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Improving Relative Entropy of Entanglement

An Min Wang

Published Jan 11, 2000• 0 citations
OpenAlex

Abstract

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.

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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.
We present the modified relative entropy of entanglement (MRE) in order to both improve the computability for the relative entropy of entanglement and avoid the problem that the entanglement of formation seems to be greater than entanglement of distillation. For two qubit system we derive out an explicit and "weak" closed expression of MRE that depends on the pure state decompositions in the case of mixed states. For more qubit system, we obtain an algorithm to calculate MRE in principle. MRE significantly improves the computability of relative entropy of entanglement and decreases the dependence and sensitivity on the pure state decompositions. Moreover it is able to inherit most of the important physical features of the relative entropy of entanglement. In addition, a kind of states, as an extension of Werner's states, is discussed constructively.
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.
We present the modified relative entropy of entanglement 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 modified relative entropy of entanglement of some important and interesting examples, we display the main behaviors and elementary properties of the modified relative entropy of entanglement and compare it with the generalized entanglement of formation. Moreover, we also propose an assistant, as the upper bound, of the modified relative entropy of entanglement.
We determine the relative entropy of entanglement of a bipartite mixed state ρABρ_{AB}ρAB​ obtained by tracing out one subsystem of a tripartite Haar-random pure state ∣ψ⟩ABC|ψ\rangle_{ABC}∣ψ⟩ABC​, finding ER(ρAB)=log⁡dAdBmax⁡(dA,dB,dC)+O(1)E_R(ρ_{AB})=\log\frac{d_Ad_B}{\max(d_A,d_B,d_C)}+O(1)ER​(ρAB​)=logmax(dA​,dB​,dC​)dA​dB​​+O(1). Equivalently, the relative entropy of entanglement nearly saturates the smaller of the entanglement of formation EF(ρAB)E_F(ρ_{AB})EF​(ρAB​) and the mutual information I(A:B)I(A:B)I(A:B). The upper bound is achieved by an explicit separable state obtained through one-sided Schmidt dephasing, which is therefore approximately closest.