Monogamy of entanglement and teleportation capability

Abstract

The monogamy inequality in terms of the concurrence, called the Coffman-Kundu-Wootters inequality [Phys. Rev. A 61, 052306 (2000)], and its generalization [T.J. Osborne and F. Verstraete, Phys. Rev. Lett. 96, 220503 (2006)] hold on general n-qubit states including mixed ones. In this paper, we consider the monogamy inequalities in terms of the fully entangled fraction and the teleportation fidelity. We show that the monogamy inequalities do not hold on general mixed states, while the inequalities hold on n-qubit pure states.PACS numbers: 03.67. Mn, 03.65.Ud, 03.67.Hk Entanglement is a crucial one of several quantum mechanical phenomena, which provide us with various quantum information processing such as quantum computation superior to any classical computation and perfectly secure quantum communication. On this account, over the last three decades, a lot of research works on entanglement have been significantly developed. However, there still exist quite a few unsolved problems and interesting aspects related to entanglement, especially multipartite entanglement.In order to understand entanglement more effectively, quantum information scientists have considered the socalled entanglement measure as a tool to quantify the degree of entanglement. Among several entanglement measures, there is a simple form of entanglement measure, called the concurrence [1], which is defined as follows: For a given pure state |φ 12 in 2 ⊗ d quantum system (d ≥ 2), the concurrence C is defined as C(|φ 12 φ|) = 2(1 − trρ 2 1 ) = 2 √ det ρ 1 , where ρ 1 = tr 2 |φ 12 φ|, and for a given mixed state ρ 12 ,where the minimum is taken over its all possible decompositions,The concurrence has been considered as one of the most important measures of entanglement, since the computable formula for the entanglement of formation in the two-qubit system can be derived from the explicit formula for the concurrence in the two-qubit system, and various properties about entanglement can be seen by virtue of the concurrence.One of those interesting properties about multipartite entanglement is the monogamy of entanglement, which can be seen by the monogamy inequality in terms of the concurrence as follows: For a given n-qubit pure states |Ψ 12···n and its reduced density operators ρ 1j ,where C 1(2···n) = C(|Ψ 1(2···n) Ψ|) and C 1j = C(ρ 1j ). The inequality is called the Coffman-Kundu-Wootters inequality [2] when n = 3, and its generalization to multiqubit states was recently proved by Osborne and Verstraete [3] as well. The monogamy inequality tells us that if one particle P in a multipartite quantum system is entangled with another particle Q as much as entanglement between P and the other particles, then P cannot be entangled with any other particles except Q.There are several useful applications of entanglement, among which a practical one is teleportation [4]. Since teleportation is to send quantum information through a classical channel assisted by entanglement, the teleportation capability essentially depends on the degree of the entanglemen...

Restricted distribution of quantum correlations in bilocal network

“…Considered to be two inequivalent resources in general, both of these features form the basis of various information processing tasks such as device independent entanglement witnesses [5], Quantum Key Distribution(QKD) [6][7][8][9], Bayesian game theoretic applications [10], private randomness generation [11,12], etc, which cannot be performed by any classical resource. One of the inherent features responsible for strengthening efficiency of quantum resources over classical ones is the existence of restrictions over shareability of quantum particles or quantum correlations in multiparty scenario [13][14][15][16][17][18][19][20][21][22][23][24]. Research activities conducted so far clearly point out the existence of limitations over shareability of both quantum nonlocality [13,15] and entanglement [14,[20][21][22][23][24].…”

“…One of the inherent features responsible for strengthening efficiency of quantum resources over classical ones is the existence of restrictions over shareability of quantum particles or quantum correlations in multiparty scenario [13][14][15][16][17][18][19][20][21][22][23][24]. Research activities conducted so far clearly point out the existence of limitations over shareability of both quantum nonlocality [13,15] and entanglement [14,[20][21][22][23][24]. Such sort of limitations are frequently referred to as monogamy of nonlocality and entanglement respectively.…”

Monogamy of Quantum Correlations - A Review

“…More precisely, within the consideration of a multiparty set-up, for example, of an editor with several reporters, if the shared quantum state between the editor and a single reporter violates a Bell inequality [55,56] or is quantum dense codeable [57], then the rest of the channels shared between the editor and the other reporters are prohibited from possessing the same quantum advantage. Analogous monogamy constraints have also been addressed in the arXiv:1610.01069v1 [quant-ph] 4 Oct 2016 context of quantum steering [58,59], quantum teleportation fidelity [60], and contextual inequalities [61,62].…”

Maximally entangled states and fully entangled fraction

“…[13] some analytical results have been derived for some special states. The monogamy relations in terms of fully entangled fraction have been proven for multiqubit pure states, but it is not true for general mixed states [14].…”