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Journal of Mathematical Physics

Realization of latent ensembles of the statistical operator

Boris Leaf

Published Jun 1, 1985• 4 citations
Publisher

Abstract

The statistical operator is a weighted sum, over eigenvalues of a complete commuting set of observables, of pure states each evolving unitarily from a microstate of the set. For every complete commuting set q̂ there is a latent ensemble, the diagonal projection of the statistical operator in q̂ representation, which determines the probabilities of observables in set q̂. When constraints imposed by a measurement situation makes these observables measurable, the latent q̂ ensemble is realized as the initial state of a new ensemble, which then evolves unitarily. Realization of a latent ensemble is an irreversible process with non-negative entropy increase. Reduced ensembles are realized when the constraints of the measurement situation permit only a subset of q̂ observables to be measurable. Successive realizations in response to changing measurement situations drive the system toward a state of maximum entropy. This mechanism explains the approach to equilibrium, and the change of an equilibrium system to an equilibrium state of increased entropy as constraints are relaxed.

DOI
10.1063/1.526942

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The statistical operator is a weighted sum, over eigenvalues of a complete commuting set of observables, of pure states each evolving unitarily from a microstate of the set. For every complete commuting set q̂ there is a latent ensemble, the diagonal projection of the statistical operator in q̂ representation, which determines the probabilities of observables in set q̂. When constraints imposed by a measurement situation makes these observables measurable, the latent q̂ ensemble is realized as the initial state of a new ensemble, which then evolves unitarily. Realization of a latent ensemble is an irreversible process with non-negative entropy increase. Reduced ensembles are realized when the constraints of the measurement situation permit only a subset of q̂ observables to be measurable. Successive realizations in response to changing measurement situations drive the system toward a state of maximum entropy. This mechanism explains the approach to equilibrium, and the change of an equilibrium system to an equilibrium state of increased entropy as constraints are relaxed.
In many cases, measurements are performed on “mixed” ensembles of realizations of a system, which cannot be associated with a single vector in its Hilbert space. In these cases the state of the system is represented by a proper “density operator.” It is instructive to associate density operators with state vectors in a vector space, termed Liouville’s space, where the Schrödinger equation is reformulated as the Liouville–Von Neumann equation. An important consequence of this equation is that the density operator of a system at equilibrium must commute with its Hamiltonian. Namely, the matrix representation of the density operator in the basis of Hamiltonian eigenstates is diagonal, where the diagonal elements are the relative populations of the system’s Hamiltonian eigenstates. The equilibrium populations are revealed by maximizing the (Von Neumann) entropy, subject to given constraints. We derive the equilibrium density operator explicitly for the cases of canonical and grand canonical ensembles.
Beginning with the statistical description of the results of a duplicated experiment as a set of configurations of some observables tagether with frequencies of occurrence of each configuration it is shown that an evolution of the system by a stochastic process is obtained when the conditioning parameters of the experiment are altered.It is then shown that any such stochastic process has a representation in terms of linear operators in an abstract vector space with a state vector evolving by an isometric operator S and commuting Hermitian operators representing observables which evolve by another, unitary, operator U.This has the structure of conventional quantum theory as in the interaction, or Dirac picture, but here S is not unitary as in conventional theory.This is shown to yield a matrix of transition probabilities that is not doubly stochastic as in conventional theory and hence the Pauli master equation does not follow.All implications of this completely irreversible quantum theory have not yet been fully explored, but it points to a new viewpoint in irreversible thermod ynamics.
Maximum-entropy ensembles are key primitives in statistical mechanics. Several approaches have been developed in order to justify the use of these ensembles in statistical descriptions. However, there is still no full consensus on the precise reasoning justifying the use of such ensembles. In this work, we provide an approach to derive maximum-entropy ensembles, taking a strictly operational perspective. We investigate the set of possible transitions that a system can undergo together with an environment, when one only has partial information about the system and its environment. The set of these transitions encodes thermodynamic laws and limitations on thermodynamic tasks as particular cases. Our main result is that the possible transitions are exactly those that are possible if both system and environment are assigned the maximum-entropy state compatible with the partial information. This justifies the overwhelming success of such ensembles and provides a derivation independent of typicality or information-theoretic measures.
Maximum-entropy ensembles are key primitives in statistical mechanics. Several approaches have been developed in order to justify the use of these ensembles in statistical descriptions. However, there is still no full consensus on the precise reasoning justifying the use of such ensembles. In this work, we provide an approach to derive maximum-entropy ensembles, taking a strictly operational perspective. We investigate the set of possible transitions that a system can undergo together with an environment, when one only has partial information about the system and its environment. The set of these transitions encodes thermodynamic laws and limitations on thermodynamic tasks as particular cases. Our main result is that the possible transitions are exactly those that are possible if both system and environment are assigned the maximum-entropy state compatible with the partial information. This justifies the overwhelming success of such ensembles and provides a derivation independent of typicality or information-theoretic measures.
The density matrix $\ensuremath{\rho}$ for an n-level system is decomposed into the minimum number of pure states necessary to calculate physical observables. The corresponding physical system is first represented by a set B\mathcal{B}B of n pure states $|{\ensuremath{\beta}}_{i}〉,$ together with their statistical weights. The time evolution of the system is therefore calculated as B(t)=UB(t0),\mathcal{B}(t)=U\mathcal{B}{(t}_{0}),B(t)=UB(t0​), with the propagator U operating on each member of the set, in contrast to the more laborious $\ensuremath{\rho}(t)=U\ensuremath{\rho}{(t}_{0}{)U}^{\ifmmode\dagger\else\textdagger\fi{}}.$ At least one of the states can be eliminated from the set by reducing its weight to zero via a simple transformation of the density matrix. When there are degenerate weights, the transformation is applied using the weight with the largest degeneracy. Thus, even if none of the original statistical weights are equal to zero, so that $\mathrm{rank}(\ensuremath{\rho})=n,$ the system can be described by a set of m states with m<n.m<n.m<n. The resulting minimal set provides a concommitant increase in calculational efficiency that depends on the fractional degeneracy. The independent evolution of the states comprising the set also provides an inherent parallelism that can be readily utilized in numerical computations. In addition to efficiency and conceptual simplification, this decomposition can provide physical insight, since the explicit contribution of each pure state to the dynamics of the system is readily apparent.
The challenge of understanding quantum measurement persists as a fundamental issue in modern physics. Particularly, the abrupt and energy-nonconserving collapse of the wave function appears to contradict classical thermodynamic laws. The contradiction can be resolved by considering measurement itself to be an entropy-increasing process, driven by the second law of thermodynamics. One such resolution explains the apparently irreversible emergence of objective outcomes in an isolated, unitarily evolving quantum system via the theory of closed-system equilibration. Working within this framework, we construct the set of “” that best encode the measurement statistics of a system in an objective manner and establish a measurement error bound to quantify the probability an observer will obtain an incorrect measurement outcome. Using this error bound, we show that the objectifying observables readily equilibrate on average under the set of Hamiltonians which preserve the outcome statistics on the measured system. Using a random matrix model for this set, we numerically determine the measurement error bound, finding that the error only approaches zero with increasing environment size when the environment is coarse grained into so-called observer systems. This indicates the necessity of coarse graining an environment for the emergence of objective, classical measurement outcomes.