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arXiv:2103.02983 · Journal of High Energy Physics

Tripartite entanglement and quantum correlation

Xingyu Guo, Chen‐Te Ma

Published May 1, 2021• 6 citations
arXivHTMLPDF

Abstract

Abstract We provide an analytical tripartite-study from the generalized R-matrix. It provides the upper bound of the maximum violation of Mermin’s inequality. For a generic 2-qubit pure state, the concurrence or R-matrix characterizes the maximum violation of Bell’s inequality. Therefore, people expect that the maximum violation should be proper to quantify Quantum Entanglement. The R-matrix gives the maximum violation of Bell’s inequality. For a general 3-qubit state, we have five invariant entanglement quantities up to local unitary transformations. We show that the five invariant quantities describe the correlation in the generalized R-matrix. The violation of Mermin’s inequality is not a proper diagnosis due to the non-monotonic behavior. We then classify 3-qubit quantum states. Each classification quantifies Quantum Entanglement by the total concurrence. In the end, we relate the experiment correlators to Quantum Entanglement.

DOI
10.1007/jhep05(2021)185

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Abstract We provide an analytical tripartite-study from the generalized R-matrix. It provides the upper bound of the maximum violation of Mermin’s inequality. For a generic 2-qubit pure state, the concurrence or R-matrix characterizes the maximum violation of Bell’s inequality. Therefore, people expect that the maximum violation should be proper to quantify Quantum Entanglement. The R-matrix gives the maximum violation of Bell’s inequality. For a general 3-qubit state, we have five invariant entanglement quantities up to local unitary transformations. We show that the five invariant quantities describe the correlation in the generalized R-matrix. The violation of Mermin’s inequality is not a proper diagnosis due to the non-monotonic behavior. We then classify 3-qubit quantum states. Each classification quantifies Quantum Entanglement by the total concurrence. In the end, we relate the experiment correlators to Quantum Entanglement.
Violation ≠\neq= Quantum
The only entanglement quantity is concurrence in a 2-qubit pure state. The maximum violation of Bell's inequality is monotonically increasing for this quantity. Therefore, people expect that pure state entanglement is relevant to the quantum violation. To understand the relation between violation and entanglement, we extend the study to three qubits. We consider all possible 3-qubit operators with a symmetric permutation. When only turning on one entanglement measure, the numerical result shows a contradiction to the expectation. The maximum violation does not have the same behavior as in 2-qubit pure states. Therefore, we conclude Violation≠\neq=Quantum. In the end, we propose the generalized RRR-matrix or correlation matrix for the new diagnosis of Quantum Entanglement. We demonstrate the evidence by restoring the monotonically increasing result.
Abstract The unique entanglement measure is concurrence in a two-qubit pure state. The maximum violation of Bell’s inequality is monotonically increasing for this quantity. Therefore, people expect that pure state entanglement is relevant to the non-locality. For justification, we extend the study to three qubits. We consider all possible three-qubit operators with a symmetric permutation. When only considering one entanglement measure, the numerical result contradicts expectation. Therefore, we conclude ‘non-locality ≠ quantum entanglement’. We propose the generalized R -matrix or correlation matrix for the new diagnosis of quantum entanglement. We then demonstrate the evidence by restoring the monotonically increasing result. PACS numbers: 00.00, 20.00, 42.10
We propose an alternative evaluation of quantum entanglement by measuring the maximum violation of the Bell’s inequality without information of the reduced density matrix of a system. This proposal is demonstrated by bridging the maximum violation of the Bell’s inequality and a concurrence of a pure state in an n-qubit system, in which one subsystem only contains one qubit and the state is a linear combination of two product states. We apply this relation to the ground states of four qubits in the Wen-Plaquette model and show that they are maximally entangled. A topological entanglement entropy of the Wen-Plaquette model could be obtained by relating the upper bound of the maximum violation of the Bell’s inequality to the generalized concurrence of a pure state with respect to different bipartitions.
We find a single parameter family of genuinely entangled three-qubit pure states, called the maximally Bell-inequality violating states (MBV), which exhibit maximum Bell-inequality violation by the reduced bipartite system for a fixed amount of genuine tripartite entanglement quantified by the so-called tangle measure. This in turn implies that there holds a complementary relation between the Bell-inequality violation by the reduced bipartite systems and the tangle present in the three-qubit states, not necessarily pure. The MBV states also exhibit maximum Bell-inequality violation by the reduced bipartite systems of the three-qubit pure states with a fixed amount of genuine tripartite correlation quantified by the generalized geometric measure, a genuine entanglement measure of multiparty pure states, and the discord monogamy score, a multipartite quantum correlation measure from information-theoretic paradigm. The aforementioned complementary relation has also been established for three-qubit pure states for the generalized geometric measure and the discord monogamy score, respectively. The complementarity between the Bell-inequality violation by the reduced bipartite systems and the genuine tripartite correlation suggests that the Bell-inequality violation in the reduced two-qubit system comes at the cost of the total tripartite correlation present in the entire system.
It is well known that the maximal violation of the Bell’s inequality for a two-qubit system is related to the entanglement formation in terms of a concurrence. However, a generalization of this relation to an [Formula: see text]-qubit state has not been found. In this paper, we demonstrate some extensions of the relation between the upper bound of the Bell’s violation and a generalized concurrence in several [Formula: see text]-qubit states. In particular, we show the upper bound of the Bell’s violation can be expressed as a function of the generalized concurrence, if a state can be expressed in terms of two variables. We apply the relation to the Wen-Plaquette model and show that the topological entanglement entropy can be extracted from the maximal Bell’s violation.
Quantities invariant under local unitary transformations are of natural interest in the study of entanglement. This paper deduces and studies a particularly simple quantity that is constructed from a combination of two standard permutations of the density matrix, namely, realignment and partial transpose. This bipartite quantity, denoted here as R12{R}_{12}R12​, vanishes on large classes of separable states including classical-quantum correlated states, while being maximum for only maximally entangled states. It is shown to be naturally related to the 3-tangle in three-qubit states via their two-qubit reduced density matrices. Upper and lower bounds on concurrence and negativity of two-qubit density matrices for all ranks are given in terms of R12{R}_{12}R12​. Ansatz states satisfying these bounds are given and verified using various numerical methods. In the rank-2 case, it is shown that the states satisfying the lower bound on R12{R}_{12}R12​ versus concurrence define a class of three-qubit states that maximize the tripartite entanglement (the 3-tangle) given an amount of entanglement between a pair of them. The measure R12{R}_{12}R12​ is conjectured, via numerical sampling, to be always larger than the concurrence and negativity. In particular, this is shown to be true for the physically interesting case of XXX states.