On the Empirical Foundations of the Quantum No ...

ON THE EMPIRICAL FOUNDATIONS OF THE QUANTUM NO-SIGNALLING PROOFS* J. B. KENNEDYt: Department of Philosophy Universityof Notre Dame I analyze a number of the quantum no-signalling proofs (Ghirardi et al. 1980, Bussey 1982, Jordan 1983, Shimony 1985, Redhead 1987, Eberhard and Ross 1989, Sherer and Busch 1993). These purport to show that the EPR correlations cannot be exploited for transmitting signals, i.e., are not causal. First, I show that these proofs can be mathematically unified; they are disguised versions of a single theorem. Second, I argue that these proofs are circular. The essential theorem relies upon the tensor product representation for combined systems, which has no physical basis in the von Neumann axioms. Historically, the construction of this representation scheme by von Neumann and Weyl built no-signalling assumptions into the quantum theory. Signalling between the wings of the EPR-Bell experiments is unlikely but is not ruled out empirically by the class of proofs considered.

1. Introduction.The EPR-Bell debate has achieved an odd, unsettling consensus. After Bell's theorem, correlations between measurement results on the distant wings demand superluminal influences but special relativity forbids them. The conflict is rendered less acute for some by the absence of causal influences, i.e., of the sort which would permit signalling between the wings. This "non-local but acausal" consensus is vague and unsatisfactory. The intractability of this conflict has led Shimony, Teller, and Healey to propose radical holistic metaphysics; it has led others like Fine and van Fraassen to urge abandoning the search for a resolution. This paper opens up a new line of attack on the problem. An extensive literature, the so-called quantum no-signalling proofs, has shown that all signalling between distant systems is forbidden by the quantum theory itself. In this paper I consider the papers by Ghirardi et al. (1980), Bussey (1982), Jordan (1983), Shimony (1985), Redhead (1987), Eberhard and Ros (1989), and Sherer and Busch (1993). In effect, they show that the quantum theory seems to contain principles which predict correlations but *Received September 1994, revised January 1995. tI would like to thank the Department of Physics at the University of Notre Dame and the History and Philosophy of Science Department at Cambridge University, where earlier versions of this paper were delivered. Portions of section 1 appeared in my thesis, Kennedy (1992), and I am grateful for the support and encouragement of my advisors. I also owe thanks to the many commentators who contributed to this essay. This work was partially supported by the NSF grant SBR 93-11567. tSend reprint requests to the author, Department of Philosophy, University of Notre Dame, Notre Dame, IN, 46556, USA Philosophyof Science,62 (1995)pp. 543-560 Copyright? 1995by the Philosophyof ScienceAssociation.

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also wisely constrains their operation in ways which make them compatible with special relativity. A new approach to the study of non-locality begins by analyzing the quantum theory itself and attempting to articulate those implicit principles which govern the behavior of non-local influences. This has led to some surprising results. In order to set the stage for arguments below, I first illustrate a claim typically made by the no-signalling theorems. Consider a pair of systems L and P which exhibit EPR correlations between the measurement results {i} and {j}. The probability for finding a result j on the right system with no prior measurement on the left system differs markedly from the probability for j given some specific measurement result i on the left system: P(10) # Pj I i)

(1)

(here P(j I 0) is read "the probability or frequency of result j on the right given no prior measurement on the left"). This surprising alteration persists even when the systems are space-like separated. Bell's theorem says these correlations are not simply pre-set values which are discovered upon measurement. Vaguely put, there is some "influence"between the systems. Initially, a number of authors speculated that these influences might be exploited for signalling, but the no-signalling theorems showed that, over many trials, the effects of these influences wash out and disappear. The probability for resultj on the right hand system, averaged over many prior results on the left system, is just the same as if no prior measurement at all were performed: P( 10)

=

P(I I L)-

P(

iP( 1 0)

(2)

i

Here P(j I L) is read "the probability for result j on the right given some unspecified result on the left" which is an average over all specific results on the left. Thus an experimenter who has access only to the right hand system cannot tell whether the other entangled system is disturbed or measured. The average statistics on the right are the same whether or not a measurement is performed on the left. These no-signalling results are extraordinarily important: they guarantee the so-called "peaceful co-existence" with relativity theory by forestalling the paradoxes associated with faster-than-light causation. Since they are derivations, they also promise to ground the prohibition in fundamentals. This is in sharp contrast to relativistic quantum field theories where the analogous no-signalling results follow directly from the bare postulation of the commutation relations between spacelike separated operators (Eberhard and Ross 1989). On this issue, at least, the non-relativistic theory appears to be deeper. On the other hand, the results are some-


