Lecture 12c: The range of classical and quantum correlations

Transcription

1 Pre-Collegiate Institutes Quantum Mechanics 015 ecture 1c: The range of classical and quantum correlations The simplest entangled case: Consider a setup where two photons are emitted from a central source to two different observers Alice and Bob that are far apart. Alice and Bob can measure the photon they receive in one of two bases, A and A for Alice, and B and B for Bob. These settings correspond to different angle settings for their polarizing beam splitters (PBS). source PBS The PBS can be orientated two different ways on each side. We will write schematically the choice by an arrow which points in the direction of orientation of each PBS. This is achieved by rotating the PBS. This setup is labelled, meaning that Alice has two possible settings and two possible outcomes () and the same for Bob. settinga setting B setting A source setting B The sourceofphotons may ormaynot be entangled. We want to considermultiple scenariosin this lecture. When Bob and Alice make a measurement the two possible outcomes will be labeled ±1. In each run of the experiment, the photon will exit either channel with total probability 1. We desire to have a measure of whether there exists a joint probability distribution (jpd) or not. Or, as more commonly stated, what correlations are possible for classical systems? (ater we will examine what are the range of correlations for quantum systems.) To begin, we will consider cases where the probability at each side is 50/50 for each outcome. In this case we have, E(A) = E(A ) = E(B) = E(B ) = 0. Because of this (and the fact that the variances are all equal to 1) we have the simplification that the correlation between two observables, ρ(xy), equals the second moment E(XY). We want to explore the range of correlations, ρ(ab),ρ(ab ),ρ(a B),ρ(A B ) = E(AB),E(AB ),E(A B),E(A B ). Note that these are the only possible observables in any one run of the experiment. For example, we can not observe the correlations E(AA ) or E(BB ). (This would correspond to Alice, say, measuring both the A setting and A setting for the same photon - she can only set her angle to one value for each run of the experiment.) Even though these can not be observed in any one run of the experiment we can still express the probability if Alice had chosen the other setting. The reason is the following: Alice and Bob have free will to chose their settings after the photons have been emitted from the source and are on their way to Alice and Bob. (Or, even one side can choose after the other half of the pair has been measured.) If we think of the photons as separable entities, that is, there is no connection between the two after they are emitted, then each half should already know what its outcome will be regardless of what is occurring on the other side. Thus the joint probability of what can occur should exist over the 4 possible settings. That is, we should be able to write consistently the atoms as p aa bb for every possible outcome. If we can not express the jpd (that is, if it does not exist), then the outcome on one side depends, in some way, on what choice is made on the other side. The system is then contextual, in that what happens on one side depends on the context (setting chosen) on the other side. Howcanwegoaboutdeterminingwhetherajpd existsornot? Theresultsofexperimentsdonotgiveprobabilities directly but give expectations and correlations. That is, we make many runs of the experiment and look at the averages. We can express the expectations for each observable (assumed to be zero here) and the correlations. We can not tabulate the unobservable correlations (E(AA ) and E(BB )) nor higher moments. For example, E(ABB ) can not be observed because we can not have outcomes for B and B in one run of the experiment.

2 So let s examine the correlations (i.e. second moments), E(AB),E(AB ),E(A B),E(A B ). Consider first that we experimentally find these four correlations in the same experiment (where only the two settings can be changed in different runs of the experiment). We might consider adding all four correlations, R = E(AB)+E(AB )+E(A B)+E(A B ). It should be clear that since the range of correlations is [,] that the range of R is [-4,+4]. It turns out that this is not of much use, for these are the maximum bounds whether there is a jpd or not. So we need to think of something else. Again, we want to make sure of what happens at the location B is independent of what setting Alice chooses (and vis-versa). Thus, consider the following construction, S = [E(AB)+E(AB )]+[E(A B) E(A B )]. (1) In any run of the experiment we have either B and B being the same value (++, - -), or being different values (+-, -+). Again, this is considering that the outcomes are determined before measurement (and before a choice of setting is made) and independent of the setting by Alice. In this case either we have outcomes B +B = ± if they are the same or B+B = 0 if they are different. If they are the same (and B+B = ±) then clearly B B = 0. ikewise, if B +B = 0 then B B = ±. And again, if what occurs at Bob s location is independent of what choice Alice makes for her setting, then we should have the following cases, A =,A = S = ()(B +B )+()(B B ) = ± A =,A = 1 S = ()(B +B )+( 1)(B B ) = ± A = 1,A = S = ( 1)(B +B )+()(B B ) = ± A = 1,A = 1 S = ( 1)(B +B )+( 1)(B B ) = ±. In an actual experiment S takes values between - and + because there may be a mixture of outcomes. The key point is that if the results are noncontextual (that is, outcomes do not depend on the context of what choice is made at the other side) then we have, S +, E(AB)+E(AB )+E(A B) E(A B ) +. () The inequality () is known as the Clauser-Horne-Shimony-Holt inequality (or CHSH inequality []). It is often labeled as a Bell inequality -there are several variants. Note also that there are 3 other variants of () where the minus sign is moved cyclically around. Again, the CHSH inequality is a condition that must be satisfied for a joint probability distribution to exist[3]. Quantum mechanical representation of S The discussion above was portrayed in terms of probability theory (expectations, correlations, nd moments). Here the description of S in terms of operators, that is quantum mechanically, is given. For each observable we introduce a Hermitian linear operator: Â,Â, ˆB, ˆB. As we can observe the four combinations of observables in S, these pairs of observables commute. That is, [Â, ˆB] = [Â, ˆB ] = [Â, ˆB] = [Â, ˆB ] = 0. (3) Those pairs which can not be observed in any one run have operators that may not commute (actually, will not), [Â, ] 0 [ˆB, ˆB ] 0. Thus when examining combinations of these operators, care must be given when writing pairs involving unobservable operators. With these observables we have the CHSH operator, Ŝ = ˆB +ˆB + ˆB  ˆB. (4) Each operator has eigenvalues of ±1 and thus each pair has the same eigenvalues. To compare with experiment we need to consider the expectation value of this CHSH operator, Ŝ = ˆB + ˆB +  ˆB  ˆB. (5) We will see shortly that the results for an outcome for Bob with settings ˆB need not be the same with the settings  ˆB.

