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arXiv:quant-ph/0209081 · Journal of Mathematical Physics

Broken symmetries in the entanglement of formation

Fabio Benatti, H. Narnhofer, A. Uhlmann

Published May 21, 2003• 2 citations
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Abstract

We compare some recent computations of the entanglement of formation in quantum information theory and of the entropy of a subalgebra in quantum ergodic theory. Both notions require optimization over decompositions of quantum states. We show that both functionals are strongly related for some highly symmetric density matrices. Indeed, for certain interesting regions the entanglement of formation can be expressed by the entropy of a commuting subalgebra, and the corresponding optimal decompositions can be obtained one from the other. We discuss the presence of broken symmetries in relation with the structure of the optimal decompositions.

DOI
10.1063/1.1570509

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We compare some recent computations of the entanglement of formation in quantum information theory and of the entropy of a subalgebra in quantum ergodic theory. Both notions require optimization over decompositions of quantum states. We show that both functionals are strongly related for some highly symmetric density matrices. Indeed, for certain interesting regions the entanglement of formation can be expressed by the entropy of a commuting subalgebra, and the corresponding optimal decompositions can be obtained one from the other. We discuss the presence of broken symmetries in relation with the structure of the optimal decompositions.
To calculate the entropy of a subalgebra or of a channel with respect to a state, one has to solve an intriguing optimalization problem. The latter is also the key part in the entanglement of formation concept, in which case the subalgebra is a subfactor. I consider some general properties, valid for these definitions in finite dimensions, and apply them to a maximal commutative subalgebra of a full matrix algebra. The main method is an interplay between convexity and symmetry. A collection of helpful tools from convex analysis for the problems in question is collected in an appendix.
Through choosing and fixing the relative density matrix within a finite decomposition set of pure states for a mixed state, the definition of the relative entropy of entanglement is improved. This improving relative entropy of entanglement is able to inherit all the physical features of the relative entropy of entanglement. It is obtained that the improving relative entropy of entanglement is a function of the polarization vectors of the reduced density matrices of the decomposition density matrices. This makes that the improving relative entropy of entanglement has the same computability as the entanglement of formation has. In addition, a kind of states, as an extension of Werner's states, is discussed.
The entanglement of formation for a class of high-dimensional quantum mixed states is investigated. A special kind of D-computable states is defined and the lower bound of entanglement of formation for a large class of density matrices whose decompositions lie in these D-computable quantum states is obtained. Moreover we present a kind of construction for this special state which is defined by a class of special matrices with two non-zero different eigenvalues and the other eigenvalues are zero. Making use of the D-computable we construct a class of bound entangled states.
We show how to simplify the computation of the entanglement of formation and the relative entropy of entanglement for states, which are invariant under a group of local symmetries. For several examples of groups we characterize the state spaces, which are invariant under these groups. For specific examples we calculate the entanglement measures. In particular, we derive an explicit formula for the entanglement of formation for $(U\ensuremath{\bigotimes}U)$-invariant states, and we find a counterexample of the additivity conjecture for the relative entropy of entanglement.
We present the improved relative entanglement entropy for multi-party systems by a given relative density matrix which is spanned by a linear combination of the direct products of so-called basis of relative density matrices and reduced density matrices for every party. For three qubit system in a pure state we derive out the explicit and closed expression of our relative density matrix which is a function of the components of the state vector of three qubits. Through computing the improved relative entanglement entropy of some important and interesting examples, we display the main behaviors and elementary properties of the improved relative entanglement entropy and compare it with the generalized entanglement of formation. Moreover, we also propose an assistant, as the upper bound, of the improved relative entanglement entropy.