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CRC Press eBooks

Mixtures

Bernard d’Espagnat

Published Mar 5, 2018• 0 citations
Publisher

Abstract

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.".

DOI
10.1201/9780429501449-9

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Along with an elementary description of the well-known density matrix formalism and of its application to the description of quantum-statistical ensembles, this chapter includes the proofs of a few propositions that are especially useful for understanding most of the developments. In the new formulation, ensembles are directly described by density matrices, observables are still described by Hermitean operators, and the rules giving the statistical distribution and the mean values of the predicted measurement results are, respectively, expressions. When stated in this manner, the rules of quantum mechanics apply directly not only to pure states but also to mixtures. This is a distinct advantage. In the case of mixtures defined by their density matrices, the new formulation also has another merit. If two proper mixtures are described by the same density matrix, any state vector representing a subensemble of one of these mixtures can be expressed as a linear combination of the state vectors representing the subensembles of the other mixture.

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Up to now, the initial state of the instrument has been ascribed a state vector. This convention can be criticized, however, either by observing that the instrument is a complicated system which is therefore imperfectly known, or, more adequately still (this is again the difference between proper and improper mixtures), by noting that the instrument has interacted with many other systems before the time when it is used for measuring L , and that consequently it can be associated, not with any definite state vector, but only with a density matrix (in other words, an ensemble of such instruments identically prepared would be an improper mixture). We may wonder whether this remark offers a clue for solving the difficulty encountered at the end of Chapter 14 . There it was shown that, if the ensemble E of instruments is initially a pure case, it is in general impossible to consider that every instrument coordinate has some definite eigenvalue after the interaction with the system has taken place. Let us investigate whether or not this is still the case when ensemble E is a mixture.

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Meaning of the Density Matrix

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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.

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Bernard d’Espagnat

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.

arXiv:quant-ph/9804063•Apr 27, 1998•4 citations
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.".
Along with an elementary description of the well-known density matrix formalism and of its application to the description of quantum-statistical ensembles, this chapter includes the proofs of a few propositions that are especially useful for understanding most of the developments. In the new formulation, ensembles are directly described by density matrices, observables are still described by Hermitean operators, and the rules giving the statistical distribution and the mean values of the predicted measurement results are, respectively, expressions. When stated in this manner, the rules of quantum mechanics apply directly not only to pure states but also to mixtures. This is a distinct advantage. In the case of mixtures defined by their density matrices, the new formulation also has another merit. If two proper mixtures are described by the same density matrix, any state vector representing a subensemble of one of these mixtures can be expressed as a linear combination of the state vectors representing the subensembles of the other mixture.
Observable dynamical quantities are described by Hermitean operators, and in the early days of quantum mechanics the correspondence was believed to be one-to-one. It was assumed that (i) every observable can be related to some Hermitean operator and (ii) every Hermitean operator corresponds to one observable. The arguments concerning the difference between proper and improper mixtures are conceived of for the benefit of the physicists with a "realist" turn of mind who, prima facie, would tend to consider that, quite on the contrary, the statement in question is meaningful and true. Two ensembles are said to be different with respect to some observable A if the statistical frequencies of the prospective measurement outcomes on observable A are different on the two ensembles. Also, two ensembles are said to be different if there exists at least one observable A with respect to which they are different.
Up to now, the initial state of the instrument has been ascribed a state vector. This convention can be criticized, however, either by observing that the instrument is a complicated system which is therefore imperfectly known, or, more adequately still (this is again the difference between proper and improper mixtures), by noting that the instrument has interacted with many other systems before the time when it is used for measuring L , and that consequently it can be associated, not with any definite state vector, but only with a density matrix (in other words, an ensemble of such instruments identically prepared would be an improper mixture). We may wonder whether this remark offers a clue for solving the difficulty encountered at the end of Chapter 14 . There it was shown that, if the ensemble E of instruments is initially a pure case, it is in general impossible to consider that every instrument coordinate has some definite eigenvalue after the interaction with the system has taken place. Let us investigate whether or not this is still the case when ensemble E is a mixture.
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.
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.