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International Journal of Quantum Information

MIXED-STATE ENTANGLEMENT IN THE LIGHT OF PURE-STATE ENTANGLEMENT CONSTRAINED BY SUPERSELECTION RULES

Stephen D. Bartlett, Howard M. Wiseman, Robert W. Spekkens, Andrew C. Doherty

Published Nov 1, 2005• 0 citations
Publisher

Abstract

We show that the classification of bi-partite pure entangled states when local quantum operations are restricted, e.g., constrained by local superselection rules, yields a structure that is analogous in many respects to that of mixed-state entanglement, including such exotic phenomena as bound entanglement and activation. This analogy aids in resolving several conceptual puzzles in the study of entanglement under restricted operations. Specifically, we demonstrate that several types of quantum optical states that possess confusing entanglement properties are analogous to bound entangled states. Also, the classification of pure-state entanglement under restricted operations can be much simpler than for mixed state entanglement. For instance, in the case of local Abelian superselection rules all questions concerning distillability can be resolved.

DOI
10.1142/s021974990500133x

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We show that the classification of bi-partite pure entangled states when local quantum operations are restricted, e.g., constrained by local superselection rules, yields a structure that is analogous in many respects to that of mixed-state entanglement, including such exotic phenomena as bound entanglement and activation. This analogy aids in resolving several conceptual puzzles in the study of entanglement under restricted operations. Specifically, we demonstrate that several types of quantum optical states that possess confusing entanglement properties are analogous to bound entangled states. Also, the classification of pure-state entanglement under restricted operations can be much simpler than for mixed state entanglement. For instance, in the case of local Abelian superselection rules all questions concerning distillability can be resolved.
We show that the classification of bipartite pure entangled states when local quantum operations are restricted yields a structure that is analogous in many respects to that of mixed-state entanglement. Specifically, we develop this analogy by restricting operations through local superselection rules, and show that such exotic phenomena as bound entanglement and activation arise using pure states in this setting. This analogy aids in resolving several conceptual puzzles in the study of entanglement under restricted operations. In particular, we demonstrate that several types of quantum optical states that possess confusing entanglement properties are analogous to bound entangled states. Also, the classification of pure-state entanglement under restricted operations can be much simpler than for mixed-state entanglement. For instance, in the case of local Abelian superselection rules all questions concerning distillability can be resolved.
Bipartite entanglement may be reduced if there are restrictions on allowed local operations. We introduce the concept of a generalized superselection rule to describe such restrictions, and quantify the entanglement constrained by it. We show that ensemble quantum information processing, where elements in the ensemble are not individually addressable, is subject to the superselection rule associated with the symmetric group (the group of permutations of elements). We prove that even for an ensemble comprising many pairs of qubits, each pair described by a pure Bell state, the entanglement per element constrained by this superselection rule goes to zero for a large number of elements.
Superselection rules severely constrain the operations which can be implemented on a distributed quantum system. While the restriction to local operations and classical communication gives rise to entanglement as a nonlocal resource, particle number conservation additionally confines the possible operations and should give rise to a new resource. In Phys. Rev. Lett. 92, 087904 (2004) we showed that this resource can be quantified by a single additional number, the superselection-induced variance (SIV)(\mathrm{SIV})(SIV), without changing the concept of entanglement. In this paper, we give the results on pure states in greater detail; additionally, we provide a discussion of mixed-state nonlocality with superselection rules where we consider both formation and distillation. Finally, we demonstrate that SIV\mathrm{SIV}SIV is indeed a resource, i.e., that it captures how well a state can be used to overcome the restrictions imposed by the superselection rule.
In recent years it has become apparent that constraints on possible quantum operations, such as those constraints imposed by superselection rules (SSRs), have a profound effect on quantum information theoretic concepts like bipartite entanglement. This paper concentrates on a particular example: the constraint that applies when the parties (Alice and Bob) cannot distinguish among certain quantum objects they have. This arises naturally in the context of ensemble quantum information processing such as in liquid NMR. We discuss how a SSR for the symmetric group can be applied, and show how the extractable entanglement can be calculated analytically in certain cases, with a maximum bipartite entanglement in an ensemble of NNN Bell-state pairs scaling as log(N)\mathrm{log}(N)log(N) as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We discuss the apparent disparity with the asymptotic $(N\ensuremath{\rightarrow}\ensuremath{\infty})$ recovery of unconstrained entanglement for other sorts of superselection rules, and show that the disparity disappears when the correct notion of applying the symmetric group SSR to multiple copies is used. Next we discuss reference frames in the context of this SSR, showing the relation to the work of von Korff and Kempe [Phys. Rev. Lett. 93, 260502 (2004)]. The action of a reference frame can be regarded as the analog of activation in mixed-state entanglement. We also discuss the analog of distillation: there exist states such that one copy can act as an imperfect reference frame for another copy. Finally we present an example of a stronger operational constraint, that operations must be noncollective as well as symmetric. Even under this stronger constraint we, nevertheless, show that Bell nonlocality (and hence entanglement) can be demonstrated for an ensemble of NNN Bell-state pairs no matter how large NNN is. This last work is a generalization of that of Mermin [Phys. Rev. D 22, 356 (1980)].
Quantum entanglement is a fascinating topic with both theoretical and technological impacts. Yet, even in the simplest scenarios, there are still various open questions involving pure state and mixed state entanglement. Here we present the simplest entangled mixed states. We consider both qubits (quantum bits) and qudits (quantum digits). We first provide our simplest yet most important result for rank-2 bi-partite states, and we then generalize our results to go beyond rank-2 and to go beyond bi-partite states.
The characterization of entanglement is a fundamental issue for Quantum Information Theory. But the definition of entanglement depends on the notion of locality, and thus on the tensor product structure of the state space of the composite system. This notion is affected by the presence of superselection rules that restrict the accessible Hilbert space to a direct sum of subspaces. Indistinguishability of particles imposes one such restriction, namely to totally symmetric or totally antisymmetric states. The entanglement can in this case be defined with respect to partitions of modes in the second quantization formalism. For fermionic systems the Fock space of m modes is isomorphic to the space of m qubits, but the action of creation and annihilation operators is not local, due to their anticommutation. Conservation of the parity of fermion number imposes another relevant superselection rule. It requires that local physical observables commute with the local parity operator. Taking into account the considerations above, it is possible to define the set of separable states or equivalently the concept of entanglement for fermionic systems in a number of ways. Here we analyze systematically these possibilities and the relation among the various sets of separable states. We also discuss the behavior of the different classes when taking several copies of the state, as well as the characterization of the sets in terms of the usual criteria regarding the tensor product.