QUANTUM NO-SIGNALLING PROOFS

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what suspicious. The proofs do not involve Planck's constant, the de Broglie relations, nor the Schr6dinger equation, i.e., none of the ruling features of quantum theory. Likewise, they do not involve the speed of light. Signalling between the correlated subsystems at any speed is prohibited. An excavation of the principles behind the proofs is imperative. I make the following claims: * The proofs considered can be unified and are trivial consequences of the way combined systems are represented, that is, of the "tensor product representation." * New physical assumptions, beyond the von Neumann axioms of standard textbooks (Cohen-Tannoudji 1976), are needed to derive the tensor product representation for combined systems. * Since these new physical assumptions are equivalent to no-signalling theorems, the proofs are circularjust in the sense that (i) the proofs assume the validity of the tensor product representation, and (ii) derivations of the tensor product representation assume the impossibility of signalling. * The empiricaljustification for these new axioms is therefore unclear. They encode the sort of "particularism"emphasized by Teller, and were plausible or obvious only before the EPR-Bell experiments ("I claim that we tend unreflectively to presuppose particularism as a facet of our conception of the world, a facet which never gets explicitly stated and yet conditions all our thinking.... Particularism states that the world is composed of individuals, that the individuals have nonrelational properties, and that all relations between individuals supervene on the nonrelational properties of the relata." (Teller: 1989, 213)). * Thus there is as yet no clear empirical justification for this class of no-signalling proofs. If these claims are upheld, their negative contribution to the EPR-Bell debate is clear. They would show that the quantum signalling proofs, now the foundation for the causal but non-local consensus, are prima facie unreliable. Just as in quantum field theory, no-signalling assumptions are simply assumed in the construction of the theory. Two sorts of positive contributions are possible. Most likely, the claims could lead to a deeper clarification of the way non-locality is built into and yet constrained by the quantum theory. Though it is much less likely, if an empirical basis for the no-signalling proofs is in fact found wanting, this paper would raise the issue of quantum signalling once more and perhaps transform it into an empirical question.


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I want to emphasize two points clearly at the outset: * Quantum signalling is highly unlikely. The point of considering quantum signalling is that it provides one of the few operational tests of quantum non-locality. The failure of those tests, if connected to the axioms of the theory, would clarify the nature of non-locality and the axioms themselves. * The tensor product representation is no doubt largely correct. Its use in a wide variety of applications is routine. The effort here aims at determining how and whether it is forced upon us by empirical data. If it can be shown complete and unique, the no-signalling proofs could be provided with an empirical basis. In addition to these disclaimers, an important limitation of the present study should be mentioned. Here I pursue the relation of the no-signalling proofs to the conventional axioms of non-relativistic quantum mechanics. There is a literature on axiomatics which proposes a wide variety of alternative axiomatic bases for the theory (see Ludwig (1985) for an extensive bibliography). One direction for future research would be to canvas such systems for clean justifications of the tensor product formalism. In this essay, however, I ignore these non-standard axioms. 2. Analysis of No-Signalling Proofs. 2.1 The Tensor Product Representation. The representation of a standard quantumsystem usually has three constituents: the Hilbert space, the operators defined by their action on a basis of the space, and the psifunction or state vector. I will consider two such systems labeled L and R. System L, for example, will be representedwith the space_cL, operators OL

(including the Hamiltonian HL), and a psi-function

tL.

In some circumstances, a standard quantum system can be represented as a subsystem of another standard quantum system, which would then be regarded as a combined system. If the systems L and R, for example, were parts of such a larger system, the representation of the combined system would just as above have three parts, but they would now be constructed as tensor (sometimes 'direct') products of the subsystem representations. Specifically, the tensorproduct representation(TPR) of a combined system has the following three components: TPS. Tensor Product Space: c4/ =

/4L

() /R

RAE. Restricted Action of Extensions. e.g., a =

9L

R

CPF. CombinedPsi-Function (or Density Operator). e.g., Iv) = -, ci, i) ? /) The no-signalling proofs use all three of these features of the TPR.


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2.2 Choice-of-ObservableProofs The proofs of Bussey (1982) and Jordan (1983) show that the distribution of results on one system does not depend upon which observable is measured on the other system; for example, they reject a signalling scheme in which repeated measurement on L of spin a, as opposed say to az affects the statistics gathered on R. For convenience, the projectors li)(iIon the left subsystem are abreviated PLi. With a minor simplification, Jordan's proof begins by considering (PLiPRj), the joint probability of result pairs: P(j I L) =

E i

(PLiPRj) = (

i

PLPR)

PL) PR) = (PR)

= ((,

(3)

