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Acta Sinica Quantum Optica

Entanglement for a Three-parameter Class of States in Bipartite Quantum System

Yao-Min Di

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

Entanglement for a three-parameter class of states in bipartite quantum system is investigated.The concurrence for these states in 2n quantum system is calculated and their lower bound of the entanglement of formation is discussed.The negativity for these states in mn(2≤m≤n) quantum system is calculated and the analytic expressions are obtained.Investigation reveals that for θ=π/4,entanglement of the states reaches its maximum,and the states come down to the two-parameter class of states researched by some authors.For these three-parameter states in 2n quantum system,all PPT states are separable,and all NPT states are entangled.But for higher dimensional three-parameter states,the similar conclusion does not exist.

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Entanglement for a three-parameter class of states in bipartite quantum system is investigated.The concurrence for these states in 2n quantum system is calculated and their lower bound of the entanglement of formation is discussed.The negativity for these states in mn(2≤m≤n) quantum system is calculated and the analytic expressions are obtained.Investigation reveals that for θ=π/4,entanglement of the states reaches its maximum,and the states come down to the two-parameter class of states researched by some authors.For these three-parameter states in 2n quantum system,all PPT states are separable,and all NPT states are entangled.But for higher dimensional three-parameter states,the similar conclusion does not exist.
Any arbitrary state in bipartite quantum system can be transformed into a two-parameter class of states by means of local quantum operation and classical communication.In this paper,the quantum relative entropy of entanglement for this kind of states is calculated,the nearest separable states are found and the analytic expression of the entanglement is obtained.The comparation with two other measures of entanglement,the negativity and the entanglement of formation,is also made.The sequence of three measures from high to low is the negativity,the entanglement of formation and the relative entropy of entanglement.
We exhibit a two-parameter class of states ρ (α,γ) in a 2 ⊗ n quantum system for n ⩾ 3, which can be obtained from an arbitrary state by means of local quantum operations and classical communication, and which are invariant under all bilateral operations on a 2 ⊗ n quantum system. We calculate the negativity of ρ (α,γ) , and a lower bound and a tight upper bound on its entanglement of formation. It follows from this calculation that the entanglement of formation of ρ (α,γ) cannot exceed its negativity.
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 construct a class of entangled states in ℋ = ℋA ⊗ ℋB ⊗ ℋC quantum systems with dim ℋA = dim ℋB = dim ℋC = 2 and classify those states with respect to their distillability properties. The states are bound entanglement for the bipartite split (AB) - C. The states are non-positive partial transpose (NPT) entanglement and 1-copy undistillable for the bipartite splits A - (BC) and B - (AC). Moreover, we generalize the results of 2 ⊗ 2 ⊗ 2 systems to the case of 2n ⊗ 2n ⊗ 2n systems.