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arXiv:1104.1675 · Journal of Physics A Mathematical and Theoretical

Unified entropy, entanglement measures and monogamy of multi-party entanglement

Jeong San Kim, Barry C. Sanders

Published Jun 20, 2011• 71 citations
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Abstract

We show that restricted shareability of multi-qubit entanglement can be fully characterized by unified-( q , s ) entropy. We provide a two-parameter class of bipartite entanglement measures, namely unified-( q , s ) entanglement with its analytic formula in two-qubit systems for q ⩾ 1, 0 ⩽ s ⩽ 1 and qs ⩽ 3. Using unified-( q , s ) entanglement, we establish a broad class of the monogamy inequalities of multi-qubit entanglement for q ⩾ 2, 0 ⩽ s ⩽ 1 and qs ⩽ 3.

DOI
10.1088/1751-8113/44/29/295303

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We show that restricted shareability of multi-qubit entanglement can be fully characterized by unified-( q , s ) entropy. We provide a two-parameter class of bipartite entanglement measures, namely unified-( q , s ) entanglement with its analytic formula in two-qubit systems for q ⩾ 1, 0 ⩽ s ⩽ 1 and qs ⩽ 3. Using unified-( q , s ) entanglement, we establish a broad class of the monogamy inequalities of multi-qubit entanglement for q ⩾ 2, 0 ⩽ s ⩽ 1 and qs ⩽ 3.
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.
Entanglement concurrence is an important bipartite entanglement measure that has found wide applications in quantum technologies. In this work, inspired by unified entropy, we introduce a two-parameter family of entanglement measures, referred to as the unified (q,s)(q,s)(q,s)-concurrence. Both the standard entanglement concurrence and the recently proposed qqq-concurrence emerge as special cases within this family. By combining the positive partial transposition and realignment criteria, we derive an analytical lower bound for this measure for arbitrary bipartite mixed states, revealing a connection to strong separability criteria. Explicit expressions are obtained for the unified (q,s)(q,s)(q,s)-concurrence in the cases of isotropic and Werner states under the constraint q>1q>1q>1 and qs≥1qs\geq 1qs≥1. Furthermore, we explore the monogamy properties of the unified (q,s)(q,s)(q,s)-concurrence for q≥2q\geq 2q≥2, 0≤s≤10\leq s\leq 10≤s≤1 and 1≤qs≤31\leq qs\leq 31≤qs≤3, in qubit systems. In addition, we derive an entanglement polygon inequality for the unified (q,s)(q,s)(q,s)-concurrence with q≥1q\geq 1q≥1 and qs≥1qs\geq 1qs≥1, which manifests the relationship among all the marginal entanglements in any multipartite qudit system.
We show that the restricted shareability and distribution of multiqubit entanglement can be characterized by Tsallis-qqq entropy. We first provide a class of bipartite entanglement measures named Tsallis-qqq 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-qqq entanglement, and we also provide a polygamy inequality using Tsallis-qqq entropy for $1\ensuremath{\leqslant}q\ensuremath{\leqslant}2$ and $3\ensuremath{\leqslant}q\ensuremath{\leqslant}4$.
Monogamy relation is one of the essential properties of quantum entanglement, which characterizes the distribution of entanglement in a multipartite system. By virtual of the unified-(q,s) entropy, we obtain some novel monogamy and polygamy inequalities in general class of entanglement measures. For the multiqubit system, a class of tighter monogamy relations are established in term of the α-th power of unified-(q,s) entanglement for α ≥ 1. We also obtain a class of tighter polygamy relations in the β-th (0 ≤ β ≤ 1) power of unified-(q,s) entanglement of assistance. Applying these results to specific quantum correlations, e.g., entanglement of formation, Renyi-q entanglement of assistance, and Tsallis-q entanglement of assistance, we obtain the corresponding monogamy and polygamy relations. Typical examples are presented for illustration. Furthermore, the complementary monogamy and polygamy relations are investigated for the α-th (0 ≤ α ≤ 1) and β-th (β ≥ 1) powers of unified entropy, respectively, and the corresponding monogamy and polygamy inequalities are obtained.
We establish a characterization of multi-qubit entanglement constraints in terms of non-negative power of entanglement measures based on unified-(q,s)(q,s)(q,s) entropy. Using the Hamming weight of the binary vector related with the distribution of subsystems, we establish a class of tight monogamy inequalities of multi-qubit entanglement based on the αααth-power of unified-(q,s)(q,s)(q,s) entanglement for α≥1α\geq 1α≥1. For 0≤β≤10 \leq β\leq 10≤β≤1, we establish a class of tight polygamy inequalities of multi-qubit entanglement in terms of the βββth-power of unified-(q,s)(q,s)(q,s) entanglement of assistance. Thus our results characterize the monogamy and polygamy of multi-qubit entanglement for the full range of non-negative power of unified entanglement.