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arXiv:quant-ph/9909020 · Physical Review A

Probability distributions consistent with a mixed state

Michael A. Nielsen

Published Oct 16, 2000• 42 citations
arXivHTMLPDF

Abstract

A density matrix \ensuremath{\rho} may be represented in many different ways as a mixture of pure states, $\ensuremath{\rho}={\ensuremath{\sum}}_{i}{p}_{i}|{\ensuremath{\psi}}_{i}〉〈{\ensuremath{\psi}}_{i}|.$ This paper characterizes the class of probability distributions ${(p}_{i})$ that may appear in such a decomposition, for a fixed density matrix \ensuremath{\rho}. Several illustrative applications of this result to quantum mechanics and quantum information theory are given.

DOI
10.1103/physreva.62.052308

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A density matrix \ensuremath{\rho} may be represented in many different ways as a mixture of pure states, \ensuremathρ=\ensuremath∑ipi∣\ensuremathψi〉〈\ensuremathψi∣.\ensuremath{\rho}={\ensuremath{\sum}}_{i}{p}_{i}|{\ensuremath{\psi}}_{i}〉〈{\ensuremath{\psi}}_{i}|.\ensuremathρ=\ensuremath∑i​pi​∣\ensuremathψi​〉〈\ensuremathψi​∣. This paper characterizes the class of probability distributions (pi){(p}_{i})(pi​) that may appear in such a decomposition, for a fixed density matrix \ensuremath{\rho}. Several illustrative applications of this result to quantum mechanics and quantum information theory are given.
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.
The concepts of classical information theory can be extended to quantum information theory. Since in general measurement yields a result with probability, we may suggest using these probabilities in classical information theory. However the probabilities do not contain phase information, which cannot be neglected. Thus the definitions are given in terms of the density operator. These probabilities depend on the basis used for measurement. A density operator ρ over a n-dimensional Hilbert space H is a positive operator with unit trace. The trace tr(A) is defined as tr(A):=∑j=1n⟨βj∣A∣βj⟩tr(A): = \sum\limits_{j = 1}^n {\left\langle {{\beta _j}|A|{\beta _j}} \right\rangle } tr(A):=j=1∑n​⟨βj​∣A∣βj​⟩ where β j for j = 1,..., n is any orthonormal basis in HThus tr(P)=1. The eigenvalues of a density operator are greater than zero. By the spectral theorem every density operator can be represented as a mixture of pure states ρ=∑j=1nPj∣⟨aj⟨aj∣\rho = \sum\limits_{j = 1}^n {Pj|} \left\langle {aj} \right.\langle aj|ρ=j=1∑n​Pj∣⟨aj⟨aj∣ where α j for j = 1,..., n are the orthonormal eigenvectors of ρ (which form a basis in H and pj∈R,pj0,∑j=1npj=1pj \in R,{p_j}0,\sum\limits_{j = 1}^n {{p_j}} = 1pj∈R,pj​0,j=1∑n​pj​=1
A mixed state may be represented in many different ways as a mixture of pure states ρ=∑pi∣ψi⟩⟨ψi∣. The mixing problem in quantum mechanics asks the characterization of the probability distribution (pi) and the mixed states (ρi) such that ρ=∑piρi for any given mixed state ρ. Some constraints based on eigenvalues of the mixed states are established in uni-party case [see Nielsen, Phys. Rev. A. 63, 052308 (2000), 63, 022144 (2000), Nielsern and Vidal Quantum Inf. Comput. 1 76 (2001)]. We develop some new invariant sets for bipartite mixed states under local unitary operations, which are independent of eigenvalues, and prove some strong constraints based on these invariant sets for the mixing problem in bipartite case. This exhibits a remarkable difference from the uni-party case.