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Topological classification of multipartite entangled states by the hyperdeterminant

Akimasa Miyake

Published Jun 18, 2002• 1 citation
OpenAlex

Abstract

We find that the hyperdeterminant $Det A$, related to an entanglement measure (the concurrence, 3-tangle for the 2,3-qubit respectively), is derived from a duality between the entangled states and separable states. In terms of $Det A$ and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similarly to the local rank in the bipartite case. This reveals that many inequivalent multipartite entangled states are partially ordered such that entanglement measures like $Det A$ as well as local ranks are needed to distinguish them. In particular, the nonzero $Det A$ distinguishes generic entangled states of the maximal dimension (the outermost class of the onion structure). It suggests that the majority of multipartite entangled states never locally converts to the maximally entangled states in Bell's inequalities even probabilistically in general (e.g., in the $n \\geq 4$-qubit), contrary to the widely known bipartite or 3-qubit cases. The classification is also useful for that of mixed states.

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We find that the hyperdeterminant DetADet ADetA, related to an entanglement measure (the concurrence, 3-tangle for the 2,3-qubit respectively), is derived from a duality between the entangled states and separable states. In terms of DetADet ADetA and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similarly to the local rank in the bipartite case. This reveals that many inequivalent multipartite entangled states are partially ordered such that entanglement measures like DetADet ADetA as well as local ranks are needed to distinguish them. In particular, the nonzero DetADet ADetA distinguishes generic entangled states of the maximal dimension (the outermost class of the onion structure). It suggests that the majority of multipartite entangled states never locally converts to the maximally entangled states in Bell's inequalities even probabilistically in general (e.g., in the ngeq4n \\geq 4ngeq4-qubit), contrary to the widely known bipartite or 3-qubit cases. The classification is also useful for that of mixed states.
We find that multidimensional determinants ``hyperdeterminants,'' related to entanglement measures (the so-called concurrence, or 3-tangle for two or three qubits, respectively), are derived from a duality between entangled states and separable states. By means of the hyperdeterminant and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similar to that by the local rank in the bipartite case. This reveals how inequivalent multipartite entangled classes are partially ordered under local actions. In particular, the generic entangled class of the maximal dimension, distinguished as the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the n> 4n>~4n> 4 qubits), contrary to the widely known bipartite or three-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical n-partite entangled states. Our classification is also useful for the mixed states.
We classify multipartite entanglement in a unified manner, focusing on a duality between the set of separable states and that of entangled states. Hyperdeterminants, derived from the duality, are natural generalizations of entanglement measures, the concurrence, 3-tangle for 2, 3 qubits respectively. Our approach reveals how inequivalent multipartite entangled classes of pure states constitute a partially ordered structure under local actions, significantly different from a totally ordered one in the bipartite case. Moreover, the generic entangled class of the maximal dimension, given by the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the n ⁣≥ ⁣4n \!\geq\! 4n≥4 qubits), contrary to the widely known bipartite or 3-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical nnn-partite entangled states. Our classification is also useful for that of mixed states.
Abstract Previously, it was shown that both the concurrence and n -tangle on 2 n -qubit pure quantum states can be expressed in terms of Cayley’s first hyperdeterminant Dür et al (2000 Phys. Rev. A 62 062314), indicating that Cayley’s first hyperdeterminant, denoted hdet , captures some aspects of a state’s 2 n -way entanglement. In this paper, we rigorously prove that on both pure and mixed states, | hdet | 2 / d is identically zero on separable states, is an local unitary invariant, and is non-increasing on average under local operations and classical communications, thus demonstrating that | hdet | d / 2 is a physically meaningful and legitimate entanglement measure. Moreover, we discuss a few key examples to illustrate the particular type of entanglement Cayley’s first hyperdeterminant is detecting: genuine full d -level GHZ-type entanglement across all 2 n parties. Combined, this establishes Cayley’s first hyperdeterminant (or | hdet | 2 / d to be precise), as a physically significant generalization of the G -concurrence and the n -tangle to 2 n -qudit states.
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.
Entanglement is a resource in quantum information theory when state manipulation is restricted to local operations assisted by classical communication (LOCC). It is therefore of paramount importance to decide which LOCC transformations are possible and, particularly, which states are maximally useful under this restriction. While the bipartite maximally entangled state is well known (it is the only state that cannot be obtained from any other and, at the same time, it can be transformed to any other by LOCC), no such state exists in the multipartite case. In order to cope with this fact, we introduce here the notion of the maximally entangled set (MES) of n-partite states. This is the set of states which are maximally useful under LOCC manipulation; i.e., any state outside of this set can be obtained via LOCC from one of the states within the set and no state in the set can be obtained from any other state via LOCC. We determine the MES for states of three and four qubits and provide a simple characterization for them. In both cases, infinitely many states are required. However, while the MES is of measure zero for 3-qubit states, almost all 4-qubit states are in the MES. This is because, in contrast to the 3-qubit case, deterministic LOCC transformations are almost never possible among fully entangled four-partite states. We determine the measure-zero subset of the MES of LOCC convertible states. This is the only relevant class of states for entanglement manipulation.