Thus the marginal probabilities for results on R are always independent of which observable lL is measured on L. This derives the result in Equation 2 above: no signalling by manipulating distant observables is possible. Bussey's proof which is not repeated here turns essentially upon the orthonormality of the rows (or columns) of the unitary matrix [/ik] which relates the normalized bases of any two measured observables on L, say {| ui)} and {[ wk)}. Jordan shows the simpler fact that, in any given basis for L, EkPLk = 1 suffices and indeed this can be used trivially to derive the orthonormality SPLk

= 1

Skk, = 2Eigkik' since (kk' = (Wk Wk,)

(4)

k

and all pairs of bases for L are related by kk' =

l Wk')

[

(Ui, I flik][:

E /AJt (ui Iue) = Se ~~~~ie

Ak

Iwk)

=

2Eiik

| u,

(5)

I ue)]

lik'

Thus Bussey's proof can be derived from Jordan's; the proofs are not distinct. Scherer and Busch (1993) merely remark that the mixture remaining on the right is the same regardless of which observable is measured on the left. If the pre-measurement density matrix p after projection on I i)(i I is pi, then the mixture on the right is the weighted sum of the reduced states: i

P(i)Tr{pi} = TrL{

P(i)

-(

JP(

p

I}

= TrL{p}

(7)

P(i)

where the RHS is just the mixture obtaining with no measurement on the left. (Note that throughout the paper LHS and RHS refer to the left and right sides of equations only.) I assert that this point is the same as that made by Jordan: both turn on the idea that projecting the psi-function


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J. B. KENNEDY

down on a complete set of states and then adding the projections will return the original state. (There is much more in Sherer and Busch (1993). They give an original argument against signalling via state amplification or 'cloning'. As they say, however, this new no-signalling theorem rests upon the sort of results considered here.) The admirably general choice of observables no-signalling result of Eberhard and Ross (1989) also turns upon the same elementary observation (see their equations 32 and 34). 2.3 Perturbation Proofs. The no-signalling proofs of Ghirardi et al. (1980), Shimony (1984), and Redhead (1987) all assume that attempts at signalling between separated but entangled systems are instigated by a perturbation on L which they presume models the interaction between a measuring instrument and the system. Instead of repeating the proofs, which use a variety of symbolisms and strategies, I simply assert that the version of Redhead is simplest and captures all the essentials of the earlier proofs. Let the time evolution operator for system L, L, be modified to include the measurement and write the new operator r/L (treating the measurement as a perturbation makes the modified system conservative and the evolution unitary). The expectation of an observable 0R on system R alone at time t can be written ((t)

I (l

) (t)) = (V(O)I (L?

4)(1 ? )(L

?

)

I (0)) (8)

The operators 4,, and r/L,by assumption affecting only system L, cancel and leave (9) R )( 0)( OR)(l (O) = ((0)|(1 ? 1 0 R) J(0)) which is just the expectation of ORwhen there is no perturbation on system L, and attempts to signal via such perturbations are thus necessarily thwarted. Eberhard and Ross (1989) provide useful general calculations for the case of a perturbation ("modified Hamiltonian"). In the end, their proof turns on the commutivity of the left perturbation and the observable measured on the right, and is equivalent to Redhead's derivation above. Their emphasis on the role of commutivity is correct but can be made more precise. The sender's perturbation must commute with all operators on the right to avoid signalling, and this means it must not merely commute with a particular observable but must have the identity extension on the right. 2.4 Unificationof the Two Classes. The essential similarity in both classes of proofs is best revealed by recasting the perturbation proofs in a


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form reminiscent of the interaction picture (Schiff, 1968). Here I split the evolution operator for the perturbed subsystem, A, = /LZL so that the perturbation is represented by kIL only (i.e., the perturbation is modeled by the addition of an interaction term to the original Hamiltonian). Now I let the psi-function evolve as before but let the perturbation operators act on the observable. Thus the probability for finding j on the right with a perturbation on the left is, using f = i i)(i i, (W1() ?(\Y I}jli)((j,L)

) (= ((

4Li)(ilk)

? IJi) I)

(10)

Writing it this way makes it clear that the perturbation is in effect merely producing a rotation on the left-hand basis, and this provides another, intuitive way of understanding the no-signalling result. The choice-of-observable proofs can be rewritten in exactly the same form. The probability of findingj on the right after a measurement on the left is the sum over the joint probabilities ;i P(ij) or i

(l(l1i)(il

Ij)>/'I) 1v) = (

I (E I i)(i 1)? (Ij)f1) I)

)

