Shi Xian. Monogamy relations of quantum entanglement for partially coherently superposed states. Chinese Physics B, 2017, 26(12): 120303
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Monogamy relations of quantum entanglement for partially coherently superposed states
Shi Xian1, 2, 3, †
Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
University of Chinese Academy of Sciences, Beijing 100190, China
UTS-AMSS Joint Research Laboratory for Quantum Computation and Quantum Information Processing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Project partially supported by the National Key Research and Development Program of China (Grant No. 2016YFB1000902), the National Natural Science Foundation of China (Grant Nos. 61232015, 61472412, and 61621003), the Beijing Science and Technology Project (2016), Tsinghua-Tencent-AMSS-Joint Project (2016), and the Key Laboratory of Mathematics Mechanization Project: Quantum Computing and Quantum Information Processing.
Abstract
Monogamy is a fundamental property of multi-partite entangled states. Recently, Kim J S [Phys. Rev. A93 032331] showed that a partially coherent superposition (PCS) of a generalized W-class state and the vacuum saturates the strong monogamy inequality proposed by Regula B et al. [Phys. Rev. Lett.113 110501] in terms of squared convex roof extended negativity; and this fact may present that this class of states are good candidates for studying the monogamy of entanglement. Hence in this paper, we will investigate the monogamy relations for the PCS states. We first present some properties of the PCS states that are useful for providing our main theorems. Then we present several monogamy inequalities for the PCS states in terms of some entanglement measures.
Quantum entanglement[1] is an essential feature of quantum information theory, which distinguishes the quantum from classical theory. One of the fundamental differences between quantum correlation and classical relations is that a quantum system entangled with one of the other systems limits its entanglement with the remaining others.[2] This property is known as monogamy of entanglement (MoE). Monogamy relations also applies in many fields of physics, such as, characterizing the entanglement structure in multipartite quantum systems,[3–10] causing the frustration effects observed in condensed matter physics;[11] moreover, MoE is a key ingredient to make quantum cryptography secure[12,13] and MoE can apply to the blackhole information problem.
Mathematically, for a tripartite system A,B, and C, the general monogamy in terms of an entanglement measure implies that the entanglement between A and BC satisfies
This relation is first proved for qubit systems with respect to the squared concurrence C2.[14,15] However, it is not true with respect to each entanglement measure. For example, entanglement of formation (EoF) Ef does not satisfy the inequality (1) for three qubit states. However, Bai et al.[16] proved that the squared EoF satisfies the monogamy inequality for n-qubit states. Moreover, Zhu and Fei[17] showed that Cα and satisfy the monogamy inequalities for multiqubit systems when and respectively. Luo et al.[18] proved that the squared Tsallis-q entanglement satisfies the monogamy inequality for multiqubit states when
The inequality (1) may not be true for a qudit tripartite pure state in terms of the squared concurrence, the author in Ref. [19] presented a counterexample in a 3 ⊗ 3 ⊗ 3 quantum system, Li and Wang showed that this counterexample satisfies the inequality (1) in terms of the squared convex roof extended negativity.[20]
Recently, the authors in Ref. [10] proposed a much sharper monogamy inequality (SMI) in an n-partite system , and the authors in Refs. [21]–[23] showed for the class of multiqudit generalized W-class states and the class of a partially coherent superposition (PCS) of a generalized W-class state and the vacuum, the SMI is valid with respect to the square of convex extended roof of negativity. Furthermore, these two classes saturate the SMI. Then it may be important to investigate the monogamy of these two classes of states. Recently, Geetha et al. analyzed the monogamous nature for the W class of symmetric N-qubit states in terms of the squared negativity and concurrence and presented that for these states, the concurrence tangle is zero, while the negativity tangle is non-zero,[24] the authors in Ref. [25] showed the generalized monogamy for the generalized W-class multiqubit states.
In this article, we will first present some preliminary knowledge needed in this article. Then we will provide the monogamy relations in terms of concurrence of assistance (CoA) and polygamy relation with respect to EoF for the PCS states when d = 2. Finally, we will consider the polygamy relations in terms of Tsallis-q entanglement for the PCS states when
and d = 2.
