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arXiv:1808.03382

Computing Entanglement Polytopes

Konstantin Wernli

Published Aug 9, 2018• 0 citations
arXivHTMLPDF

Abstract

In arXiv:1208.0365 entanglement polytopes where introduced as a coarsening of the SLOCC classification of multipartite entanglement. The advantages of classifying entanglement by entanglement polytopes are a finite hierarchy for all dimensions and a number of parameters linear in system size. In arXiv:1208.0365 a method to compute entanglement polytopes using geometric invariant theory is presented. In this thesis we consider alternative methods to compute them. Some geometrical and algebraical tools are presented that can be used to compute inequalities giving an outer approximation of the entanglement polytopes. Furthermore we present a numerical method which, in theory, can compute the entanglement polytope of any given SLOCC class given a representative. Using it we classify the entanglement polytopes of $2 \times 3 \times N$ systems.

DOI
10.48550/arxiv.1808.03382

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In arXiv:1208.0365 entanglement polytopes where introduced as a coarsening of the SLOCC classification of multipartite entanglement. The advantages of classifying entanglement by entanglement polytopes are a finite hierarchy for all dimensions and a number of parameters linear in system size. In arXiv:1208.0365 a method to compute entanglement polytopes using geometric invariant theory is presented. In this thesis we consider alternative methods to compute them. Some geometrical and algebraical tools are presented that can be used to compute inequalities giving an outer approximation of the entanglement polytopes. Furthermore we present a numerical method which, in theory, can compute the entanglement polytope of any given SLOCC class given a representative. Using it we classify the entanglement polytopes of 2×3×N2 \times 3 \times N2×3×N systems.
Quantum Entanglement is one of the key manifestations of quantum mechanics that separate the quantum realm from the classical one. Characterization of entanglement as a physical resource for quantum technology became of uppermost importance. While the entanglement of bipartite systems is already well understood, the ultimate goal to cope with the properties of entanglement of multipartite systems is still far from being realized. This paper covers characterization of multipartite entanglement using algebraic-geometric tools. First, we establish an algorithm to classify multipartite entanglement by [Formula: see text]-secant varieties of the Segre variety and [Formula: see text]-multilinear ranks that are invariant under Stochastic Local Operations with Classical Communication (SLOCC). We present a fine-structure classification of multiqubit and tripartite entanglement based on this algorithm. Another fundamental problem in quantum information theory is entanglement transformation that is quite challenging regarding to multipartite systems. It is captivating that the proposed entanglement classification by algebraic geometry can be considered as a reference to study SLOCC and asymptotic SLOCC interconversions among different resources based on tensor rank and border rank, respectively. In this regard, we also introduce a new class of tensors that we call persistent tensors and construct a lower bound for their tensor rank. We further cover SLOCC convertibility of multipartite systems considering several families of persistent tensors.
Abstract We approach multipartite entanglement classification in the symmetric subspace in terms of algebraic geometry, its natural language. We show that the class of symmetric separable states has the structure of a Veronese variety and that its k -secant varieties are SLOCC invariants. Thus SLOCC classes gather naturally into families. This classification presents useful properties such as a linear growth of the number of families with the number of particles, and nesting, i.e. upward consistency of the classification. We attach physical meaning to this classification through the required interaction length of parent Hamiltonians. We show that the states W N and GHZ N are in the same secant family and that, effectively, the former can be obtained in a limit from the latter. This limit is understood in terms of tangents, leading to a refinement of the previous families. We compute explicitly the classification of symmetric states with N ⩽ 4 qubits in terms of both secant families and its refinement using tangents. This paves the way to further use of projective varieties in algebraic geometry to solve open problems in entanglement theory.
In quantum information theory, understanding the complexity of entangled states within the context of SLOCC (stochastic local operations and classical communications) involving d qubits (or qudits) is essential for advancing our knowledge of quantum systems. This complexity is often analyzed by classifying the states via local symmetry groups. The resulting classes can be distinguished using invariant polynomials, which serve as a measure of entanglement. This paper introduces a novel method for obtaining invariant polynomials of the smallest degrees, which significantly enhances the efficiency of characterizing SLOCC classes of entangled quantum states. Our method not only simplifies the process of identifying these classes but also provides a robust tool for analyzing the entanglement properties of complex quantum systems. As a practical application, we demonstrate the derivation of minimal degree invariants in specific cases, illustrating the effectiveness of our approach in real-world scenarios. This advancement has the potential to streamline various processes in quantum information theory, making it easier to understand, classify, and utilize entangled states effectively.
Abstract Entanglement polytopes have been recently proposed as a way of witnessing the stochastic local operations and classical communication (SLOCC) multipartite entanglement classes using single particle information. We present first asymptotic results concerning the feasibility of this approach for a large number of qubits. In particular, we show that entanglement polytopes of the L -qubit system accumulate in the distance O ( 1 L ) from the point corresponding to the maximally mixed reduced one-qubit density matrices. This implies existence of a possibly large region where many entanglement polytopes overlap, i.e. where the witnessing power of entanglement polytopes is weak. Moreover, we argue that the witnessing power cannot be strengthened by any entanglement distillation protocol, as for large L the required purity is above current capability.
Classification of entanglement is an important problem in quantum resource theory. In this paper we discuss an embedding of this problem in the context of topological quantum field theories (TQFTs). This approach allows classifying entanglement patterns in terms of topological equivalence classes. In the bipartite case a classification equivalent to the one by stochastic local operations and classical communication (SLOCC) is constructed by restricting to a simple class of connectivity diagrams. Such diagrams characterize quantum states of TQFT up to braiding and tangling of the ``connectome.'' In the multipartite case the same restricted topological classification only captures a part of the SLOCC classes, in particular, it does not see the WWW entanglement of three qubits. Nonlocal braiding of connections may solve the problem, but no finite classification is attempted in this case. Despite incompleteness, the connectome classification has a straightforward generalization to any number and dimension of parties and has a very intuitive interpretation, which might be useful for understanding specific properties of entanglement and for design of new quantum resources.