(11)

i

Measuring a new observable on the left is equivalent to replacing the sum on the RHS with a sum over the new eigenstates Yi, I i')(i' 1,which again would merely transform the left hand basis as in the perturbation proofs. This finally allows us to see that the no-signalling proofs rely upon a single, rather straightforwardmathematical fact and therefore are susceptible to unification in a single formulation. It is claimed that the following is mathematically equivalent to all the proofs and can fairly be said to reveal the central idea involved. Rather than introduce another equivalent symbolism, it is stated here for clarity in geometric terms: Given the usual tensor product representation for combined systems, arbitrary interactions or instrument reconfigurations which in effect only transform the basis of one subsystem do not change projections onto the basis of a distant subsystem. The choice-of-observable proofs alter the basis in effect when they project on a variety of sets of eigenstates; the perturbation proofs can be construed as altering the basis through the unitary transformation which models the perturbation. The proofs are more or less elaborate restatements of the elementary theorem (Cohen-Tannoudji, 1977) that probabilities for measurement results on the distant system R are independent of the basis used for the measured system L. 2.5 Discussion. There are severe limitations on the implications of the no-signalling proofs treated here. They do show that the various schemes


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J. B. KENNEDY

cannot be exploited by us for signalling; it is nonetheless possible that a measurement on L could causally influence or even determine a subsequent result on the distant subsystem. Since the measurement result on L is random, an experimenter cannot deliberately produce some particular, initial measurement result in order to set the 'input' of the putative signalling device. This leads to a garbage-in, garbage-out (GIGO) effect, as recognized already by Jordan. These statistical tests for signalling cannot distinguish between the absence of causal signals and an uncontrollable, random mixture of causal signals. Second, it is important to emphasize that the proofs apply only to a variety of specific signalling schemes. Third, I note that if the Aharonov-B6hm effect turns out to be a manifestation of quantum non-locality (which is controversial, but Aharonov (1984) so claims), then we arguably already have working examples of subluminal quantum signalling. Consider the enclosed flux to be one subsystem and the passing particles to be the other. Alteration in the field strength produces phase shifts in the passing particles long after all transients have vanished (even though the two subsystems are by design isolated). Fourth, I note that modelling the measurement process by a perturbation, although common in some contexts, seems to presume some resolution of the measurement problem (i.e., by treating 'collapse' as Schr6dinger evolution). 3. Derivations of the TPR. 3.1 Relation to von Neumann Axioms. Since the no-signalling theorems rest on the TPR, I proceed to investigate their relation to the "von Neumann axioms" for non-relativistic quantum theory. A standard textbook (Cohen-Tannoudji 1977) presents them as: QMI. State Representation: yle/ QMII. Observables are Hermitian Operators QMIII. Measurement Results are Eigenvalues QMIV. Probability of Results Given by Born Rule QMV. Psi-Function Collapse on Measurement QMVI. Time Evolution Given by Schrodinger Equation. The history of these axioms and their claim to represent "the" quantum theory is complex. They were codified in 1932 by von Neumann (1968), and many recent textbooks present the theory axiomatically in their terms. Even cursory study of the texts, however, reveals that the axioms are not complete: important parts of the theory are introduced incidentally or as supplementary definitions. For example, the so-called quantization rules, the construction of the Hamiltonian observable, and the so-called symmetrization postulate are needed beyond the usual axioms. Together with



551

such additions, however, the von Neumann axioms do capture the theoretical core of the quantum theory. I claim that the tensor product representation of combined systems is just such an incidental addition to the theory: the TPR cannot be derived from the six central axioms. This is surprising, and problematizes the foundations of the TPR and the no-signalling theorems. The TPR enters the quantum theory along at least three distinct routes. First, more recent textbooks tend simply to introduce it as mathematical definition or simply as the "appropriate" representation for combined systems. Second, in only one text I have encountered, the TPR is presented as a new and important axiom. B6hm's QuantumMechanics (1979) states: ... we can formulate the basic assumption about the physical combination of two quantum-mechanical systems: IVa. Let one physical system be described by an algebra of operators, Al, in the space R1, and the other physical system by an algebra A2in R2.The direct product space R1( R2is then the space of physical states of the physical combination of these two systems, and its observables are operators in the direct-product space. The particular observables of the first system alone are given by A, ? I, and the observables of the second system alone by I ? A2 (I = identity operator). We reemphasize that IVa is a basic assumption of quantum mechanics and can only be justified by the fact that such physical systems exist. (p. 113) The third route whereby the TPR enters the quantum theory is the most important. I claim that historically the TPR was derivedfrom other, now largely neglected, physical assumptions. Scrutiny of these derivations should both connect the TPR to the rest of the quantum theory and expose the new empirical content required for its annexation. Isolating this content would put us hot on the trail of the empirical basis of the no-signalling theorems. 3.2 Framing the Problem. Von Neumann's discussion is the locus classicus for derivations of the TPR and influenced most or all subsequent treatments (London and Bauer 1982; Jauch 1968). Von Neumann's analysis is driven by a kind of universality assumption which remains implicit: the subsystems and the combined system will each have representations consisting of the three standard components (Hilbert space, operators, state vectors). That is, he assumes all systems are standard quantum systems, regardless of the nature of their composition. Given this sort of universality, he frames his problem as an attempt to derive functional relations between the components at the two levels. For example, given 4L


