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arXiv:0911.5181 · Physical Review A

Tsallis entropy and entanglement constraints in multiqubit systems

Jeong San Kim

Published Jun 21, 2010• 135 citations
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Abstract

We show that the restricted shareability and distribution of multiqubit entanglement can be characterized by Tsallis-$q$ entropy. We first provide a class of bipartite entanglement measures named Tsallis-$q$ entanglement, and provide its analytic formula in two-qubit systems for $1\ensuremath{\leqslant}q\ensuremath{\leqslant}4$. For $2\ensuremath{\leqslant}q\ensuremath{\leqslant}3$, we show a monogamy inequality of multiqubit entanglement in terms of Tsallis-$q$ entanglement, and we also provide a polygamy inequality using Tsallis-$q$ entropy for $1\ensuremath{\leqslant}q\ensuremath{\leqslant}2$ and $3\ensuremath{\leqslant}q\ensuremath{\leqslant}4$.

DOI
10.1103/physreva.81.062328

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Monogamy and polygamy relations characterize the distributions of entanglement in multipartite systems. We provide classes of monogamy and polygamy inequalities of multiqubit entanglement in terms of concurrence, entanglement of formation, negativity, Tsallis-qqq entanglement and R\'{e}nyi-α\alphaα entanglement, respectively. We show that these inequalities are tighter than the existing ones for some classes of quantum states.
Monogamy and polygamy relations characterize the distributions of entanglement in multipartite systems. We provide a characterization of multiqubit entanglement constraints in terms of unified-(q, s) entropy. A class of tighter monogamy inequalities of multiqubit entanglement based on the α-th power of unified-(q, s) entanglement for α ≥ 1 and a class of polygamy inequalities in terms of the β-th power of unified-(q, s) entanglement of assistance are established in this paper. Our results present a general class of the monogamy and polygamy relations for bipartite entanglement measures based on unified-(q, s) entropy, which are tighter than the existing ones. What is more, some usual monogamy and polygamy relations, such as monogamy and polygamy relations based on entanglement of formation, Renyi-q entanglement of assistance and Tsallis-q entanglement of assistance, can be obtained from these results by choosing appropriate parameters (q, s) in unified-(q, s) entropy entanglement. Typical examples are also presented for illustration.
We provide generalized entanglement constraints in multi-qubit systems in terms of Tsallis entropy. Using quantum Tsallis entropy of order qqq, we first provide a generalized monogamy inequality of multi-qubit entanglement for q=2q=2q=2 or 333. This generalization encapsulates multi-qubit CKW-type inequality as a special case. We further provide a generalized polygamy inequality of multi-qubit entanglement in terms of Tsallis-qqq entropy for 1≤q≤21 \leq q \leq21≤q≤2 or 3≤q≤43 \leq q \leq 43≤q≤4, which also contains the multi-qubit polygamy inequality as a special case.