Jeffrey Bub

Philosophical debate on the measurement problem of quantum mechanics has, for the most part, been confined to the non-relativistic version of the theory. Quantizing quantum field theory, or making quantum mechanics relativistic, yields a conceptual framework capable of dealing with the creation and annihilation of an indefinite number of particles in interaction with fields, i.e. quantum systems with an infinite number of degrees of freedom. I show that a solution to the standard measurement problem is available if we exploit the properties of the infinite quantum models available in this broader conceptual framework.

I show that the quantum state ω can be interpreted as defining a probability measure on a subalgebra of the algebra of projection operators that is not fixed (as in classical statistical mechanics) but changes with ω and appropriate boundary conditions, hence with the dynamics of the theory. This subalgebra, while not embeddable into a Boolean algebra, will always admit two-valued homomorphisms, which correspond to the different possible ways in which a set of “determinate” quantities (selected by ω and the boundary conditions) can have values. The probabilities defined by ω (via the Born rule) are probabilities over these two-valued homomorphisms or value assignments. So any universe of interacting systems, including those functioning as measuring instruments, can be modelled quantum mechanically without the projection postulate.

A quantum algorithm succeeds not because the superposition principle allows ‘the computation of all values of a function at once’ via ‘quantum parallelism’, but rather because the structure of a quantum state space allows new sorts of correlations associated with entanglement, with new possibilities for information‐processing transformations between correlations, that are not possible in a classical state space. I illustrate this with an elementary example of a problem for which a quantum algorithm is more efficient than any classical algorithm. I also introduce the notion of ‘pseudotelepathic’ games and show how the difference between classical and quantum correlations plays a similar role here for games that can be won by quantum players exploiting entanglement, but not by classical players whose only allowed common resource consists of shared strings of random numbers (common causes of the players’ correlated responses in a game). *Received October 2008. †To contact the author, please write to: Department of Philosophy, University of Maryland, College Park, MD 20742; e‐mail: [email protected].

Emilio Santos has argued (Santos, Studies in History and Philosophy of Physics http: //arxiv-org/abs/quant-ph/0410193) that to date, no experiment has provided a loophole-free refutation of Bell’s inequalities. He believes that this provides strong evidence for the principle of local realism, and argues that we should reject this principle only if we have extremely strong evidence. However, recent work by Malley and Fine (Non-commuting observables and local realism, http: //arxiv-org/abs/quant-ph/0505016) appears to suggest that experiments refuting Bell’s inequalities could at most confirm that quantum mechanical quantities do not commute. They also suggest that experiments performed on a single system could refute local realism. In this paper, we develop a connection between the work of Malley and Fine and an argument by Bub from some years ago [Bub, The Interpretation of Quantum Mechanics, Chapter VI(Reidel, Dodrecht,1974)]. We also argue that the appearance of conflict between Santos on the one hand and Malley and Fine on the other is a result of differences in the way they understand local realism

In a series of lectures written around 1952, Schrödinger refers to von Neumann’s account of measurement in quantum mechanics as follows:I said quantum physicists bother very little about accounting, according to the accepted law, for the supposed change of the wave-function by measurement. I know of only one attempt in this direction, to which Dr. Balazs recently directed my attention. You find it in John von Neumann’s well-known book. With great acuity he constructs one analytical example. It does not refer to any actual experiment, it is purely analytical. He indicates in a simple case a supplementary operator which, when added to the internal wave operator, would with any desired approximation turn the wave function as time goes on into an eigenfunction of the observable that is measured. He found it necessary to show that such a mechanism is analytically possible. The idea has not been taken up and worked out since — in about twenty years or more. Indeed I do not think it would pay. I do not believe any real measuring device is of this kind.

The aim of cognitive neuropsychology is to articulate the functional architecture underlying normal cognition, on the basis of congnitive performance data involving brain-damaged subjects. Throughout the history of the subject, questions have been raised as to whether the methods of neuropsychology are adequate to its goals. The question has been reopened by Glymour [1994], who formulates a discovery problem for cognitive neuropsychology, in the sense of formal learning theory, concerning the existence of a reliable methodology. It appears that the discovery problem may be insoluble in principle! I propose a modified formulation of Glymour's discovery problem and argue that a sceptical conclusion about the possiblity of cognitive neuropsychology as an empirical science is not warranted.

We prove a uniqueness theorem showing that, subject to certain natural constraints, all 'no collapse' interpretations of quantum mechanics can be uniquely characterized and reduced to the choice of a particular preferred observable as determine (definite, sharp). We show how certain versions of the modal interpretation, Bohm's 'causal' interpretation, Bohr's complementarity interpretation, and the orthodox (Dirac-von Neumann) interpretation without the projection postulate can be recovered from the theorem. Bohr's complementarity and Einstein's realism appear as two quite different proposals for selecting the preferred determinate observable--either settled pragmatically by what we choose to observe, or fixed once and for all, as the Einsteinian realist would require, in which case the preferred observable is a 'beable' in Bell's sense, as in Bohm's interpretation (where the preferred observable is position in configuration space).