552

J. B. KENNEDY

and 4R, he asks which larger space d/ is the appropriate space for the combined system. This analysis will omit some of his simpler calculations, which are clear enough in the original, and focus on the implicit and explicit premises of von Neumann's argument. I will show that a variety of particularistic assumptions infect his discussion, and finally claim that some assumptions are equivalent to no-signalling conditions. 3.3 Tensor Product Space. For Von Neumann, the tensor product of the subsystems's Hilbert spaces is the appropriate space for representing the combined system. If, he says, { i)} and {Ij)} are the bases for the left and right systems respectively, then the product basis {I i)? Ij)} or, abbreviating, { ij)} is "obviously" (offenbar) the basis for the combined system. In effect, this mathematical claim determines the size of the combined space and the interpretation of its basis vectors. This move depends upon a theorem from function analysis which von Neumann leaves unstated. If {f(x)} and {gj(y)} are sets of orthonormal functions in the domains of x and y, which can be used to express any arbitrary function as a series expansion (i.e., are complete), then the set of product functions {f(x)gj(y)} also forms a complete, orthonormal set and can be used for series expansions of arbitrary, bivariate functions in the domain (x, y). This result, for example, appears in Courant and Hilbert's Methods of Mathematical Physics (1924), and is easy to prove. Any well-behaved bivariate function F(x, y) becomes, if the second variable is fixed at Yo,a function of x alone. It can therefore be expanded on the {f} basis, and is, in the discrete case, F(x, y) =

(12)

ai(y0)f i

where a,(y0)is some number depending upon Yoalone (and the function is assumed well-behaved enough to ensure convergence). Now for free and arbitraryy, the coefficient ai(y) becomes a function of y, and can itself be expanded on the basis gj. Substituting this new expansion, F(x, y) =

( i

j

b,gj)f = E b g,

(13)

ij

and the product functions are thus a complete basis for the bivariate function. If the functions f and gj are associated with vectors in the subsystem Hilbert spaces, this expansion is equivalent to use of the tensor product basis for the combined Hilbert space. This proof, however, makes explicit the connection to universality: the assumption of a combined psi-function appears to lead immediately to the use of tensor products. Thus, for many, tensor product spaces have become deceptively familiar and natural. Textbooks typically encourage this with examples like the decomposition of a 3-D position space: I xyz) = I x)) I y)() z). This



553

suggests that any vector space may be regarded as a tensor product space, that their invocation is an inconsequential choice of description. Even apart from universality, however, tensor product spaces encode several strong physical assumptions. Simply put, von Neumann assumes that the combined states are just pairs of subsystem states or superpositions of such pairs. The intuition here is classical: the subsystems possess their own individual characteristics and combining the systems merely combines their characteristics. Weyl is clear about this: The totality of quantities (i.e., measurable properties) of the composite system c is obtained by starting from the quantities of the component systems a and b and multiplying them and adding them together in all possible ways. The quantities a of a commute with the quantities fi of b ... We refer to the content of these last two sentences when we say that c consists of two kinematically independentparts a and b. The two systems are dynamicallyindependentif the energy of the composite system H is the sum of the energies ... of the partial systems.... (1931:91-92) This physical assumption Weyl calls 'kinematic independence' (I discuss commuting operators below) lies behind the use of tensor product spaces; it has several implicit consequences. The possibility that wholly new degrees of freedom or new subsystem states might appear in the combined system is ruled out, along with the possibility that the subsystem states might be somehow incompatible. No states are gained or lost in the combination. Once these particularistic elements are given, the mathematical result engages and leads to the tensor product space. 3.4 Extended Operators. Here von Neumann seeks a relation between the operator AL in c4/ and the corresponding extension A in the tensor product space A. He will conclude, of course, that the appropriate extension is AL? i, so that the extended operator acts non-trivially only in cL. I will call this principle for extending operators the restricted action of extensions (RAE) principle. Jauch's justification for this principle is refreshingly brief, and I begin with him before moving to von Neumann. Jauch's entire discussion of extensions is: Since it is an observable on L alone, it must be the identity operator fin 4/R. That is, it must have the form AL? . (Jauch: 1968, 179, my notation) There is a formal sense in which this statement is trivial. Subsystem operators do belong to subsystems. It is not immediately clear, however, that