3 The range of classical correlations In classical physics widely separated objects can not have any influence on each other. This is a result of special relativity: no interaction, energy, matter, or information may travel faster than the speed of light. Thus if Alice and Bob measure their photons within a time that it would take a signal between the two events to travel faster than the speed of light, there can be no interaction between the photons. In this case, the outcome for Bob can not, in any way, depend on the context of what setting Alice chooses - the system is non-contextual. As we have stated above, the CHSH inequality is a constraint on non-contextual, and thus classical correlations. The space of all correlations for the setup can be represented in a space of correlations. Within this space, the CHSH inequality defines a region within which classical correlations can exist. Since linear inequalities divide spaces in two by straight lines or planes, this region is a polytope (and is often called the local polytope)[7]. A typical picture is shown below, The local polytope has extreme points, vertices,, which are cases where the correlations are all either ±1. They are called the deterministic vertices. This image is only a schematic of the polytope as it actually resides in a higher dimensional space. As we proceed, we will only examine one edge of the polytope. Here we examine one edge of the local polytope. The deterministic vertices are, 1 S = 1 = [E(AB) =,E(AB ) =,E(A B) =,E(A B ) = ], = [E(AB) =,E(AB ) =,E(A B) = 1,E(A B ) = 1]. The horizontal line which saturates the inequality, S =, is defined by setting E(A B) = E(A B ) = x where x : [ 1]. Quantum correlations and the CHSH inequality Of course the main question of this lecture is whether quantum mechanical correlations always satisfy this inequality or not. For an entangled pair of photons we have, ψ = ψ 1 ψ = 1 [ ], (6) where we have chosen the basis of the operator Â. That is, the direction for  = is θ = 0. If Bob chooses a setting with the same angle as Alice, there will be perfect anti-correlations. If Bob chooses his axis 90 o from Alice s, there will be perfect correlations. If Alice chooses setting A and Bob chooses setting B that is at some angle θ B from 0 o, the correlation between the two will be, ˆB = cos((θ A θ B )) = cosθ B. (7) Consider the following settings, [θ A = 0, θ A = 45o,θ B =.5 o,θ B =.5o ], see figure. The four correlations are ˆB = cos(.5 o ) = 1 ˆB  ˆB then, ˆB = cos(.5 o ) = 1   ˆB = cos(.5 o ) = 1  ˆB = cos( 67.5 o ) = + 1 The CHSH operator violates the inequality in this case, θ θ θ θ =.5 o S = (+ 1 ) = <. (8)

