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Europhysics Letters (EPL)

Monogamy of finite-dimensional entanglement induced by coherence

Chang‐shui Yu, Dong-mo Li, Nan-nan Zhou

Published Apr 9, 2019• 5 citations
Publisher

Abstract

One fundamental feature of quantum entanglement is the monogamy which shows that the entanglement of two systems limits the ability of either system to entangle with a third one. Such an understanding is only well described in the qubit systems, but it remains an open question for high-dimensional quantum systems. Here the relative entropy of entanglement and the negativity are respectively used to quantify the entanglement. Based on the resource theory of coherence, we found that the monogamy inequalities in arbitrarily finite-dimensional systems can be successfully established for the entanglement induced by quantum coherence. Moreover, the similar inequalities are also constructed for the distribution of quantum coherence.

DOI
10.1209/0295-5075/125/50001

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One fundamental feature of quantum entanglement is the monogamy which shows that the entanglement of two systems limits the ability of either system to entangle with a third one. Such an understanding is only well described in the qubit systems, but it remains an open question for high-dimensional quantum systems. Here the relative entropy of entanglement and the negativity are respectively used to quantify the entanglement. Based on the resource theory of coherence, we found that the monogamy inequalities in arbitrarily finite-dimensional systems can be successfully established for the entanglement induced by quantum coherence. Moreover, the similar inequalities are also constructed for the distribution of quantum coherence.
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Characterizing entanglement, including quantifying and distribution of entanglement, which lies at heart of the quantum resource theory, have been investigated extensively ever since Bennett \etal proposed three seminal measures of entanglement in 1996. Up to now, there are numerous measures of entanglement that have been proposed from different point of view and plenty of monogamy relations have been explored which make the distribution of entanglement became more and more clear. While this is relatively easy in the case of pure states, it is much more intricate for the case of mixed quantum states especially with higher dimension and more particles in the system. We present here an overview of the theory along this line. We outline most of the results in this field historically and focus on the finite-dimensional systems. In particular we emphasize the point of view that (i) which yardsticks haven been applied in quantifying entanglement and its distribution, (ii) what are the substantive characteristics and interrelations of these measures and their monogamy relations mathematically by comparing, and (iii) which concepts should be improved or revised and how they were developed accordingly.
We investigate the monogamy nature of entanglement in a three-qubit system. A monogamy inequality is presented to describe the exclusive relation between the $A\text{\ensuremath{-}}B$ two-qubit concurrence CAB{C}_{AB}CAB​ and the $AB\text{\ensuremath{-}}C$ three-qubit concurrence C(AB)C{C}_{(AB)C}C(AB)C​, which represents the entanglement between qubits A\mathit{A}A and B\mathit{B}B as a whole and the third qubit C\mathit{C}C. It is found that the entanglement between any two qubits in a three-qubit system is limited by the entanglement between these two qubits and another qubit. As a consequence, we present the upper bounds for the concurrence CAB{C}_{AB}CAB​, when the concurrence between qubits A\mathit{A}A and C\mathit{C}C (CAC)({C}_{AC})(CAC​) and the concurrence between qubits B\mathit{B}B and C\mathit{C}C (CBC)({C}_{BC})(CBC​) are both given or one of the two is provided.
We investigate monogamy relations related to quantum entanglement for n−n-n−qubit quantum systems. General monogamy inequalities are presented to the β\betaβth (β∈(0,2))(\beta\in(0,2))(β∈(0,2)) power of concurrence, negativity and the convex-roof extended negativity, as well as the β\betaβth (β∈(0,2))(\beta\in(0,\sqrt{2}))(β∈(0,2​)) power of entanglement of formation. These monogamy relations are complementary to the existing ones with different regions of parameter β\betaβ. In additions, new monogamy relations are also derived which include the existing ones as special cases.
We establish duality for monogamy of entanglement: whereas monogamy of entanglement inequalities provide an upper bound for bipartite sharability of entanglement in a multipartite system, as quantified by linear entropy, we prove that the same quantity (namely, linear entropy) provides a lower bound for distribution of bipartite entanglement in a multipartite system. Our theorem for monogamy of entanglement is used to establish relations between bipartite entanglement that separate one qubit from the rest versus separating two qubits from the rest.
The entanglement in multipartite quantum system is hard to characterize and quantify, although it has been intensively studied in bipartite systems. The monogamy of entanglement, as a special property of multipartite systems, shows the distribution of entanglement in the system. In this paper, we investigate the monogamy relations for multi-qubit systems. By using two entangled measures, namely the concurrence C and the negativity Nc, we establish tighter monogamy inequalities for their α-th power than those in all the existing ones. We also illustrate the tightness of our results for some classes of quantum states.