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arXiv:0710.4061

The quantum entanglement contained in density matrices

Yong Heng Zhou

Published Oct 22, 2007• 0 citations
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Abstract

We point out that density matrices can only be used to describe quantum states, so the entanglement contained in a density matrix is just quantum entanglement. This means a bipartite state described by a density matrix contains quantum entanglement, unless the density matrix has the form $ρ_{AB}=ρ_A\otimes ρ_B$.

DOI
10.48550/arxiv.0710.4061

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We point out that density matrices can only be used to describe quantum states, so the entanglement contained in a density matrix is just quantum entanglement. This means a bipartite state described by a density matrix contains quantum entanglement, unless the density matrix has the form ρAB=ρA⊗ρBρ_{AB}=ρ_A\otimes ρ_BρAB​=ρA​⊗ρB​.
Linear maps of matrices describing the evolution of density matrices for a quantum system initially entangled with another are identified and found to be not always completely positive. They can even map a positive matrix to a matrix that is not positive, unless we restrict the domain on which the map acts. Nevertheless, their form is similar to that of completely positive maps. Only some minus signs are inserted in the operator-sum representation. Each map is the difference of two completely positive maps. The maps are first obtained as maps of mean values and then as maps of basis matrices. These forms also prove to be useful. An example for two entangled qubits is worked out in detail. The relation to earlier work is discussed.
We obtain a new inequality for arbitrary Hermitian matrices. We describe particular linear maps called the matrix portrait of arbitrary NxN matrices. The maps are obtained as analogs of partial tracing of density matrices of multipartite qudit systems. The structure of the maps is inspired by "portrait" map of the probability vectors corresponding to the action on the vectors by stochastic matrices containing either unity or zero matrix elements. We obtain new entropic inequalities for arbitrary qudit states including a single qudit and an discuss entangled single qudit state. We consider in detail the examples of N=3 and 4. Also we point out a possible use of entangled states of systems without subsystems (e.g., a single qudit) as a resource for quantum computations.