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.
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.
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.
We show that probabilities of results of all possible measurements performing on a quantum system depend on the system's state only through its density matrix. Therefore all experimentally available information about the state contains in the density matrix. In this study, we do not postulate that measurements obey some given formalism (such as observables, positive-operator valued measures, etc.), and do not use Born rule. The process of measurement is considered in a fully operational manner---as an interaction of a measured system with some black-box apparatus. The key point of our approach is the proof that, for improper mixtures, the expected value of any measurement depends linearly on the reduced density function. Such a proof is achieved by considering appropriate thought experiments. We demonstrate that Born rule can be derived as a natural consequence of our results.
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.
From a classical perspective, a quantum state is a surprising mixture of information and ignorance. If a system is sitting in an eigenstate of an observable, we can predict with confidence that a measurement result will yield the appropriate eigenvalue. With a superposition, certainty is not granted to us. Unlike the closest possible classical case, where a superposition represents a genuine blending of classical states, the quantum superposition is a strange overlapping of states simultaneously occupied by the system. On measurement, this ontologically curious animal collapses into one of the overlapping states, a process that is not well understood to this day. The collapse is random, a-causal, but governed by quantum amplitudes. Heading into a measurement, we do not know what we are going to get out. Given the evident link between entropy and information, the probabilities and uncertainties baked into a quantum state may well be a fertile ground for another form of entropy. The density matrix gives us a flexible tool for studying the properties of quantum ensembles of any kind. In this chapter, we will see how von Neumann helped develop and use the density matrix to study the problem of measurement and the information/entropy content of quantum states .