Bell inequalities in translational-invariant systems

Thursday started in Paraty with a beautiful sunny morning, after somewhat less spectacular weather on Wednesday. The Paraty IV Workshop resumed at 9 am with a talk titled “Nonviolation of Bell’s inequality in translational invariant systems”, given by Thiago R. de Olveira, my colleague at UFF.

Thiago works at the interface between condensed matter and quantum information. He reviewed some problems in which quantum information brings something new to the study of condensed matter. For example, in quantum phase transitions there are drastic changes in the system’s macroscopic properties, even at zero temperature. At these transitions, long-range correlations appear, and the question of quantifying how the different measures of correlations behave is a very active field. Measures of entanglement, for example, often signal quantum phase transitions, as they may inherit the non-analytical behaviour of the energy at the transition.

In his talk, he discussed the issue of whether, and when, the spin systems he studies violate some Bell inequality. Bell inequalities would work as entanglement witnesses in these systems. The situation, however, is not so clear-cut, as in many examples one can show there is entanglement, even though no Bell inequality is violated, as is the case of Werner states. This seemed to be a common situation, requiring a better understanding.

The key to understanding the phenomenon are the monogamy relations obeyed by some multipartite measures of entanglement. One important example, by Coffman, Kundu ad Wootters (2000), bounds above the sum of pairwise concurrences in a multiqubit system. Thiago and co-workers showed how a similar result for quantum discord fails to hold. A related result, by Toner and Verstraete (2006), shows that Bell inequality violation is also monogamous.

The main result is: a N-qubit system with translational invariance will never violate the CHSH inequality. Translation invariance forces each qubit’s system to be in the same state rho. This means if two subsystems violate a Bell inequality, so should all other pairs, but this is forbidden by the monogamy relations. This was hinted at, in thicker mathematical language, in earlier work by Terhal et al (2003) and Werner (1989). This main result, however, only actually holds for an even number of qubits, as noted in later work by Sun, Wu, Huang, Chen and Wang (arXiv:1308.0131 [quant-ph]).

Thiago and collaborators have also extended this work, proving more general bounds and also analysing Bell inequality violation by random states.

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One thought on “Bell inequalities in translational-invariant systems

  1. Pingback: Random Interlacements | Eventually Almost Everywhere

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