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arXiv:quant-ph/0608056 · Physical Review A

Entanglement and symmetry: A case study in superselection rules, reference frames, and beyond

S. J. Jones, Howard M. Wiseman, Stephen D. Bartlett, Joan A. Vaccaro, Damian T. Pope

Published Dec 20, 2006• 23 citations
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Abstract

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 $N$ Bell-state pairs scaling as $\mathrm{log}(N)$ 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 $N$ Bell-state pairs no matter how large $N$ is. This last work is a generalization of that of Mermin [Phys. Rev. D 22, 356 (1980)].

DOI
10.1103/physreva.74.062313

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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→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞. We discuss the apparent disparity with the asymptotic (N\ensuremath→\ensuremath∞)(N\ensuremath{\rightarrow}\ensuremath{\infty})(N\ensuremath→\ensuremath∞) 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)].
Microscopic physical systems requiring a quantum description are increasingly being harnessed to develop useful technologies. The appeal of such an approach lies in the fact that quantum systems possess inherently different properties from those observed on a macroscopic scale. Uniquely quantum phenomena, such as entanglement, offer new tools for performing experimental tasks. For instance, the use of quantum systems for implementing cryptographic protocols and performing computation has caused significant excitement in recent years as it seems likely that the quantum approach offers an advantage over purely classical implementations. The development of quantum technologies leads to the concept of quantum resources. One key quantum resource that lies at the heart of the divergence between quantum and classical descriptions is that of entanglement. However, for entanglement to act as a resource, it must be possible for the parties (we will restrict ourselves to only two) sharing the entanglement to verify its presence. This may seem like a simple task, however, verifying the properties of quantum systems can be nontrivial. To make matters more complicated, there are often constraints under which the parties sharing the quantum systems find themselves which further restrict the ability of the quantum systems to act as a resource. One family of constraints arises when the two parties are dealing with quantum systems which have properties constrained by a conservation law, or some other constraint which limits knowledge of the properties of the local subsystems. These constraints can be expressed using the well studied formalism of superselection rules. When a superselection rule applies to quantum systems it can be ambiguous as to whether entanglement survives between the systems. This thesis considers a situation that arises naturally in a nuclear magnetic resonance experiment where two parties share many copies of identical quantum systems. In this case a superselection rule associated with the finite symmetric group applies due to the indistinguishability of the systems. The concept of a reference frame for the symmetric group is used to alleviate the effect of the superselection rule and demonstrate when entanglement survives despite the constraint of indistinguishability. In addition to this, a situation which goes beyond the superselection formalism is considered, one where the operations performable by the two parties are further constrained. It is shown that despite this stronger constraint, it is still possible in principle to verify the existence of entanglement experimentally as demonstrated by the violation of a Bell inequality. Methods for verifying the presence of entanglement can also prove useful in characterising the properties of entangled quantum systems. Such characterisation may offer insight into the possible advantages offered by quantum systems over their classical counterparts. It becomes possible to more accurately characterise entangled systems by considering a conceptual constraint under which the two parties may find themselves. This is the situation that arises when one of the parties, say Bob, distrusts the other, Alice. In this situation, Alice can only convince Bob of the presence of their entanglement if the entangled system they share possesses a certain property. We term this property steerability, after Schrödinger’s term “steering” [Proc. Camb. Phil. Soc. 31, 553 (1935)] for the Einstein-Podolsky-Rosen effect. This thesis offers a rigorous characterisation of steering in a quantum information context and probes a number of questions relating to this property. First, a number of examples are presented which demonstrate state steerability which is a uniquely quantum property possessed by certain entangled states. Second, the task of experimentally detecting steerability, and thus entanglement, is addressed through the introduction of steerability criteria. Finally two more detailed examples are explored which illustrate various aspects of the measurement dependence of steerability in realistic experiments.
Entanglement is a resource central to quantum information (QI). In particular, entanglement shared between two distant parties allows them to do certain tasks that would otherwise be impossible. In this context, we study the effect on the available entanglement of physical restrictions on the local operations that can be performed by the two parties. We enforce these physical restrictions by generalized superselection rules (SSRs), which we define to be associated with a given group of physical transformations. Specifically the generalized SSR is that the local operations must be covariant with respect to that group. Then we operationally define the entanglement constrained by a SSR, and show that it may be far below that expected on the basis of a naive (or ``fluffy bunny'') calculation. We consider two examples. The first is a particle number SSR. Using this we show that for a two-mode BEC (with Alice owning mode AAA and Bob mode BBB), the useful entanglement shared by Alice and Bob is identically zero. The second, a SSR associated with the symmetric group, is applicable to ensemble QI processing such as in liquid-NMR. We prove that even for an ensemble comprising many pairs of qubits, with each pair described by a pure Bell state, the entanglement per pair constrained by this SSR goes to zero for a large ensemble.
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.
Superselection rules (SSRs) limit the mechanical and quantum processing resources represented by quantum states. However SSRs can be violated using reference systems to break the underlying symmetry. We show that there is a duality between the ability of a system to do mechanical work and to act as a reference system. Further, for a bipartite system in a globally symmetric pure state, we find a triality between the system's ability to do local mechanical work, its ability to do ``logical work'' due to its accessible entanglement, and its ability to act as a shared reference system.
Recently, there has been much interest in a new kind of ``unspeakable'' quantum information that stands to regular quantum information in the same way that a direction in space or a moment in time stands to a classical bit string: the former can only be encoded using particular degrees of freedom while the latter are indifferent to the physical nature of the information carriers. The problem of correlating distant reference frames, of which aligning Cartesian axes and synchronizing clocks are important instances, is an example of a task that requires the exchange of unspeakable information and for which it is interesting to determine the fundamental quantum limit of efficiency. There have also been many investigations into the information theory that is appropriate for parties that lack reference frames or that lack correlation between their reference frames, restrictions that result in global and local superselection rules. In the presence of these, quantum unspeakable information becomes a new kind of resource that can be manipulated, depleted, quantified, etc. Methods have also been developed to contend with these restrictions using relational encodings, particularly in the context of computation, cryptography, communication, and the manipulation of entanglement. This paper reviews the role of reference frames and superselection rules in the theory of quantum-information processing.