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.
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.
In quantum physics, we often need to consider ensembles that are not pure cases, that is, that cannot be described by state vectors. This is so, for instance, when an ensemble is composed of several subensembles that are described by nonequivalent state vectors. It also happens in most cases in which we consider ensembles of definite subsystems of some larger systems. The density matrix—or "statistical operator"—formalism is one that makes it possible to deal with such situations swiftly and efficiently. The rules apply only to ensembles that can be described by state vectors. These rules entail as consequences new expressions for quantum probabilities and mean values, which can be called the—derived— "statistical operator rules." However, it is also possible to formulate the statistical operator rules directly instead. The Heisenberg "uncertainty" relations imply that in general f(x,p) cannot be interpreted as really being a joint probability distribution.
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.
Abstract A formulation of quantum mechanics is presented based on the theory of semigroups and the associated enveloping algebras of functions defined on countable subsemigroups. The existence of a unique *‐involution is not assumed. The fundamental elements of a semigroup are identified with experimental precedures for the separation of subensembles from a given ensemble of experimental systems. Observables are represented as elements of enveloping algebras, and ensembles as density matrices within an enveloping algebra. The statistical properties of ensembles are expressed in terms of traces defined on the semigroup and its enveloping algebras. The elements and generators of the Poincaré group can be defined and interpreted in the usual way. A variety of applications is described, in which the theory of the density matrix plays an essential or effective role. Advantages associated with the resulting freedom from the limitations of Hilbert space are illustrated.
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.".
The density matrix ρ is defined and its basic properties are derived in detail: ρ is a Hermitian non-negative operator with unit trace, its eigenvalues are non-negative and lie in the closed interval [0, 1]; if ρ is a projector, it projects onto a one-dimensional sub-space, the trace of ρ 2 is ≤ 1, the equality holding only if ρ is a projector. A necessary and sufficient condition for ρ to be a projector is that all vectors in its definition are identical up to a phase. Pure cases are sets that can be described by a state vector, unlike mixtures . The density matrix of a mixture is not a projection operator, and this may be viewed as its distinguishing feature with respect to pure cases. The mean value of an observable can be reexpressed with the help of the density matrix, and a Schrödinger equation for density matrices holds. Last, but not least, pure cases cannot unitarily evolve into mixtures . The applications of the formalism presented in this chapter deal with orientation of spin-½ particles, polarization of pairs of photons, thermal-equilibrium states. In the second part, quantum entanglement is presented, with emphasis on the existence of state vectors for entangled pairs of particles that belong to the tensor product of Hilbert spaces H 1 ⊗ H 2 , but are not of the simple product form ψ 1 ⊗ ψ 2 . The only correct interpretation of such states has a statistical nature, associated to series of repeated measurements performed by two observers on the constituents of the entangled pair. Hidden-variable theories, with the associated Bell inequalities, are introduced, and a way of using entangled quantum states is described which makes it possible to transfer the polarization state of a photon to another photon. This property provides the foundation of the modern investigations of quantum teleportation.
In this chapter the basic concepts of quantum theory are formulated via a self-consistent set of six postulates. These six postulates deal with the following: operator images of physical observables; properties of state vectors and wave functions; the connection with experimental measurements; coordinate-space forms of those operators having classical analogs; the primary dynamical equation of quantum mechanics; and the completeness of the eigenvectors of observables. The postulates are the framework on which the edifice of quantum mechanics is constructed. A separate section is devoted to the statement of each postulate and to a few examples chosen to illustrate or apply it. A comprehensive set of applications is presented in Chapters 6 and 7, where various 1-D systems are used as the vehicles for illustrating the postulates. The basic theory of this chapter is extended and developed in Chapters 8 and 9. Since probability is a key interpretational concept, a brief introduction to this topic is given in an appendix. We emphasize that our formulation of quantum theory (in this chapter and throughout this book) is restricted to the case of pure states. This is not a major restriction. It means that, when we consider ensembles of individual quantum systems (e.g., single particles, atoms, nuclei, etc.), all of the systems in the ensemble are assumed to be in the same quantum state, not necessarily an eigenstate of a quantal operator. Ensembles for which this assumption does not hold are described by a density operator or density matrix rather than by a pure state; treatments of such cases can be found in references cited later.