Friedman and Putnam have argued (Friedman and Putnam 1978) that the quantum logical interpretation of quantum mechanics gives us an explanation of interference that the Copenhagen interpretation cannot supply without invoking an additional ad hoc principle, the projection postulate. I show that it is possible to define a notion of equivalence of experimental arrangements relative to a pure state φ, or (correspondingly) equivalence of Boolean subalgebras in the partial Boolean algebra of projection operators of a system, which plays a role in the Copenhagen explanation of interference analogous to the role played by the material equivalence, given φ, of certain propositions in the Friedman-Putnam quantum logical analysis. I also show that the quantum logical interpretation and the Copenhagen interpretation are equally capable of avoiding the paradoxical conclusion of the Einstein-Podolsky-Rosen argument (Einstein, Podolsky, and Rosen 1935). Thus, neither interference phenomena nor the correlations between separated systems provide a test case for distinguishing between the relative acceptability of the Copenhagen interpretation and the quantum logical interpretation as explanations of quantum effects.

J. S. Bell's argument that only “nonlocal” hidden variable theories can reproduce the quantum statistical correlations of the singlet spin state in the case of two separated spin-1/2 particles is examined in terms of Wigner's formulation. It is shown that a similar argument applies to a single spin-1/2 particle, and that the exclusion of hidden variables depends on an obviously untenable assumption concerning conditional probabilities. The problem of completeness is discussed briefly, and the grounds for rejecting a phase-space reconstruction of the quantum statistics are clarified.

The aim of cognitive neuropsychology is to articulate the functional architecture underlying normal cognition, on the basis of cognitive performance data involving brain-damaged subjects. Glymour (forthcoming) formulates a discovery problem for cognitive neuropsychology, in the sense of formal learning theory, concerning the existence of a reliable methodology, and argues that the problem is insoluble: granted certain apparently plausible assumptions about the form of neuropsychological theories and the nature of the available evidence, a reliable methodology does not exist! I argue for a reformulation of the discovery problem in terms of an alternative characterization of relevant evidence in neuropsychology.

I formulate the interpretation problem of quantum mechanics as the problem of identifying all possible maximal sublattices of quantum propositions that can be taken as simultaneously determinate, subject to certain constraints that allow the representation of quantum probabilities as measures over truth possibilities in the standard sense, and the representation of measurements in terms of the linear dynamics of the theory. The solution to this problem yields a modal interpretation that I show to be a generalized version of Bohm's hidden variable theory. I argue that unless we alter the dynamics of quantum mechanics, or accept a for all practical purposes solution, this generalized Bohmian mechanics is the unique solution to the problem of interpretation.

Bohr's complementarity interpretation is represented as the relativization of the quantum mechanical description of a system to the maximal Boolean subalgebra (in the non-Boolean logical structure of the system) selected by a classically described experimental arrangement. Only propositions in this subalgebra have determinate truth values. The concept of a minimal revision of a Boolean subalgebra by a measurement is defined, and it is shown that the nonmaximal measurement of spin on one subsystem in the spin version of the Einstein—Podolsky—Rosen experiment actually selects an appropriate maximal Boolean subalgebra ℛ′ in the logical structure of the composite system, via a minimal revision of the maximal Boolean subalgebra & associated with the preparation of the singlet spin state. This provides an explanation for the determinate truth values of propositions in ℛ′ referring to the second subsystem within the framework of the complementarity interpretation.

The properties of classical and quantum systems are characterized by different algebraic structures. We know that the properties of a quantum mechanical system form a partial Boolean algebra not embeddable into a Boolean algebra, and so cannot all be co-determinate. We also know that maximal Boolean subalgebras of properties can be (separately) co-determinate. Are there larger subsets of properties that can be co-determinate without contradiction? Following an analysis of Bohrs response to the Einstein-Podolsky-Rosen objection to the complementarity interpretation of quantum mechanics, a principled argument is developed justifying the selection of particular subsets of properties as co-determinate for a quantum system in particular physical contexts. These subsets are generated by sets of maximal Boolean subalgebras, defined in each case by the relation between the quantum state and a measurement (possibly, but not necessarily, the measurement in terms of which we seek to establish whether or not a particular property of the system in question obtains). If we are required to interpret quantum mechanics in this way, then predication for quantum systems is quite unlike the corresponding notion for classical systems.

Bell's problem of the possibility of a local hidden variable theory of quantum phenomena is considered in the context of the general problem of representing the statistical states of a quantum mechanical system by measures on a classical probability space, and Bell's result is presented as a generalization of Maczynski's theorem for maximal magnitudes. The proof of this generalization is shown to depend on the impossibility of recovering the quantum statistics for sequential probabilities in a classical representation without introducing a randomization process for the hidden variables. Hidden variable theories that exclude such a randomization process are termed “strict,” and it is shown that the class of local hidden variable theories is included in the class of strict theories. A counterargument by Freedman and Wigner is evaluated with reference to Clauser's extension of a hidden variable model proposed by Bell.