Abstract
We investigate possible generalizations of the Coffman-Kundu-Wootters monogamy inequality to four qubits, accounting for multipartite entanglement in addition to the bipartite terms. We show that the most natural extension of the inequality does not hold in general, and we describe the violations of this inequality in detail. We investigate alternative ways to extend the monogamy inequality to express a constraint on entanglement sharing valid for all four-qubit states, and perform an extensive numerical analysis of randomly generated four-qubit states to explore the properties of such extensions.
Highlights
The conceptual and computational difficulties of quantifying the entanglement of larger systems make the complete description of multipartite entanglement one of the biggest challenges of quantum information theory [1,2]
We investigate alternative ways to extend the monogamy inequality to express a constraint on entanglement sharing valid for all four-qubit states, and perform an extensive numerical analysis of randomly generated four-qubit states to explore the properties of such extensions
We begin by noting that while four-qubit states can be divided into infinitely many stochastic local operations and classical communication (SLOCC)-inequivalent classes [8,31], there is, a convenient way to classify them into nine classes representing nine different ways in which they can be entangled [31,32], where the classes can be subdivided further into a three-dimensional characteristic entanglement vector [10]
Summary
The conceptual and computational difficulties of quantifying the entanglement of larger systems make the complete description of multipartite entanglement one of the biggest challenges of quantum information theory [1,2]. The concept was first formalized for a system of three qubits in the seminal work of Coffman, Kundu, and Wootters (CKW) [5], showing that the entanglement of a qubit with another pair of qubits (quantified with the tangle τ ) places a bound on the total amount of pairwise entanglement between the considered qubit and each qubit from the pair This relation, referred to as the CKW monogamy inequality, can be expressed for a pure state |φ of three qubits as τ1(|12) (|φ ) τ1(|22) (|φ ) + τ1(|23) (|φ ),. It admits a straightforward closed-form expression as a polynomial of homogeneous degree 4 in the coefficients of the state |φ [5,9,10] Even though it is tailored for particular choices of the entanglement measure, the CKW monogamy inequality is often regarded as a quantitative constraint capturing a characteristic property of entanglement itself, which distinguishes it from other, weaker forms of nonclassical correlations [11,12]. We present other possible methods of extending the inequality and consider their implications on the validity of a possible general strong monogamy relation by performing an extensive numerical investigation of arbitrary fourqubit states and analyzing some example states in more detail
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