4 Thus we see in this case that quantum mechanics violates the CHSH inequality and no joint probability distribution exists for this choice of settings. We will see shortly that this is the maximum violation that quantum mechanics allows. Other settings may or may not allow a jpd. The important point is that there are experimental arrangements that are contextual -the outcomes on one side depends on the context of choice of setting at the other side. The range of quantum correlations We can go further and ask the question of what is the limit of possible quantum mechanical correlations? This question is still an active area of current research as their is no definitive answer. Here, we will mention two known limitations. A main question in the 1990s was whether there exist any principles that defines quantum mechanics. In terms of entanglement Popescu and Rohrlich[5] proposed that the condition that you should not be able to signal from one observer to the other might be a principle to define quantum correlations. In terms of probabilities, the condition of non-signaling can be stated as follows, Non-signaling condition: P(a A) = b p(a,b A,B) = b p(a,b A,B ) P(b B) = a p(a,b A,B) = a p(a,b A,B) (9) The first expression says that what occurs at Alice s side does not depend on the setting at Bob, and the second expression is the same for Bob (is independent of Alice s setting). They found a maximal case now called a box. The box has correlations E(AB) = 1, E(A,B) = 1, E(AB ) = 1, E(A B ) = 1, (10) and satisfies the non-signaling condition 1 Note well that the box is the greatest value that S = 4 can take for correlations (all [ 1, ]). Thus it is a highly contextual case. The question is, can quantum mechanics implement a box? The answer is no. Tsirelson bound on quantum mechanical correlations There was a result in 1980[4] that placed an upper bound on the value of the value of S for quantum mechanical correlations. First note that quantum mechanics describes systems in terms of (linear, Hermitian) operators representing observables. The observables here are our four operators for different settings. To find the Tsirelson bound, simply operate Ŝ on itself. To do so, we need to recall that any A operator will commute with any B operator -they are simultaneously observable. Also recall that we can not observe two settings on one side for a single side. That means that [Â, ] 0,[ˆB, ˆB ] 0, thus we need to be careful of the ordering of these pairs (though we can rearrange As and Bs as we wish). Ŝ =  ˆB + ˆB + ˆB + ˆB +ˆB ˆB + ˆBˆB ˆB ˆB + ˆBˆB + ˆBˆB  ˆB ˆB +ˆB ˆB +ˆB  ˆB ˆB  ˆB  ˆB ˆB  ˆB  ˆB  ˆB ˆB = 4ˆ1+ + ˆB ˆB  ˆB ˆB + Â+ ˆB ˆB ˆB ˆB +ˆB ˆB +  ˆB ˆB   ˆB ˆB ˆB ˆB   = 4ˆ1  ˆB ˆB + ˆB ˆB + ˆB ˆB  ˆB ˆB = 4ˆ1+[Â, ] [ˆB, ˆB ]. (1) 1 The explicit form for the probabilities are, P(++ AB) = 1, P(++ A B) = 1 P(++ AB ) = 1, P(++ A B ) = 0 P(+ AB) = 0, P(+ A B) = 0 P(+ AB ) = 0, P(+ A B ) = 1 P( + AB) = 0, P( + A B) = 0 P( + AB ) = 0, P( + A B ) = 1 P( AB) = 1, P( A B) = 1 P( AB ) = 1, P( A B ) = 0 (11)

5 Because the remaining commutators are [Â, ]   Â, and each operator takes values ±1, the maximum value for each commutator is. Thus we get Ŝ max = 8ˆ1 S (13) This is the Tsirelson bound and is a sufficient but not necessary condition for the existence of a quantum correlation. There was a later inequality that limits quantum correlations further but again is not necessary[8], sin 1 (E(AB))+sin 1 (E(A B))+sin 1 (E(AB )) sin 1 (E(A B )) π. (14) (We will not derive this relation as it is a bit of a challenge.) What we have arrived at is an extension in the space of correlations defining the non-signaling correlations and quantum correlations. It is known that the the former form a polytope while the later do not. P Q S = In the figure, the points labeled are the maximal violating boxes and are the (classical) deterministic correlations. The blue region P is the non-signaling polytope. Within that is the set of quantum correlations Q. The exact for of the range of quantum correlations is not exactly known for this simple case (and less is known for more complex experiments, say three settings on each side for example). References [1] V. Scarani, ecture Notes on Quantum Information 007. [] Clauser, 1969: J. F. Clauser, M.A. Horne, A. Shimony and R. A. Holt, Proposed experiment to test local hiddenvariable theories, Phys. Rev. ett. 3, 880 (1969). [3] P. Suppes, M. Zanotti, When are probabilistic explanations possible? Synthese 48, 191 (1981). [4] B.S. Cirel son, ett. Math. Phys. 4, 93 (1980). (Cirel son changed the spelling of his name to Tsirelson in the 1980s[1]). [5] S. Popescu,. Rohrlich, Found. Phys. 4, 379 (1994). [6] J. Barrett, N. inden, et. al. Nonlocal correlations as an information-theoretic resource, Phys. Rev. A 71, 0101 (005). [7] I. Pitowski The range of quantum probability, J. Math. Phys. 7 (6). p1566. June 1986 [8] B. Tsirelson, J. SOv. Math. 36, 557 (1987)..andau, Found. Phys. 18, 449 (1988).. Masanes, quant-ph/ (003). [9] I. Pitowski, K. Svozil Optimal tests of quantum nonlocality, Phys. Rev. A 64, (001).