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Physical Review A

Representation of a quantum ensemble as a minimal set of pure states

Thomas E. Skinner, Steffen J. Glaser

Published Sep 25, 2002• 5 citations
Publisher

Abstract

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 $\mathcal{B}$ of n pure states $|{\ensuremath{\beta}}_{i}〉,$ together with their statistical weights. The time evolution of the system is therefore calculated as $\mathcal{B}(t)=U\mathcal{B}{(t}_{0}),$ 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 $m<n.$ 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.

DOI
10.1103/physreva.66.032112

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The density matrix \ensuremathρ\ensuremath{\rho}\ensuremathρ 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 ∣\ensuremathβi〉,|{\ensuremath{\beta}}_{i}〉,∣\ensuremathβi​〉, 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 \ensuremathρ(t)=U\ensuremathρ(t0)U\ifmmode†\else\textdagger\fi.\ensuremath{\rho}(t)=U\ensuremath{\rho}{(t}_{0}{)U}^{\ifmmode\dagger\else\textdagger\fi{}}.\ensuremathρ(t)=U\ensuremathρ(t0​)U\ifmmode†\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 rank(\ensuremathρ)=n,\mathrm{rank}(\ensuremath{\rho})=n,rank(\ensuremathρ)=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.
An analytical expression for optimal Lewenstein-Sanpera (L-S) decomposition of a generic two qubit density matrix is given. By evaluating the L-S decomposition of Bell decomposable states, the optimal decomposition for arbitrary full rank state of two qubit system is obtained via local quantum operations and classical communications (LQCC). In Bell decomposable case the separable state optimizing L-S decomposition, minimize the von Neumann relative entropy as a measure of entanglement. The L-S decomposition for a generic two-qubit density matrix is only obtained by using Wootters's basis. It is shown that the average concurrence of the decomposition is equal to the concurrence of the state. It is also shown that all the entanglement content of the state is concentrated in the Wootters's state |x_1> associated with the largest eigenvalue \lambda_1 of the Hermitian matrix \sqrt{\sqrt{rho}\tilde{rho}\sqrt{rho}} . It is shown that a given density matrix rho with corresponding set of positive numbers \lambda_i and Wootters's basis can transforms under SO(4,c) into a generic 2x2 matrix with the same set of positive numbers but with new Wootters's basis, where the local unitary transformations correspond to SO(4,r) transformations, hence, \rho can be represented as coset space SO(4,c)/SO(4,r) together with positive numbers lambda_i.
For a configurational space of arbitrary dimension a strict form of the uncertainty principle has been obtained, which takes into account the dependence of the inequality limit on the effective number of pure states present in a given statistical mixture. It is shown that in a state with minimal uncertainty the density operator eigenfunctions coincide with the stationary wavefunctions of a multi-dimensional harmonic oscillator. The spectrum of eigenvalues of the density matrix is degenerate and its terms with the same uncertainty have identical weights in the expansion. This leads to additional `packing of states' (i.e. decreasing of the specific phase volume per particular state in the mixture as compared with the limit of the system in a pure state). The analogy between the generalized uncertainty principle and one of the main postulates of statistical physics on phase-space partitioning into cells corresponding to one quantum state is discussed.
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.