Nonviolation of Bell’s inequality in translation invariant systems

Abstract

The nature of quantum correlations in strongly correlated systems has been a subject of intense research. In particular, it has been realized that entanglement and quantum discord are present at quantum phase transitions and are able to characterize them. Surprisingly, it has been shown for a number of different systems that qubit pairwise states, even when highly entangled, do not violate Bell's inequalities, being in this sense local. Here we show that such a local character of quantum correlations traces back to the the monogamy trade-off obeyed by bipartite Bell correlations, being in fact general for translation invariant systems. We illustrate this result in a quantum spin chain with a soft breaking of translation symmetry. In addition, we provide an extension of the monogamy inequality to the N -qubit scenario, showing that the bound increases with N and providing examples of its saturation through uniformly generated random pure states.

Bell-Type Correlation at Quantum Phase Transitions in Spin-1 Chain

“…The reduced density matrices on two sites, which display mixed entangled states, are not sufficient to violate the CGLMP inequality in the spin-1 chain, although many-body ground states are pure and highly entangled in this model. As the CHSH correlation has monogamy trade-off relation [ 15 ], monogamy of the CGLMP correlation can be a candidate to explain the non-violation of CGLMP inequality in the spin-1 chain. Therefore, in order to detect nonlocality in this model, one should consider multipartite Bell correlations for arbitrary dimensions.…”

“…At first, Bell correlations for many-body systems are applied to two-site reduced density states which are generated from the ground state in one-dimensional (1D) spin-1/2 XY model [ 13 ] and spin-1/2 XXZ model [ 14 ]. As a result, it has been known that the Bell correlation demonstrates the non-analyticity at the critical point while, due to the monogamy characteristics of the correlation, any bipartite Bell inequality is not violated by the translational invariant many-body systems [ 15 ].…”

Monogamy of Quantum Correlations - A Review

“…discord in the case of tripartite pure states [47]. Assuming that the optimizations for both quantum discord and quantum work deficit of the bipartite state ρ AB takes place for the same ensemble {p k , ρ k AB }, from the definition of quantum discord and quantum work deficit, one can show that (51) where H({p k }) is the Shannon entropy originating from the local measurement on the party B. Since H({p k }) ≥ S(ρ B ), W ← (ρ AB ) ≥ D ← (ρ AB ).…”

“…There exists no-go theorems which place parallel restrictions such as monogamy of Bell inequality violation [48,49] and exclusion principle of classical information transmission over quantum channels [50] (cf. [51][52][53][54]). More precisely, within the consideration of a multiparty set-up, for example, of an editor with several reporters, if the shared quantum state between the editor and a single reporter violates a Bell inequality [55,56] or is quantum dense codeable [57], then the rest of the channels shared between the editor and the other reporters are prohibited from possessing the same quantum advantage.…”

Lectures on General Quantum Correlations and their Applications

“…There exists no-go theorems which place parallel restrictions such as monogamy of Bell inequality violation [48,49] and exclusion principle of classical information transmission over quantum channels [50] (cf. [51][52][53][54]). More precisely, within the consideration of a multiparty set-up, for example, of an editor with several reporters, if the shared quantum state between the editor and a single reporter violates a Bell inequality [55,56] or is quantum dense codeable [57], then the rest of the channels shared between the editor and the other reporters are prohibited from possessing the same quantum advantage.…”