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arXiv:quant-ph/9804063

Reply to Aharonov and Anandan's "Meaning of the Density Matrix"

Bernard d’Espagnat

Published Apr 27, 1998• 4 citations
arXivHTMLPDF

Abstract

Aharonov and Anandan's claim that the notion of ``proper mixture'' is improper is shown to be unjustified. The point is made that if a purely operationalist standpoint is taken the three difficulties these authors describe relatively to the conventional interpretation of density matrices in fact vanish. It is noted that nevertheless it is very difficult for us to do without any form of realism, in particular when the quantum measurement problem is considered, and it is stressed that the proper mixture notion comes in precisely at this level. The more general question of the real bearing of Aharonov and Anandan's ideas on the interpretation of quantum mechanics problem is considered.

DOI
10.48550/arxiv.quant-ph/9804063

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Aharonov and Anandan's claim that the notion of ``proper mixture'' is improper is shown to be unjustified. The point is made that if a purely operationalist standpoint is taken the three difficulties these authors describe relatively to the conventional interpretation of density matrices in fact vanish. It is noted that nevertheless it is very difficult for us to do without any form of realism, in particular when the quantum measurement problem is considered, and it is stressed that the proper mixture notion comes in precisely at this level. The more general question of the real bearing of Aharonov and Anandan's ideas on the interpretation of quantum mechanics problem is considered.
The relevant mathematical theory was developed with regard to mixtures and their description. That theory could of course be expanded to a considerable degree. However, an abundant literature exists already on that subject. As a first, elementary way of clarifying them this chapter introduces an unconventional but useful distinction between improper mixtures and proper mixtures. It then investigates the question of the homogeneity or heterogeneity of ensembles, as well as two related questions, which bear on the measurability of the density matrix and on the relation between Hermitean operators and observables, respectively. More precisely, the assumptions were made that every observable can be related to some Hermitean operator, and every Hermitean operator corresponds to one observable. The chapter includes a discussion of the meaning that can be given to this sentence: "Such and such an observable has such and such a value on a system.".
Protective measurement, which was proposed as a method of observing the wavefunction of a single system, is extended to the observation of the density matrix of a single system. d'Espagnat's definition of `proper mixture' is shown to be improper because it does not allow for appropriate fluctuations. His claim that there could be different mixtures corresponding to the same density matrix is critically examined. These results provide a new meaning to the density matrix, which gives it the same ontological status as the wavefunction describing a pure state. This also enables quantum entropy to be associated with a single system.
An experiment that would measure non--commuting quantum mechanical observables without collapsing the wave function has been recently proposed by Y Aharonov and J Anandan. These authors argue that this "protected measurement" may give indication on "the reality of the wave function". We argue that, depending of the precise version of the experiment considered, either the author's prediction is incorrect and the wave function does collapse, or the measurement is not a measurement on a quantum system. In either case, the experiment does not provide a way for measuring non--commuting observables without collapse, and it does not bear on the issue of the "reality of the wave function".
Abstract The fundamentals of the theory of density matrices are reviewed. The density matrix is introduced first in terms of state vectors for pure states over which a probability distribution is given. It is then shown that the applications of the density matrix do not require knowledge of the particular probability distribution over pure states that may have been used for its original determination. It is stressed that density matrices, like state vectors, describe ensembles rather than single systems, as determinism in quantum theory (except for some conservation laws) is of a statistical character only. The entropy of a quantum‐mechanical system is defined as the average per system of the entropy of an ensemble of systems described by the quantum‐mechanical (pure or mixed) state. For pure states this entropy is zero, in agreement with the fact that state vectors constitute the maximal information that can be given about a quantum‐mechanical system. As the entropy should be independent of the particular probability distribution over pure states that may have been used to form the mixed state, it is calculated from statistical considerations about an imaginary ensemble in which the eigenvalues of the density matrix are probabilities over pure states given by the eigenfunctions of the density matrix. The entropy is calculated for a mixture of two nonorthogonal states. The time evolution of the density matrix, in general, is not a unitary transformation, as observable systems are not closed and therefore in the Hamiltonian contain unknown external parameters that are not the same for the entire mixture. Therefore transitions between pure states and mixed states take place. Those from pure to mixed states take place spontaneously and irreversibly under increase of entropy. Transitions from mixed states to pure states can be fabricated; an example is given. As Wigner's thought experiment of mixing coherently the two beams from a Stern‐Gerlach apparatus according to estimates of Bohm should be impossible by loss of coherence in the gradient magnetic field, it is indicated how Wigner's ideas may be realized by using, instead, polarized light beams from birefringent crystals. After a brief review of the quantum theory of measurements and the various arguments usually given for the reducibility of state after the measurement, Wigner's argument against blind use of these arguments is interpreted as a limit of validity of the familiar probability rule of quantum theory, which should not be applied after an unsuccessful measurement. By splitting light beams coherently by glass plates, one can by the same apparatus have measurements both of the relative intensities of two coherent beams, and of interference effects between the two beams. This shows that complementarity applies to the individual systems in the beam, as they reach the glass plate where a choice must be made between two complementary possibilities of future measurement. Finally, by describing Schrödinger's cat by a density matrix and asking about its time dependence, it is shown that, if it were possible to measure the interference effects between living and dead cats used by some people as a justification for writing down a state vector for the cat, then this measurement could be used to revive some dead cats.
In a recent paper [12], we discussed the serious inconsistency present within the operational and mathematical definition(s) of the notion of pure state. Continuing this analysis, in this work we attempt to address the role of 'purity' and 'mixtures' within two different categorical approaches to QM, namely, the topos approach originally presented by Chris Isham and Jeremy Butterfield [27, 28, 29] and the more recent logos categorical approach presented by the authors of this article [10, 11, 13]. While the first approach exposes the difficulties to produce a consistent understanding of pure states and mixtures, the latter approach presents a new scheme in which their reference is erased right from the start in favor of an intensive understanding of projection operators and quantum superpositions. This new account of the theory, grounded on an intensive interpretation of the Born rule, allows us not only to avoid the orthodox interpretation of projection operators --either as referring to definite valued properties or measurement outcomes-- but also to consider all matrices (of any rank) on equal footing. It is from this latter standpoint that we conclude that instead of distinguishing between pure and mixed states it would be recommendable --for a proper understanding of the theory of quanta-- to return to the original matrix formulation of quantum mechanics presented by Werner Heisenberg in 1925.