General Monogamy Inequality for Bipartite Qubit Entanglement

Abstract

We consider multipartite states of qubits and prove that their bipartite quantum entanglement, as quantified by the concurrence, satisfies a monogamy inequality conjectured by Coffman, Kundu, and Wootters. We relate this monogamy inequality to the concept of frustration of correlations in quantum spin systems. DOI: 10.1103/PhysRevLett.96.220503 PACS numbers: 03.65.UdQuantum mechanics, unlike classical mechanics, allows the existence of pure states of composite systems for which it is not possible to assign a definite state to two or more subsystems. States with this property are known as entangled states. Entangled states have a number of remarkable features, a fact which has inspired an enormous literature in the years since their discovery. These properties have led to suggestions that the propensity of multipartite quantum systems to enter nonlocal superposition states might be the defining characteristic of quantum mechanics [1,2].It is becoming clear that entanglement is a physical resource. The exploration of this idea is a central goal in the burgeoning field of quantum information theory. As a consequence, the study of the mathematics underlying entanglement has been a very active area and has led to many operational and information-theoretic insights. As for now, only the pure-state case of entanglement shared between two parties is thoroughly understood and quantified; progress on the multipartite setting has been much slower.A key property, which may be as fundamental as the nocloning theorem, has been discovered recently in the context of multipartite entanglement: entanglement is monogamous [3,4]. More precisely, there is an inevitable trade-off between the amount of quantum entanglement that two qubits A and B, in Alice's and Bob's possession, respectively, can share and the quantum correlation that Alice's same qubit A can share with Charlie, a third party, C [3]. In the context of quantum cryptography, such a monogamy property is of fundamental importance because it quantifies how much information an eavesdropper could potentially obtain about the secret key to be extracted. The constraints on shareability of entanglement lie also at the heart of the success of many information-theoretic protocols, such as entanglement distillation.In the context of condensed-matter physics, the monogamy property gives rise to the frustration effects observed in, e.g., Heisenberg antiferromagnets. Indeed, the perfect ground state for an antiferromagnet would consist of singlets between all interacting spins. But, as a particle can only share one unit of entanglement with all its neighbors (this immediately follows from the dimension of its local Hilbert space), it will try to spread its entanglement in an optimal way with all its neighbors leading to a strongly correlated ground state. The tools developed in this Letter will allow us to turn such qualitative statements into quantitative ones.The problem of fully quantifying the constraints on distributed entanglement should be seen as analogous to the N represe...

Geometric characterization of separability and entanglement in pure Gaussian states by single-mode unitary operations

“…However, we would like to remark that, among the three possible CV versions of the tangle, only τ G , Eq. (15), satisfies the CV generalization of the Coffman-Kundu-Wootters monogamy inequality [10,19], as proved in Ref. [21] for all, pure and mixed, N -mode Gaussian states.…”

“…The fact that the linear entropy of the reduced state does not coincide exactly with the minimum distance achieved under local symplectic operations may be traced back to the non uniqueness in the definition of the "tangle" for Gaussian states of CV systems. For qubits, at least four different definitions coalesce into the same entanglement monotone: (i) squared concurrence [10]; (ii) local linear entropy [19]; (iii) squared negativity (negativity equals concurrence for pure qubit states [20]); (iv) minimum distance under single-qubit unitary transformation [8]. On the other hand, while the concurrence is not well defined in CV systems, the other definitions of the tangle all give rise to different (yet equivalent) entanglement quantifiers in these systems.…”

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“…(See the following section for the definition.) One can show that the various concurrences with qubit A are restricted by the inequality [4,5] C 2 AB 1 + · · · + C 2 ABn ≤ 1.…”

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“…The monogamy relation in Eq. (8) is further generalized to the N -qubit quantum state, which has the form C 2 A1|A2···AN ≥ C 2 A1A2 + · · · + C 2 A1AN [17]. Moreover, for the two-qubit partition…”