This paper in organized as follows. In Section 2, we will recall preliminary knowledge on the definition of concurrence, EoF and Tsallis-q entanglement for bipartite states, and then we will recall their respective monogamy inequalities for multi-qubit systems. In Section 3, we will present some properties of PCS states that are necessary for the proof of their monogamy relations. Here we also present the monogamy relations in terms of CoA and polygamy relations with respect to EoF and Tsallis-q entanglement for the PCS states. In Section 4, we will end up with a conclusion.
2. Preliminary knowledge
For a bipartite pure state |ψ⟩AB in Hilbert space , the concurrence is defined as
where ρA is the reduced density matrix by tracing over the subsystem B of |ψ⟩AB. For a mixed state ρAB, its concurrence is defined as
where the minimization takes o ver all the decompositions of ρAB. For a bipartite state ρAB, its CoA is defined as[26]
where the maximization takes over all the decompositions of ρAB = Σipi|ψi⟩AB⟨ψi|. For an n-qubit state , the monogamy inequality was generalized to[17]
where α ≥ 2, β ≤ 0. The inequality in terms of CoA for a pure state was proved in Ref. [26]
Then let us recall the definition of negativity. For a bipartite pure state |ψ⟩AB, its negativity is defined as
where |ψ⟩AB⟨ψ|TB is the partial transposition of |ψ⟩AB, ||·||1 is the 1-norm. Similarly, the negativity for a mixed state ρAB is defined as
where the minimum is taken over all the possible pure state decompositions of ρAB = ∑ipi⟨ψi|AB⟨ψi|.
Afterwards, we will recall the EoF for bipartite states. For a pure state , the EoF of |ψ⟩AB is defined as
where ρA = trB|ψ⟩AB⟨ψ|, S(ρ) = − tr ρ log ρ. For a mixed state ρAB, the EoF is defined as
where the minimization takes over all the decompositions ρAB = Σipi|ψi⟩AB⟨ψi| with Σipi = 1, pi ≥ 0. It has been shown that the entanglement of formation satisfies that[17]
where.
At last, we will present Tsallis-q entanglement for bipartite states. Firstly, we will recall the definition of Tsallis-q entropy. The Tsallis-q entropy for a state ρ is defined as
for any q > 0, q ≠ 1. When q → 1, the Tsallis-q entropy converges to von Neumann entropy: . For any pure state |ψ⟩AB, its Tsallis-q entanglement is defined as: Tq(|ψAB⟩) = Tq(ρA), for any q> 0, q≠ 1, here ρA = trB|ψ⟩AB⟨ψ|. For a mixed state ρAB, its Tsallis-q entanglement (TE) can be defined as:
for any q > 0, q ≠ 1, where the minimum is taken over all possible pure state decompositions {pi,|ψi⟩AB} of ρAB. Recently, Luo et al.[18] proved that for any 2⊗d states ρAB, the TE of ρAB has an analytical expression
for
3. Partially coherent superposition of multiqudit generalized W-class states and vacuum
In this section, we will first recall the definition and some properties of the PCS states. The PCS states ρA1A2… An are defined as follows:
with p, λ ∈ [0,1]. when λ = 1, the PCS states become the W-class states. We also find a decomposition of the PCS states:
with
Note that we assume d = 2 for the residual part of this article. Next we will first bring a lemma proposed in Ref. [27] about the ensemble for density matrices.
Fig. 1. (color online) This line is the lower bound of Ca(ρA1|A2A3) as functions of x ≥ 2.
From Fig.1, we see the optimal bound is when x = 2, that is, Ca(ρA1|A2A3) ≥ (0.122 + 0.0722)1/2 ≈ 0.1399. Here we can make a comparison with the upper bound given in Ref. [28]:
Next we will consider the polygamy with respect to the EoF for the PCS states.
Note that when β = γ = 2, Corollary 1 becomes that
and here we give a polygamy relation with respect to for PCS states in an n-partite system. From another point of view, we also find a new bound for the of the PCS states in an n-partite system.
4. Conclusion
Monogamy of entanglement is a fundamental property of multipartite entangled states. In this article, we prove that the n-qubit PCS states satisfy the monogamy inequality in terms of the x-power of CoA for its m-qubit reduced density matrices with 2 ≤ m ≤ n. Then we also investigate the monogamy inequalities in terms of EoF and TE for PCS states. We believe this investigation will provide a better clarification of the usefulness of PCS states in understanding the constraints on the description of entanglement distribution. Due to the importance of multi-party entanglement, our results may be meaningful for future work on the study of multi-party quantum entanglement.
