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arXiv:1610.03916 · Physical Review A

Upper bound on three-tangles of reduced states of four-qubit pure states

S. Shelly Sharma, Naresh Kumar Sharma

Published Jun 6, 2017• 3 citations
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Abstract

Closed formulas for upper bounds on three-tangles of three-qubit reduced states in terms of three-qubit--invariant polynomials of pure four-qubit states are obtained. Our results offer tighter constraints on total three-way entanglement of a given qubit with the rest of the system than those used by Regula et al. [Phys. Rev. Lett. 113, 110501 (2014) and Phys. Rev. Lett. 116, 049902(E) (2016)] to verify monogamy of four-qubit quantum entanglement.

DOI
10.1103/physreva.95.062311

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Closed formulas for upper bounds on three-tangles of three-qubit reduced states in terms of three-qubit--invariant polynomials of pure four-qubit states are obtained. Our results offer tighter constraints on total three-way entanglement of a given qubit with the rest of the system than those used by Regula et al. [Phys. Rev. Lett. 113, 110501 (2014) and Phys. Rev. Lett. 116, 049902(E) (2016)] to verify monogamy of four-qubit quantum entanglement.
We report a set of monogamy constraints on one-tangle, two-tangles, three-tangles and four-way correlations of a general four-qubit pure state. It is found that given a two-qubit marginal ρ\rhoρ of an NNN qubit pure state $\left\vert \Psi_{N}\right\rangle $, the non-Hermitian matrix ρρ~\rho \widetilde{\rho}ρρ​ where ρ~\widetilde{\rho}ρ​ $=\left( \sigma_{y}\otimes \sigma_{y}\right) \rho^{\ast}\left( \sigma_{y}\otimes\sigma_{y}\right) $, contains information not only about the entanglement properties of the two-qubits in that state but also about the correlations of the pair of qubits with $\left( N-2\right) $ qubits in $\left\vert \Psi_{N}\right\rangle $. To extract information about tangles of a four-qubit state $\left\vert \Psi_{4}\right\rangle $, the coefficients in the characteristic polynomial of matrix ρρ~\rho\widetilde{\rho}ρρ​ are analytically expressed in terms of 2×22\times22×2 matrices of state coefficients. A general monogamy relation satisfied by one-tangle, two-tangles, three-tangles, and residual correlations in a multiqubit state $\left\vert \Psi_{N}\right\rangle $ is also given.
Given a multipartite quantum system that consists of two-level particles (qubits), one may or may not have access to all the subsystems. What can we know about the entanglement of the multiqubit system and residual correlations beyond two-tangle if we have access only to two-qubits at a time? Algebraic analysis of two-qubit states yields monogamy constraints on distribution of entanglement between sub-systems of an N-qubit state and criterion to determine if the state has multipartite entanglement. Monogamy constraints, reported in this letter, are relations between well known entanglement measures such as one-tangle, two-tangles and three-tangles of an N-qubit pure state.
I calculate the mixed three-tangle ${\ensuremath{\tau}}_{3}[\ensuremath{\rho}]$ for the reduced density matrices of the four-qubit representant states found in Phys. Rev. A 65, 052112 (2002). In most of the cases, the convex roof is obtained, except for one class, where I provide with an upper bound, which is assumed to be very close to the convex roof. I compare with results published in Phys. Rev. Lett. 113, 110501 (2014). Since the method applied there usually results in higher values for the upper bound, in certain cases it can be understood that the convex roof is obtained exactly, namely, when the zero-polytope where ${\ensuremath{\tau}}_{3}$ vanishes shrinks to a single point.
Monogamy is a defining feature of entanglement, having far-reaching applications. Recently, Regula et al. [Phys. Rev. Lett. 113, 110501 (2014)] proposed a stronger version of monogamy relation for concurrence. We have extended the strong monogamy inequality for another entanglement measure, viz., negativity. In particular, we have concentrated on the four-qubit system and provided a detailed study on the status of strong monogamy on pure states. Further, we have analytically provided some classes of states for which negativity and squared negativity satisfy strong monogamy. Numerical evidences have also been shown in proper places. Our analysis also provides cases where strong monogamy is violated.
Entanglement is an important resource for quantum technologies that allow us to go beyond the classical world. Quantum systems can be entangled in various ways. Also, the degree of entanglement between the subsystems of a quantum system can be calculated in different ways. In our recent paper [Prog. Theor. Exp. Phys. (2022)], we directly measure the three-tangle of a pure three-qubit quantum state. In this paper, we extend this method to pure N-qubit quantum states (N is any odd number greater than 1) by considering polynomial invariant of degree 4 as a measure of entanglement. In this algorithm, we use quantum gates and output probabilities to calculate polynomial invariant of degree 4. The importance of this method lies in the possibility of experimental implementation of quantum calculations in the language of quantum gates.
We investigate the lower bound obtained from experimental data of a quantum state $\ensuremath{\rho}$, as proposed independently by O. G\"uhne et al. [Phys. Rev. Lett. 98, 110502 (2007)] and J. Eisert et al. [New J. Phys. 9, 46 (2007)], and apply it to mixed states of three qubits. The measure we consider is the convex-roof extended three-tangle. Our findings highlight an intimate relation to lower bounds obtained recently from so-called characteristic curves of a given entanglement measure. We apply the bounds to estimate the three-tangle present in recently performed experiments aimed at producing a three-qubit Greenberger-Horne-Zeilinger (GHZ) state. A nonvanishing lower bound is obtained if the GHZ fidelity of the produced states is larger than 3/43/43/4.