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arXiv:1303.6403 · Physical Review Letters

Multipartite Entanglement Witnesses

Jan Sperling, W. Vogel

Published Sep 13, 2013• 163 citations
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Abstract

We derive a set of algebraic equations, the so-called multipartite separability eigenvalue equations. Based on their solutions, we introduce a universal method for the construction of multipartite entanglement witnesses. We witness multipartite entanglement of 10(3) coupled quantum oscillators, by solving our basic equations analytically. This clearly demonstrates the feasibility of our method for studying ultrahigh orders of multipartite entanglement in complex quantum systems.

DOI
10.1103/physrevlett.111.110503

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We derive a set of algebraic equations, the so-called multipartite separability eigenvalue equations. Based on their solutions, we introduce a universal method for the construction of multipartite entanglement witnesses. We witness multipartite entanglement of 10(3) coupled quantum oscillators, by solving our basic equations analytically. This clearly demonstrates the feasibility of our method for studying ultrahigh orders of multipartite entanglement in complex quantum systems.
Entanglement in multipartite systems is a key resource for quantum information and communication protocols, making its verification in complex systems a necessity.Because an exact calculation of arbitrary entanglement probes is impossible, we derive and implement a numerical method to construct multipartite witnesses to uncover entanglement in arbitrary systems.Our technique is based on a substantial generalization of the power iteration-an essential tool for computing eigenvalues-and it is a solver for the separability eigenvalue equations, enabling the general formulation of optimal entanglement witnesses.Beyond our rigorous derivation and direct implementation of this method, we apply our approach to several examples of complexly quantum-correlated states and benchmark its general performance.Consequently, we provide a generally applicable numerical tool for the identification of multipartite entanglement.
This cumulative dissertation introduces new methods of the topic of entanglement testing based on the separability eigenvalue equations. In principle, the separability eigenvalue equations represent a method to construct entanglement tests via witnessing approaches. An approach is presented that enables finding reliable entanglement tests even for experimental states. A newly introduced algorithm, the Separability Power Iteration, enables finding solutions to the separability eigenvalue equations numerically, thereby broadening the range on the introduced approach.
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We study the problem of witnessing entanglement among indistinguishable particles. For this purpose, we derive a set of equations which results in necessary and sufficient conditions for probing multipartite entanglement between arbitrary systems of bosons or fermions. The solution of these equations yields the construction of optimal entanglement witnesses for partial and full entanglement in discrete and continuous variable systems. Our approach unifies the verification of entanglement for distinguishable and indistinguishable particles. We provide general solutions for certain observables to study quantum entanglement in systems with different quantum statistics in noisy environments.
We present a method to detect entanglement partitions of multipartite quantum systems, by exploiting their inherent symmetries. Structures like genuinely multipartite entanglement, mmm-separability and entanglement depth are detected as very special cases. This formulation enables us to characterize all the entanglement partitions of all three- and four- partite states and witnesses with unitary and permutation symmetry. In particular, we find and parametrize a complete set of bound entangled states therein. For larger systems, we provide a large family of analytical witnesses detecting many-body states of arbitrary size where none of the parties is separable from the rest. This method relies on weak Schur sampling with projective measurements, and thus can be implemented in a quantum computer. Beyond physics, our results extend to the mathematical literature: we establish new inequalities between matrix immanants, and characterize the set of such inequalities for matrices of size three and four.