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arXiv:2106.00850 · Physical Review A

Entanglement classification via a single entanglement measure

Adam Burchardt, Gonçalo M. Quinta, Rui André

Published Mar 22, 2024• 3 citations
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Abstract

We show that a single polynomial entanglement measure is enough to verify equivalence between generic $n$-qubit states under stochastic local operations with classical communication (SLOCC). SLOCC operations may be represented geometrically by M\"obius transformations on the roots of the entanglement measure on the Bloch sphere. Moreover, we show how the roots of the three-tangle measure classify four-qubit generic states, and propose a method to obtain the normal form of a four-qubit state, which bypasses the possibly infinite iterative procedure.

DOI
10.1103/physreva.109.032424

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We show that a single polynomial entanglement measure is enough to verify equivalence between generic nnn-qubit states under stochastic local operations with classical communication (SLOCC). SLOCC operations may be represented geometrically by M"obius transformations on the roots of the entanglement measure on the Bloch sphere. Moreover, we show how the roots of the three-tangle measure classify four-qubit generic states, and propose a method to obtain the normal form of a four-qubit state, which bypasses the possibly infinite iterative procedure.
We show that the invariance of an entanglement measure under Stochastic Local Operations with Classical Communication (SLOCC) expands to its convex roof extension. SLOCC may be represented by Mobius transformations on the roots of the entanglement measure on the Bloch sphere. We derive efficient conditions for n-qubit states to be SLOCC-equivalent. We apply our approach on 4-qubit states and show that roots of the 3-tangle measure classify 4-qubit generic states. Moreover, we propose a method to obtain the normal form of a 4-qubit state which bypasses the possibly infinite iterative procedure.
We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those invariants with entanglement properties to be ground in future analytical work.
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.
We study entanglement and nonlocality of connected four-qubit hypergraph states. One obtains the Stochastic Local Operations and Classical Communication (SLOCC) classification from the known local unitary (LU)-orbits. We then consider Mermin’s polynomials and show that the nonlocal behavior of hypergraph states can be detected by a suitable choice of Mermin’s inequality. Finally, we implement some of the corresponding inequalities on the IBM Quantum Experience and report the detection of the nonlocality of a three-qubit hypergraph states.
Multipartite pure states are equivalent under stochastic local operations and classical communication (SLOCC) whenever they can be mapped into one another by invertible local operations. It is shown that this is equivalent to the interconvertibility through finite sequences of elementary local operations. A multipartite version of the Gauss-Jordan elimination strategy is then obtained, enabling the analytical SLOCC classification of previously unknown examples. It is argued that the problem of SLOCC classification is equivalent to the problem of classifying the multipartite fully reduced forms that the coefficient matrix of a state can assume after being subjected to this Gauss-Jordan elimination procedure. The method is applied to examples of so-called locally maximally entangleable and hypergraph states, showing different SLOCC equivalences. Moreover, possible physical implications for states with a Dicke-like structure are sketched.