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arXiv:2208.13901 · Physical review. D/Physical review. D.

Entanglement classification from a topological perspective

Dmitry Melnikov

Published Jun 12, 2023• 9 citations
arXivHTMLPDF

Abstract

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 $W$ 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.

DOI
10.1103/physrevd.107.126005

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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.
Abstract Topological quantum field theories (TQFT) encode properties of quantum states in the topological features of abstract manifolds. One can use the topological avatars of quantum states to develop intuition about different concepts and phenomena of quantum mechanics. In this paper we focus on the class of simplest topologies provided by a specific TQFT and investigate what the corresponding states teach us about entanglement. These “planar connectome” states are defined by graphs of simplest topology for a given adjacency matrix. In the case of bipartite systems the connectomes classify different types of entanglement matching the classification of stochastic local operations and classical communication (SLOCC). The topological realization makes explicit the nature of entanglement as a resource and makes apparent a number of its properties, including monogamy and characteristic inequalities for the entanglement entropy. It also provides tools and hints to engineer new measures of entanglement and other applications. Here the approach is used to construct purely topological versions of the dense coding and quantum teleportation protocols, giving diagrammatic interpretation of the role of entanglement in quantum computation and communication. Finally, the topological concepts of entanglement and quantum teleportation are employed in a simple model of information retrieval from a causally disconnected region, similar to the interior of an evaporating black hole.
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.
With the rapid development of quantum technology, understanding the properties of entanglement states has become an urgent task.Various classifications of entanglement states have been proposed to cater to distinct quantum applications, highlighting the significance of entanglement classification. This paper presents a variational quantum classifier for entanglement classification discrimination which can effectively reduce the measurement resources, specifically for four-qubit entanglement equivalence classifications under SLOCC (Stochastic Local Operations and Classical Communication). Through training one-to-one and one-to-many models, our proposed method achieves excellent performance in the entanglement classification. We further investigate the impact of parameterized quantum circuit layers and various measurement settings on the classifier’s performance. Considering the inherent noise present in near-term quantum computers, we also analyze how noise affects the model’s performance. Moreover, we explore the classifier’s capabilities in the context of multi-qubit entanglement systems. Numerical simulations demonstrate that high-precision entanglement classification tasks can be accomplished with shallow circuit depths, and the model exhibits resilience to noise. Additionally, the results reveal the versatility and potential application of our proposed model in larger-scale quantum systems by effectively handling entanglement classification tasks involving an increased number of qubits.
Chapter 14 transitions the focus to multipartite entanglement, a realm that broadens the discussion from bipartite systems to those involving multiple parties. This complex form of entanglement plays a crucial role in quantum computing, cryptography, and communication networks. The chapter introduces the foundational concepts of multipartite entanglement, including its characterization and the challenges associated with its classification. Significant attention is given to the classification of multipartite entangled states through SL-invariant polynomials, which provide tools for understanding the structure and properties of these states. Stochastic Local Operations and Classical Communication (SLOCC) are introduced as a means to classify entanglement. Furthermore, the chapter explores the entanglement of assistance and the monogamy of entanglement, two concepts that illustrate the limitations and potential for distributing entanglement among multiple parties. Through detailed explanations and examples in three and four qubits, this chapter offers insights into the intricate world of multipartite entanglement, revealing both its potential and challenges.
Kauffman and Lomonaco (New J Phys 4:73.1–73.18, 2002. arXiv:quant-ph/0401090 , New J Phys 6:134.1–134.40, 2004) explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In Alagic et al. (Yang–Baxter operators need quantum entanglement to distinguish knots, 2015. arXiv:1507.05979v1 ), it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang–Baxter equation. We show that the arguments used by Alagic et al. (2015) generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang–Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of Alagic et al. (2015). We also give a class of R matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give an example of an SU(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three-strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing. The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement and the ER=EPRER = EPRER=EPR E R = E P R hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.
Abstract Classification of multipartite pure states via the local unitary (LU) equivalence or the stochastic local operations and classical communication (SLOCC) equivalence is a fundamental task in investigating multipartite entanglement. It is shown in the seminal paper (2000 Phys . Rev . A 62 062314) that pure states of three qubits can be partitioned into six SLOCC classes. We explore here a necessary and sufficient condition for LU equivalence of pure states for any SLOCC class via Acín et al ’s Schmidt coefficients (ASC) of three qubits in (2000 Phys. Rev. Lett. 85 1560). We show that the LU equivalent states in W SLOCC class have the same ASC irrespective of phases while there are generally two different ASCs for the LU equivalent states in GHZ SLOCC class (irrespective of phases for some subclass). Consequently, we obtain a complete LU classification of three qubit pure states in the sense of ASC. Moreover, we discuss the choice among the different ASCs and then propose an approach to find a unique ASC for any given three qubit pure state. At last we analyse the entanglement/separability of the marginal states of the two inequivalent genuine entangled SLOCC classes. We find that any bipartite marginal states of the W SLOCC class are entangled, and that of the GHZ SLOCC class has eight different cases which are depend on the ASCs.