly, Alice and Bob each have copies of the S-list and the T -list, generated by some suitable random process. Picking a pair of bananas that share a value of is like Alice and Bob each picking the same pair of corresponding bits from the S-list and the T -list for a round in the simulation game. Einstein s argument is simply that, if you have perfect correlations of this sort, and you can exclude a direct causal influence between the two systems (because the correlations persist if the systems are separated by any distance), then the only explanation for the correlations is the existence of a common cause. Since the quantum description of the correlated systems lacks a representation of the common cause, quantum mechanics must be incomplete. What s missing in at least some quantum state descriptions (like + i) is something corresponding to the banana variable or the simulation variable with values defined by a pair of corresponding bits from the S-list and the T -list. 59

13 3 Bananaworld The bottom line The Einstein Podolsky Rosen correlation can be simulated with local resources. A local simulation shows that it is possible for two systems to exhibit an Einstein Podolsky Rosen correlation if they both have their own being thus, in Einstein s sense, and if the separate being thuses originate from a common source as the common cause of the correlation. In the case of Einstein Podolsky Rosen bananas, this will be the case if they originate on a banana tree as a particular pair, each with a shared value of some banana variable, the being thus of the bananas. The tastes for all possible pairs of peelings are determined by the shared value of, and the distribution of shared values for many Einstein Podolsky Rosen pairs of bananas provides an explanation of the correlation. In a simulation of the correlation, the variable corresponds to the simulation variable. The Einstein Podolsky Rosen argument is that (i) if you have a perfect correlation of this sort, and you can exclude a direct causal influence between the two systems (because the correlations persist if the systems are separated by any distance), then the only explanation for the correlation is the existence of a common cause or shared randomness (ii) since the description of the correlated systems in quantum mechanics lacks a representation of the common cause, quantum mechanics must be incomplete. What s missing in a quantum state description like + i is something corresponding to the banana variable, or the simulation variable in a local simulation. 3.2 Popescu Rohrlich Bananas and Bell s Theorem The Einstein Podolsky Rosen argument lingered for about thirty years until John Bell showed that any common cause explanation of a probabilistic correlation between measurement outcomes on two separated systems would have to satisfy an inequality, now called Bell s inequality. He also showed that the inequality is violated by measurements of certain two-valued observables of a pair of quantum systems in an entangled state. So Einstein s intuition about the correlations of entangled quantum states, and the Einstein Podolsky Rosen argument that depends on this intuition, turn out to be wrong. There are correlations between quantum systems, where a causal influence from one system to the other can be excluded, that have no common cause explanation. The result is known as Bell s theorem. It took a while before the implications of Bell s theorem penetrated mainstream physics, probably around the time that Alain Aspect and colleagues confirmed the violation of Bell s inequality in a series of experiments on entangled photons in the early 1980 s. It took several more years for 60