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Journal of Mathematical Physics

Constraints on the mixing of bipartite states

Hao Chen

Published Dec 1, 2005• 1 citation
Publisher

Abstract

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.

DOI
10.1063/1.2138048

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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.
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 (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.
Any set of pure states living in a given Hilbert space possesses a natural and unique metric—the Haar measure—on the group U(N) of unitary matrices. However, there is no specific measure induced on the set of eigenvalues, Δ, of any density matrix ρ. Therefore, a general approach to the global properties of mixed states depends on the specific metric defined on Δ. In the present work we shall employ a simple measure on Δ that has the advantage of possessing a clear geometric visualization whenever discussing how arbitrary states are distributed according to some measure of mixedness. The degree of mixture will be that of the participation ratio R=1/Tr(ρ2) and the concomitant maximum eigenvalue, λm. The cases studied will be the qubit–qubit system and the qubit–qutrit system, whereas some discussion will be made on higher-dimensional bipartite cases in both the R-domain and the λm-domain.
We introduce algebriac sets in the products of complex projective spaces for multipartite mixed states, which are independent of their eigenvalues and only measure the "position" of their eigenvectors, as their non-local invariants (ie. remaining invariant after local untary transformations). These invariants are naturally arised from the physical consideration of checking multipartite mixed states by measuring them with multipartite separable pure states. The algebraic sets have to be the sum of linear subspaces if the multipartite mixed state is separable, and thus we give a new separability criterion of multipartite mixed states. A continuous family of 4-party mixed states, whose members are separable for any 2:2 cut and entangled for any 1:3 cut (thus bound entanglement if 4 parties are isolated), is constructed and studied from our invariants and separability criterion. Examples of LOCC-incomparable entangled tripartite pure states are given to show it is hopeless to characterize the entanglement properties of tripartite pure states by only using the eigenvalue speactra of their partial traces. We also prove that at least n2+n−1n^2+n-1n2+n−1 terms of separable pure states, which are "orthogonal" in some sence, are needed to write a generic pure state in HAn2⊗HBn2⊗HCn2H_A^{n^2} \otimes H_B^{n^2} \otimes H_C^{n^2}HAn2​⊗HBn2​⊗HCn2​ as a linear combination of them.