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J. B. KENNEDY

this formal fact would justify any claim about their appropriate extensions on the combined system. Instead, I propose that there are some murky physical intuitions at work here, which I will discuss below. Von Neumann's introduction of RAE treats the psi-functions as functions of the classical degrees of freedom. If these are labeled m in L and n in R, then he would write the respective psi-functions as VyL(m),y,R(n), and t,(m, n) for the combined system. He proceeds: Every physical observable in L is naturally also one in L + R, and indeed the [extension] A can be ascertained from [the observable] AL as follows: in order to evaluate A V(m, n), regard n as a constant, and apply AL to the m-function y/(m, n). This association rule is at any rate correct for the position and momentum operators,. . . and is in harmony with the [rules for associating functions of observables with functions of their operators]; we therefore postulate this rule to be general. (It is generally customary [gebrduchich]in quantum mechanics.) (von Neumann: 1968, 225, my notation) Von Neumann's conventional recipe, the "constant rule" is really just stated here, not justified. He does not state what "correct" might mean. The remark that the rule is in harmony with his previously stated rules for functions of operators (von Neumann: 1968, 167) refers to facts like "the square of an operator's extension is the same as the extension of its square." This is comforting, but does not itself suffice to justify the RAE. Von Neumann does not clearly improve upon Jauch's simple assertion. Given the tensor product space and a combined psi-function, it is easy to show that the assumption of RAE rules out many choice-of-observable signalling schemes. That is, the denial of RAE permits some forms of signalling and its enforcement rules out just these schemes. Suppose the extension of OLis not simply 6L 0 Y, but is extended by some arbitrary operator: 6' = (I a, i)(i () ? ( xj j)(/ 1) (14) i

j

where the {ai} are the eigenvalues of L, and coefficients {x;} determine the extension. Using this extension, the average statistics on the right are disturbed by measurements on the left. In the extreme case, extensions can be constructed so that any measurement result on L uniquely determines the result for some observable on R. Although RAE appears prima facie plausible, it is a form of no-signalling premise, as I discuss further below. 3.5 CombinedPsi-Function. In this case, von Neumann explicitly states the physical intuition used to relate the representations of the two levels. He calls his principle "an invariant condition" (IC), and, for lack of a



555

better term, I will follow suit. He seeks a rule for associating the density operators of the subsystems, PLand PR,and the combined system, p: A statistical ensemble L + R is represented by its density operator p ... This determines the statistical properties of all magnitudes in L + R, and therefore in particular the properties of magnitudes in L. There is also, therefore, a statistical ensemble in L alone which corresponds to [L + R]: indeed (in der Tat), an experimenter, who could only observe L, and not R, would apprehend (auffassen) the ensemble of systems L + R as an ensemble of systems L. What now is the density operator which corresponds to this L-ensemble? (von Neumann: 1968, 226, my notation) Von Neumann soon remarks that the calculations which proceed from this physical assumption were ... derived from an invariant condition: ... i.e., the agreement of the expectation values for ALand [its extension] A, and BRand [its extension] B, respectively. Jauch picks up and repeats this assumption almost verbatim: The states of the component system shall be given by their respective density operators .... We wish to know what the state is if we consider the two components together as a joint system. The criterion for answering this question is a physical one: If we measure observables which refer to only one of the components we must obtain the same result whether we consider them measured on the joint system or on the component system. (Jauch: 1968, 179) In my notation, this invariance condition, the new physical assumption introduced by von Neumann and Jauch, is written: Tr{pRAR}= Tr{pA}

(15)

where as before A is the extension of ARinto the tensor product space. It is important to unravel the physical content of this assumption. The expression can be expanded by invoking the two elements of the TPR previously discussed, the use of the tensor product space with basis {\ij)} and of extensions with restricted action. This shows that the invariance condition is a relation between the matrix elements of the density operator of the combined system and those of the subsystem. Since the cj element of PRis, in their language, the probability of finding resultj when R "alone" is measured, it is in my notation P(ji 0). Likewise, cij is P(ij). Thus, the invariance condition is I aj,P( I X

= E a,( X

i

P(ij))

(16)


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J. B. KENNEDY

Since this last must be true for all density operators PRand p, the invariance condition must hold term by term: P( 10) =

P(ij)

P(j L)

(17)

but this is identical to Equation 2 above (and not a theorem from probability theory, see below). Surprisingly, the invariance condition used to construct the TPR is identical to a no-signalling theorem later derivedfrom the TPR. Since, as Jauch says, this is a physical criterion, it is further historical evidence for considering the no-signalling theorems circular. 3.6 Diagnosis. Surely von Neumann did not intend to impose a nosignalling condition. I first articulate the intuition behind the IC, which is innocent, but secondly show that the mathematicsembodies a far stronger, no-signalling claim. In this section, I need a more precise, supplementary notation to handle some subtle distinctions. I consider two kinds of measurements: an individual measurement on a subsystem or a joint measurement on a combined system. They may be described by three sorts of probability distributions: univariate (u) for individual measurements, bivariate (b) for joint measurements, and marginal (m) for sums over the bivariate distributions. To complicate things, there are two sorts of univariate distributions, calculated from the two sorts of representations. A univariate distribution for individual results i on the left can be calculated either from the L representation (PuL(i)) or from the representation of the combined system (PuLR(iI0)). Now the key point is that both these univariate distributions may differ from the marginal distribution. (This is mentioned in some elementary discussions of probability theory (Korn and Korn 1968, 589).) This happens just in case a subsystem behaves differently during individual and joint measurements, e.g., when the distant measurement has some sort of influence. This may have important consequences. For example, it is a theorem in probability theory that P(ij) = PmLR(i)P( I i) since all the distributions here arise in the same sample space of joint measurement events. It is, however, generally false that P(ij) = PULR(i)P(J I i), where here the

