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arXiv:1012.0758 · Journal of Physics A Mathematical and Theoretical

Entanglement for multipartite systems of indistinguishable particles

Janusz Grabowski, Marek Kuś, Giuseppe Marmo

Published Mar 31, 2011• 37 citations
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Abstract

We analyze the concept of entanglement for a multipartite system with bosonic and fermionic constituents and its generalization to systems with arbitrary parastatistics. We use the representation theory of symmetry groups to formulate a unified approach to this problem in terms of simple tensors with an appropriate symmetry. For an arbitrary parastatistics, we define the S-rank generalizing the notion of the Schmidt rank. The S-rank, defined for all types of tensors, serves for distinguishing entanglement of pure states. In addition, for Bose and Fermi statistics, we construct an analog of the Jamiołkowski isomorphism.

DOI
10.1088/1751-8113/44/17/175302

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We analyze the concept of entanglement for a multipartite system with bosonic and fermionic constituents and its generalization to systems with arbitrary parastatistics. We use the representation theory of symmetry groups to formulate a unified approach to this problem in terms of simple tensors with an appropriate symmetry. For an arbitrary parastatistics, we define the S-rank generalizing the notion of the Schmidt rank. The S-rank, defined for all types of tensors, serves for distinguishing entanglement of pure states. In addition, for Bose and Fermi statistics, we construct an analog of the Jamiołkowski isomorphism.
We analyze a general bipartite-like representation of arbitrary pure states of N indistinguishable partcles, valid for both bosons and fermions, based on MMM- and $(N\ensuremath{-}M)$-particle states. It leads to exact $(M,N\ensuremath{-}M)$ Schmidt-like expansions of the state for any M<NM<NM<N and is directly related to the isospectral reduced MMM- and $(N\ensuremath{-}M)$-body density matrices ${\ensuremath{\rho}}^{(M)}$ and ${\ensuremath{\rho}}^{(N\ensuremath{-}M)}$. The formalism also allows for reduced yet still exact Schmidt-like decompositions associated with blocks of these densities, in systems having a fixed fraction of the particles in some single-particle subspace. Monotonicity of the ensuing MMM-body entanglement under a certain set of quantum operations is also discussed. Illustrative examples in fermionic and bosonic systems with pairing correlations are provided, which show that in the presence of dominant eigenvalues in ${\ensuremath{\rho}}^{(M)}$, approximations based on a few terms of the pertinent Schmidt expansion can provide a reliable description of the state. The associated one- and two-body entanglement spectrum and entropies are also analyzed.
The usefulness of entanglement as a resource in quantum technologies increases for larger systems, that is, if more particles or higher-dimensional quantum systems are considered. Yet, the interplay between dimensionality and multiparticle entanglement is not well understood. Only for two-particle systems an unambiguous and coherent notion of entanglement dimensionality, based on the Schmidt decomposition, is known. We introduce a concept to characterize the entanglement dimensionality of multiparticle states based on decompositions of pure states into superpositions of states without genuine multiparticle entanglement. We provide constructive methods to characterize the resulting partition rank for pure and mixed states. This allows the identification of novel maximally correlated states as well as a discrete classification of quantum states under stochastic local operations and classical communication. From a mathematical perspective, our approach can be formulated in terms of the slice rank and partition rank of tensors and our results allow to characterize these by connecting them to a generalized injective tensor norm.
We relate the notion of entanglement for quantum systems composed of two identical constituents to the impossibility of attributing a complete set of properties to both particles. This implies definite constraints on the mathematical form of the state vector associated with the whole system. We then analyze separately the cases of fermion and boson systems, and we show how the consideration of both the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition of the global state vector and the von Neumann entropy of the one-particle reduced density operators can supply us with a consistent criterion for detecting entanglement. In particular, the consideration of the von Neumann entropy is particularly useful in deciding whether the correlations of the considered states are simply due to the indistinguishability of the particles involved or are a genuine manifestation of the entanglement. The treatment leads to a full clarification of the subtle aspects of entanglement of two identical constituents which have been a source of embarrassment and of serious misunderstandings in the recent literature.
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 introduce geometric measures of entanglement for indistinguishable particles, which apply to mixed states, multipartite systems, and arbitrary dimensions. They are based on generalized (i.e., not necessarily finite) norms on the set of quantum states and lead to the first necessary and sufficient computational separability criterion in this general setting. The coherent approach developed in the paper allows us to compare, in particular, entanglement for fermionic and distinguishable particles: The entanglement measure for fermionic particles coincides with the corresponding entanglement measure for distinguishable particles up to a factor of k!k!k! where kkk is the number of particles involved. By this result the amount of entanglement emerging from fermi statistics alone is clearly separated from the overall amount of entanglement. Finally, our techniques are applied to entanglement related to Schmidt and Slater numbers.