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arXiv:1004.5184 · New Journal of Physics

Reference frames for Bell inequality violation in the presence of superselection rules

Tomasz Paterek, P Kurzyński, D K L Oi, D Kaszlikowski

Published Apr 19, 2011• 9 citations
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Abstract

Superselection rules (SSRs) constrain the allowed states and operations in quantum theory. They limit preparations and measurements and hence impact our ability to observe non-locality, in particular the violation of Bell inequalities. We show that a reference frame compatible with a particle number SSR does not allow observers to violate a Bell inequality if and only if it is prepared using only local operations and classical communication. In particular, jointly prepared separable reference frames are sufficient for obtaining violations of a Bell inequality. We study the size and non-local properties of such reference frames using superselection-induced variance. These results suggest the need for experimental Bell tests in the presence of superselection.

DOI
10.1088/1367-2630/13/4/043027

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Superselection rules (SSRs) constrain the allowed states and operations in quantum theory. They limit preparations and measurements and hence impact our ability to observe non-locality, in particular the violation of Bell inequalities. We show that a reference frame compatible with a particle number SSR does not allow observers to violate a Bell inequality if and only if it is prepared using only local operations and classical communication. In particular, jointly prepared separable reference frames are sufficient for obtaining violations of a Bell inequality. We study the size and non-local properties of such reference frames using superselection-induced variance. These results suggest the need for experimental Bell tests in the presence of superselection.
Superselection rules (SSRs) constrain the allowed states and operations in quantum theory. They limit preparations and measurements and hence impact our ability to observe non-locality, in particular the violation of Bell inequalities. We show that a reference frame compatible with a particle number SSR does not allow observers to violate a Bell inequality if and only if it is prepared using only local operations and classical communication. In particular, jointly prepared separable reference frames are sufficient for obtaining violations of a Bell inequality. We study the size and non-local properties of such reference frames using superselection-induced variance. These results suggest the need for experimental Bell tests in the presence of superselection.
Superselection rules (SSRs) constrain the allowable physical operations and consequently the ability to observe violations of local realism. Not only do SSRs restrict the type of states which can be prepared, equally they restrict the reference frames which are necessary for measurements. Here, we derive quantitative conditions under which a Bell inequality is violated in the presence of SSRs which reveal the critical role of reference frames. In particular, references prepared only locally in accordance with SSRs cannot be used to violate a Bell inequality. We discuss the optimal reference frames, their minimal size and reusability. Finally we comment on experiments related to Bell inequality and single photon non-locality, and propose a new experiment showing violation under an induced SSR. ‡ Corresponding author. ar X iv :1 00 4. 51 84 v2 [ qu an tph ] 1 6 Ju n 20 10 Violation of Bell’s inequality in the presence of superselection rules 2
Proposals for Bell-inequality tests on systems restricted by the particle-number-superselection rule often require operations that are difficult to implement in practice. In this article, we derive a Bell inequality, where measurements on pairs of states are used as a method to bypass this superselection rule. In particular, we focus on mode entanglement of an arbitrary number of massive particles and show that our Bell inequality detects the entanglement in an identical pair of states when other inequalities fail. However, as the number of particles in the system increases, the violation of our Bell inequality decreases due to the restriction in the measurement space caused by the superselection rule. This Bell test can be implemented using techniques that are routinely used in current experiments.
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.
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)].
Superselection rules severely alter the possible operations that can be implemented on a distributed quantum system. Whereas the restriction to local operations imposed by a bipartite setting gives rise to the notion of entanglement as a nonlocal resource, the superselection rule associated with particle number conservation leads to a new resource, the superselection induced variance of the local particle number. We show that, in the case of pure quantum states, one can quantify the nonlocal properties by only two additive measures, and that all states with the same measures can be asymptotically interconverted into each other by local operations and classical communication. Furthermore we discuss how superselection rules affect the concepts of majorization, teleportation, and mixed state entanglement.