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Journal of Mathematical Physics

Dynamical Mappings of Density Operators in Quantum Mechanics

Thomas F. Jordan, E. C. G. Sudarshan

Published Nov 1, 1961• 65 citations
Publisher

Abstract

The most general dynamical law for a quantum mechanical system is studied with particular reference to the necessary and sufficient conditions for such a law to represent Hamiltonian dynamics. The main results are stated in the form of three theorems.
DOI
10.1063/1.1724221

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The most general continuous time-dependent evolution of a physical system is represented by a continuous one-parameter semi-group of linear mappings of density operators to density operators. It is shown that if these dynamical mappings form a group they can be represented by a group of unitary operators on the Hilbert space of state vectors. This proof does not assume that the absolute values of inner products of state vectors or ``transition probabilities'' are preserved but deduces this fact from the requirement that density operators are mapped linearly to density operators. An example is given of a continuous one-parameter semi-group of dynamical mappings which is not a group.
For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates { q i }. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement p i → ( ħ /i)(∂/∂ q i ). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ (∂Ψ/∂ t ) = H Ψ which requires as initial condition the knowledge of the wave function Ψ({ q i }, t ) at some initial time t 0 . (Boundary conditions on Ψ({ q i }, t ) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψ ν ( q ), is a basis for the Hilbert space.
Suppose that several parties jointly possess a pure multipartite state, |ψ⟩. Using local operations on their respective systems and classical communication (i.e. LOCC), it may be possible for the parties to transform deterministically |ψ⟩ into another joint state |ϕ⟩. In the bipartite case, the Nielsen majorization theorem gives the necessary and sufficient conditions for this process of entanglement transformation to be possible. In the multipartite case, such a deterministic local transformation is possible only if both the states are in the same stochastic LOCC (SLOCC) class. Here, we generalize the Nielsen majorization theorem to the multipartite case, and find necessary and sufficient conditions for the existence of a local separable transformation between two multipartite states in the same SLOCC class. When such a deterministic conversion is not possible, we find an expression for the maximum probability to convert one state to another by local separable operations. In addition, we find necessary and sufficient conditions for the existence of a separable transformation that converts a multipartite pure state into one of a set of possible final states all in the same SLOCC class. Our results are expressed in terms of (i) the stabilizer group of the state representing the SLOCC orbit and (ii) the associate density matrices (ADMs) of the two multipartite states. The ADMs play a similar role to that of the reduced density matrices when considering local transformations that involve pure bipartite states. We show, in particular, that the requirement that one ADM majorizes another is a necessary condition but is, in general, far from also being sufficient as it happens in the bipartite case. In most of the results the twirling operation with respect to the stabilizer group (of the representative state in the SLOCC orbit) plays an important role that provides a deep link between entanglement theory and the resource theory of reference frames.
Abstract For present purposes, the first thing to recall about quantum mechanics is that all dynamical quantities, instead of being functions of the location of a system in phase space as in classical mechanics, are now represented by operators in a Hilbert space of quantum states. In particular, the quantum analog of the classical phase space distribution function is the quantum mechanical density matrix. Since this is seldom discussed in introductory texts on either quantum mechanics or statistical mechanics, it may be helpful to summarize some of its basic properties.
Based on the Bohr's correspondence principle it is shown that relativistic mechanics and quantum mechanics may be considered as generalizations of classical mechanics. A comparative description of relativistic and classical mechanics is given using three main principles: the definition of momentum, the main law of dynamics (Newton's second law) and the law of conservation of energy. The difference of relativistic mechanics from classical mechanics is due to the new definition of momentum that is proportional to energy and velocity. It is shown that new relativistic laws of dynamics make it necessary to change kinematical relations of classical mechanics such as the law of velocity composition, coordinate transformation etc. It is demonstrated that quantum mechanics may be considered as a rational statistical generalization of classical mechanics. Statistical conformities in quantum mechanics are fundamental and are not due to the incomplete information about the system. From all possible multi-parametric statistical models the root model is most remarkable. Constructing of a root multi-parametric statistical model leads to obtaining such frequencies and base functions in Fourier decomposition so that classical laws of motion are fulfilled in average. Root model leads to a consistent condition that connects eigenvectors and eigenvalues of a mechanical system that is described by a matrix equation of Heisenberg. Matrix equation of Heisenberg leads to an operator equation. The solution of the operator equation may be considered as the construction of Hamiltonian of the system and a move to the Schrodinger representation. The considered approach naturally leads to the notion of momentum operator, fundamental commutation relations, density matrix and Liouville equation construction etc.
We first introduce and discuss density operator interpretations of quantum theory as a special case of a more general class of interpretations, giving special attention to a version that we call the `atomic version'. We then review some crucial parts of the theory of stochastic processes (the proper context in which to discuss dynamics), and develop a general framework for specifying a dynamics for density operator interpretations. This framework admits infinitely many empirically equivalent dynamics. We give some examples, and discuss some of the properties of one of them.