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arXiv:quant-ph/0012029

Modified relative entropy of entanglement for multi-party systems

An Min Wang

Published Dec 6, 2000• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0012029

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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 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 generalization of the entanglement of formation for three-party systems in a pure state. For three qubit system we derive out its explicit and closed expression which is a linear combination of the binary entropy functions with various arguments, and these arguments are clearly determined in terms of the components of state vector of three qubits. As a reasonable measure of entanglement, the main behaviors and elementary properties of this generalized entanglement of formation are showed through discussing some important and interesting examples. Moreover, we propose how to extend our definition to a mixed state in according to the familiar idea. Then, we suggest the generalization of the entanglement of formation for multi-party systems which is consistent with the regular definition for two-party systems and our definition for three-party systems.
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.
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 a general method to find the upper and lower bounds on the generalized entanglement of formation for multi-party systems. The upper and lower bounds can be expressed in terms of the bi-partite entanglements of formation and/or entropies of various subsystems. The examples for tri- and four-party systems in the both pure states and mixed states are given. We also suggest a little modified definition of generalized entanglement of formation for multi-party systems if EPR pairs are thought of belonging to the set of maximally entangled states.