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Quantum Information and Computation

Entanglement of formation and concurrence

William K. Wootters

Published Jul 1, 2001• 361 citations
Publisher

Abstract

This paper reviews our current understanding of entanglement of formation and the related concept of concurrence, including discussions of additivity, the problem of finding explicit formulas, and connections between concurrence and other propertis of bipartite states.
DOI
10.26421/qic1.1-3

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Quantum information and computation is one of the new fields of physics and information science. Entanglement is one of the most significant resources in quantum information processing. The most basic quantitative measure of entanglement, particularly for mixed states of a bipartite system is the entanglement of formation. This entanglement of formation is used in the concurrence, another measurement of entanglement. Here I have briefly reviewed the entanglement of formation and the concurrence of bipartite systems. I have discussed that for a three qubit pure state, if A, B and C are the three qubits, then sum of the squared concurrence between A and B and the squared concurrence between A and C never exceeds the squared concurrence between A and BC pair. The residual entanglement, obtained from the aforementioned inequality, is also discussed here. Keywords- Quantum information, entanglement, qubit.
We prove that the squared R\'enyi-$\ensuremath{\alpha}$ entanglement ($\mathrm{SR}\ensuremath{\alpha}\mathrm{E}$), which is the generalization of entanglement of formation, obeys a general monogamy inequality in an arbitrary NNN-qubit mixed state. Furthermore, for a class of R\'enyi-$\ensuremath{\alpha}$ entanglement, we prove that the monogamy relations of the $\mathrm{SR}\ensuremath{\alpha}\mathrm{E}$ have a hierarchical structure when the NNN-qubit system is divided into kkk parties. As a by-product, the analytical relation between the R\'enyi-$\ensuremath{\alpha}$ entanglement and the squared concurrence is derived for bipartite $2\ensuremath{\bigotimes}d$ systems. Based on the monogamy properties of $\mathrm{SR}\ensuremath{\alpha}\mathrm{E}$, we can construct the corresponding multipartite entanglement indicators, which still work well even when the indicators based on the squared concurrence and EOF lose their efficacy. In addition, the monogamy property of the $\ensuremath{\mu}\mathrm{th}$ power of R\'enyi-$\ensuremath{\alpha}$ entanglement is analyzed.