univariate distribution arises from individual measurements on the lefthand subsystem of L + R. Von Neumann and Jauch base the invariance condition on the intuition that, when a theory can represent one and the same physical system in two ways, the predictions issuing from the two representations should match exactly. In particular, von Neumann is concerned to calculate the distribution for an individualmeasurement first using the representation of the standard quantum system L and then using the representation of



557

the combined system L + R. The two univariate distributions should correspond. Thus the intended content of the invariance condition is IC-UNIUNI:

PL(i)

= PULR(i I 0)

This is generally true and unobjectionable, but I argue von Neuman's mathematics belies this intention. Equation 15 is a functional relation between the two levels of representation. As von Neumann and Jauch say, the unknown in this equation is the subsystem density operator PL, and this must be the case since the combined density operator cannot generally be calculated from the subsystem operators. Thus p is assumed known. This, however, can only be found by joint measurements on the combined system (or by operations on both subsystems at the time of preparation), and these lead to marginal not univariate distributions. The trace operator in Equation 15 is the smoking gun; it just extracts the marginal distribution from the presumedjoint distribution, as Equation 17 makes explicit. Thus while the IC is intended to compare two univariate distributions, the mathematics makes the far stronger claim that IC-UNIMARG:

PU(i)

= PmLR(i I R)

In short, the mathematics actually compares two different sorts of experiments. The probabilities on the left, it asserts, are the same whether or not the right hand side is measured too. This is another no-signalling premise. The intuitive appeal of the invariance condition as a free choice of the mode of representation actually conceals much stronger comparisons between modes of behavior when entirely different kinds of measurements are made. 3.7 Strength of Invariance Condition. Given the tensor product space and the combined psi-function, IC-UNIUNI and IC-UNIMARG together imply the RAE. The question is which operator A on L + R gives the same probabilities for individual measurements on L + R that AL gives for the same individual measurements on L. By transitivity, the two conditions assert PuL(i

0) = PmLR( I R) = PuLR(i I 0)

(18)

Thus the operator which extracts the marginal (second expression) will in effect also extract the appropriate univariate distribution for individual measurements on L + R (third expression). But the marginal is extracted by the identity extension: r Tr{p [ i)({il} = Tr{p( J

I ij)(ij I)} = Tr{p I i)(i I? 4

(19)

J

Thus when an individual measurement is performed, the extension AL?

f


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J. B. KENNEDY

will return the same probabilities on the combined system that ALgives on the subsystem representation. Since the assumption of the IC-UNIMARG is suspect, this is another way of showing that no-signalling prohibitions are built into RAE. IC-UNIMARG is, however, a no-signalling assumption which goes beyond RAE. Pilot wave theories sometimes envisage a psi-function which is dependent upon measurement device settings, i.e., upon which measurements will be made. In effect, this says that the combined psi-function varies with the distant observable measured. If this were the case, choiceof-observable signalling might be possible even if RAE held. It is the implicit assumption of IC-UNIMARG in the ordinary formalism which guarantees that such shifts in r/ have no detectable consequences. In retrospect, it is easy to see the centrality of RAE and IC-UNIMARG to the no signalling proofs. In Jordan's, for example, the last inequality in Equation 3 is licensed by RAE and the entire line amounts to a statement of IC-UNIMARG. 3.8 Weyl's Derivation of the TPR. Von Neumann's derivation of the TPR is similar to the treatment in Weyl: (1931, 89-93), which permits a very brief discussion here. It is prefaced by a discussion of the classical 'problem of several bodies,' with two 'particles'-Weyl is certainly working with pictures laden with particularisticassumptions. Very briefly:Weyl simply asserts the use of the tensor product space for the combined system. Regarding the RAE, he recommends the identity extension since it maintains the functional relationships among the subsystem operators (as above). He is quite clear about the physical content of the RAE assumption: the RAE implies a commutivity between operators from different subsystems, and this is equivalent to the 'kinematic independence' of the subsystems. In effect, RAE rules out signalling. Weyl's discussion of the combined psi-function merely remarks that it is not fully determined by knowledge of the subsystems, and this implies for him some sort of 'holism'. Together Weyl's and von Neumann's presentations of the tensor product representation set the stage for later treatments. 4. Summary and Conclusion. Briefly put, this essay is a case study supporting Teller's broad claim, referred to above, that our physical theories "tend unreflectively to presuppose particularism," i.e., that physical systems are given as distinct individuals with non-relational properties. My more restrictedclaim is to have discovered exactly where such assumptions are built into the formalism of quantum mechanics. Over and over again, the devisers of the tensor product representation supposed quantum systems were individuals which combined as unproblematically as classical particles. In summary their assumptions were:



