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Laser Physics

Generalized monogamy linear entropy relations for multi-qubit pure states

Xin-Wei Zha, Irfan Ahmed, Da Zhang, Yanpeng Zhang

Published Feb 14, 2020• 3 citations
Publisher

Abstract

Abstract Monogamy of entanglement is an interesting property discovered recently in the context of multi-qubit entanglement. Here, we present a new kind of monogamous relations for pure quantum states of N qubits. By comparing the entanglement monogamy equality proposed by Coffman, Kundu, and Wootters, we prove that there exists a strict triangle relation for the three-qubit system. Since the entropy defines the entanglement of a pure state of a pair of quantum systems; therefore, using linear entropy, we conjecture that there is a polygon relation of many-qubit systems. We also formalize the general triangle relation for the three-qubit system.

DOI
10.1088/1555-6611/ab70b6

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Abstract Monogamy of entanglement is an interesting property discovered recently in the context of multi-qubit entanglement. Here, we present a new kind of monogamous relations for pure quantum states of N qubits. By comparing the entanglement monogamy equality proposed by Coffman, Kundu, and Wootters, we prove that there exists a strict triangle relation for the three-qubit system. Since the entropy defines the entanglement of a pure state of a pair of quantum systems; therefore, using linear entropy, we conjecture that there is a polygon relation of many-qubit systems. We also formalize the general triangle relation for the three-qubit system.
Abstract In this paper, we determine the complementarity relations for pure quantum states of N qubits by presenting the definition of local and non-local forms. By comparing the entanglement monogamy equality proposed by Coffman, Kundu, and Wootters, we prove that there exist strict monogamy laws for quantum correlations in all many-qubit systems. Further, the proper form of general entanglement monogamy equality for arbitrary quantum states is found with the characterization of total quantum correlation of qubits. These results may open a new window for multi-qubit entanglement.
We propose a method to detect genuine quantum correlation for multi-qubit pure states. We then derive a complementarity relations for pure quantum states of N qubits. We prove that in all many-qubit systems there exist strict monogamy laws for quantum correlations. On the other hand, it is known that the entanglement monogamy equality proposed by Coffman, Kundu, and Wootters is in general not true for multiqubit states. Inducing from the CKW equality, we find a proper form of entanglement monogamy equality for arbitrary quantum states. The total quantum correlation of qubit k with the remaining qubits Rk can be characterizes. Furthermore, the quantum correlation of qubit mn with the remaining qubits Rmn can also be obtained. Furthermore, some monogamy relations have been obtained.
We discuss the possibility to interpret the residual entanglement for more than three qubits in terms of distributed multipartite entanglement or, in other words, possible extensions of the Coffman-Kundu-Wootters monogamy equality to higher qubit numbers. Existing knowledge on entanglement in multipartite systems puts narrow constraints on the form of such extensions. We study various examples for families of pure four-qubit states for which the characterization of three-qubit and four-qubit entanglement in terms of polynomial invariants is known. These examples indicate that, although families with such extensions do exist, a generalized monogamy equality cannot be found along those lines.
The monogamy of entanglement is generally discussed using a bipartite entanglement measure as an upper bound. Here we discuss a new kind of monogamous relation where the upper bound is given by a multipartite measure of entanglement, the generalized concurrence. We show a monogamous equality involving the multipartite concurrence, all the bipartite concurrences, and the genuine tripartite entanglement for pure three-qubit systems. The result extends to mixed states in an inequality involving the generalized concurrence and all the bipartite concurrences. We provide a counterexample showing that the result cannot be extended for systems with more than three qubits.
For any three-qubit quantum systems ABC, Oliveira et al. numerically found that both the concurrence and the entanglement of formation (EoF) obey the linear monogamy relations in pure states. They also conjectured that the linear monogamy relations can be saturated when the focus qubit A is maximally entangled with the joint qubits BC. In this work, we prove analytically that both the concurrence and EoF obey linear monogamy relations in an arbitrary three-qubit state. Furthermore, we verify that all three-qubit pure states are maximally entangled in the bipartition A|BC when they saturate the linear monogamy relations. We also study the distribution of the concurrence and EoF. More specifically, when the amount of entanglement between A and B equals to that of A and C, we show that the sum of EoF itself saturates the linear monogamy relation, while the sum of the squared EoF is minimum. Different from EoF, the concurrence and the squared concurrence both saturate the linear monogamy relations when the entanglement between A and B equals to that of A and C.