We study the time-dependent electron-electron and electron-hole correlations in a mesoscopic device which is splitting an incident current of free fermions into two spatially separated particle streams. We analyze the appearance of entanglement as manifested in a Bell inequality test and discuss its origin in terms of local spin-singlet correlations already present in the initial channel and the action of post-selection during the Bell-type measurement. The time window over which the Bell inequality is violated is determined in the tunneling limit and for the general situation with arbitrary transparencies. We compare our results with alternative Bell inequality tests based on coincidence probabilities.We study the time-dependent electron-electron and electron-hole correlations in a mesoscopic device which is splitting an incident current of free fermions into two spatially separated particle streams. We analyze the appearance of entanglement as manifested in a Bell inequality test and discuss its origin in terms of local spin-singlet correlations already present in the initial channel and the action of post-selection during the Bell-type measurement. The time window over which the Bell inequality is violated is determined in the tunneling limit and for the general situation with arbitrary transparencies. We compare our results with alternative Bell inequality tests based on coincidence probabilities.

arXiv:cond-mat/0311423Physical Review BJan 11, 200547 citations

BELL INEQUALITIES

WORLD SCIENTIFIC eBooksJun 1, 20000 citations

General Iterative Formulas of Bell's Inequalities.

Xing-Yan Fan, Hongye Liu, Jia-Le Miao +5 more

Newton's second law aids us in predicting the location of a classical object after knowing its initial position and velocity together with the force it experiences at any time, which can be seen as a process of continuous iteration. When it comes to discrete problems, e.g. building Bell inequalities, as a vital tool to study the powerful nonlocal correlations in quantum information processing. Unless having known precisely the general formula of associated inequalities, iterative formulas build a bridge from simple examples to all elements in the set of Bell inequalities. Although exhaust all entities in the set, even in the subset of tight individuals, is a NP hard problem, it is possible to find out the evolution law of Bell inequalities from few-body, limited-setting and low-dimension situations to arbitrary $(n,k,d)$ constructions, i.e. $n$ particles, $k$ measurements per particle, and $d$ outcomes per measurement. In this work, via observing Śliwa's 46 tight (3,2,2) Bell inequalities [{Phys. Lett. A}. 317, 165-168 (2003)], uniting the root method [{Phys. Rev. A}. 79, 012115, (2009)] and the idea of degeneration, we discover an iterative formula of Bell inequalities containing all $(n,k,2)$ circumstances, which paves a potential way to study the current Bell inequalities in terms of iterative relations combining root method on the one hand, and explore more interesting inequalities on the other.Newton's second law aids us in predicting the location of a classical object after knowing its initial position and velocity together with the force it experiences at any time, which can be seen as a process of continuous iteration. When it comes to discrete problems, e.g. building Bell inequalities, as a vital tool to study the powerful nonlocal correlations in quantum information processing. Unless having known precisely the general formula of associated inequalities, iterative formulas build a bridge from simple examples to all elements in the set of Bell inequalities. Although exhaust all entities in the set, even in the subset of tight individuals, is a NP hard problem, it is possible to find out the evolution law of Bell inequalities from few-body, limited-setting and low-dimension situations to arbitrary $(n,k,d)$ constructions, i.e. $n$ particles, $k$ measurements per particle, and $d$ outcomes per measurement. In this work, via observing Śliwa's 46 tight (3,2,2) Bell inequalities [{Phys. Lett. A}. 317, 165-168 (2003)], uniting the root method [{Phys. Rev. A}. 79, 012115, (2009)] and the idea of degeneration, we discover an iterative formula of Bell inequalities containing all $(n,k,2)$ circumstances, which paves a potential way to study the current Bell inequalities in terms of iterative relations combining root method on the one hand, and explore more interesting inequalities on the other.

Sep 12, 20210 citations

All the Bell Inequalities

Asher Peres

Foundations of PhysicsApr 1, 1999222 citations

Robust self-testing and certified randomness based on an arbitrary-input Bell inequality

Rajdeep Paul, Sneha Munshi, A. K. Pan

Self-testing is the strongest certification procedure that uniquely characterizes the physical system based on the observed statistics, without any knowledge of the inner workings of the devices. The optimal quantum violation of a Bell inequality enables such a device-independent (DI) self-testing of the source and the measurement devices. In this work, we demonstrate the DI self-testing based on the arbitrary-input chained Bell inequality. We devise a systematic and elegant sum-of-squares (SOS) technique enabling dimension-independent optimization of the quantum violation. Our approach enables the derivation of the state along with the relationship between the local observables directly from the optimization condition. One significant aspect is the robustness of such self-testing in real experimental situations involving noise and imperfection, leading to deviation from the optimal quantum violation. We provide an analytical technique for robust self-testing in the presence of noise. As an application of our scheme, we demonstrate the generation of two bit DI randomness and analyze the robustness of such randomness. Our optimization method is both simple and elegant, making it suitable for deriving the optimal quantum violation of various arbitrary-input Bell inequalities.Self-testing is the strongest certification procedure that uniquely characterizes the physical system based on the observed statistics, without any knowledge of the inner workings of the devices. The optimal quantum violation of a Bell inequality enables such a device-independent (DI) self-testing of the source and the measurement devices. In this work, we demonstrate the DI self-testing based on the arbitrary-input chained Bell inequality. We devise a systematic and elegant sum-of-squares (SOS) technique enabling dimension-independent optimization of the quantum violation. Our approach enables the derivation of the state along with the relationship between the local observables directly from the optimization condition. One significant aspect is the robustness of such self-testing in real experimental situations involving noise and imperfection, leading to deviation from the optimal quantum violation. We provide an analytical technique for robust self-testing in the presence of noise. As an application of our scheme, we demonstrate the generation of two bit DI randomness and analyze the robustness of such randomness. Our optimization method is both simple and elegant, making it suitable for deriving the optimal quantum violation of various arbitrary-input Bell inequalities.

arXiv:2505.19917Physical Review AFeb 27, 20260 citations

(n,k,d)

constructions, i.e.

n

particles,

k

measurements per particle, and

d

outcomes per measurement. In this work, via observing Śliwa's 46 tight (3,2,2) Bell inequalities [{Phys. Lett. A}. 317, 165-168 (2003)], uniting the root method [{Phys. Rev. A}. 79, 012115, (2009)] and the idea of degeneration, we discover an iterative formula of Bell inequalities containing all

(n,k,2)

circumstances, which paves a potential way to study the current Bell inequalities in terms of iterative relations combining root method on the one hand, and explore more interesting inequalities on the other.