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(i) Tensor Product Space: assumes the impossibility of wholly new states in the combined system (apart from combinations of subsystem states and their correlations), and assumes the compatibility of all subsystems states-in short, no gain or loss of states. (ii) RestrictedAction of Extensions: assumes extended observables are really observables on one subsystem 'alone' (Jauch), or the 'kinematic independence' (Weyl) of the subsystems-in short, no quantum signalling. (iii) CombinedPsi-Function: assumes only modes of behavior and not modes of representation can differ, i.e., assumes an invariance condition which is strictly equivalent to a no-signalling condition. These physical assumptions go beyond the six von Neumann axioms and should properly be formulated as new, presupposed axioms of the quantum theory. They were immediately plausible only in an atmosphere still colored by the worldview of classical mechanics. Clearly the claims here are severely limited. I have not explored the many non-standard axioms for quantum theory. I have not by any means cast doubt on the tensor product representation itself, but only upon its derivations. I hope, however, to have pushed the debate over quantum signalling in a new direction. It is no longer possible to suppose that 'proofs' of quantum no-signalling theorems can be simply derived from the standard formalism. Such proofs are circular. In the face of the EPRBell results, the empirical justification for these extra, implicit physical assumptions needs to be articulated and assessed. I suggest that future research seek such an empirical basis in the wide range of phenomena correctly described by the tensor product representation, or seek some new theoretical derivation solidly connected to the central axioms. In particular, I especially propose that RAE and IC-UNIMARG be tested. REFERENCES

Aharonov, Y. (1984), "Non-local phenomena and the AB effect", Foundationsof Physics. Bohm, A. (1979), QuantumMechanics. New York: Springer Verlag. Bussey, P. J. (1982), "Superluminal Communication in the EPR Experiments", Physics Letters 90A: p. 9. Cohen-Tannoudji, C., Diu, B. and Laloe, F. (1977), QuantumMechanics.New York: J. Wiley and Sons. Ghirardi, G. C. et al. (1980), "A General Argument Against Superluminal Transmission Through the Quantum Mechanical Measurement Process", Lettre Al Nuovo Cimento 27:10, 8 March 1980: p. 293. Jauch, J. F. (1968), Foundationsof QuantumMechanics. Reading, MA: Addison-Wesley, Inc. Jordan, T. F. (1983), "Quantum Correlations Do Not Transmit Signals", Physics Letters 94A:6, 7, 21 March 1983: p. 264. Kennedy, J. B. (1992), The Aharonov-BohmEffect and the Non-Locality Debate, Ph.D. dissertation, Stanford University. Korn, G. and Korn, T. (1968). Mathematical Handbookfor Scientists and Engineers. New York: J. Wiley and Sons.


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London, F. and Bauer, E. (1982), "The Theory of Observation in Quantum Mechanics", in Wheeler, J. and Zurek, A. (eds.), Quantum Theory and Measurement. Princeton: Princeton University Press, pp. 217-259. Ludwig, G. (1985), An Axiomatic Basis for QuantumMechanics. Berlin: Springer-Verlag. Redhead, M. (1987), Incompleteness,Non-locality, and Realism. Oxford: Clarendon Press. Sherer, H. and Busch, P. (1993), "Problem of Signal Transmission via Quantum Correlations and Einstein Incompleteness in Quantum Mechanics", Physical Review A47:3, March 1993, pp. 1647-1651. Shimony, A. (1984), "Controllable and Uncontrollable Non-Locality", in The Foundations of QuantumMechanics: in the Light of New Technology. Tokyo: Hitachi, Ltd. Teller, P. (1989), "Relativity, Relational Holism, and the Bell Inequalities", in Cushing, J. and McMullin, E. (eds.), Philosophical Consequences of the Quantum Theory. Notre Dame, IN: University of Notre Dame Press, pp. 208-223. Von Neumann, J. (1968), Mathematische Grundlagen der Quantenmechanik.New York: Springer Verlag. Weyl, H. (1931), The Theory of Groups and QuantumMechanics. New York: Dover Publications, Inc.
