Greenberger–Horne–Zeilinger states: Their identifications and robust violations

Abstract

The [Formula: see text]-qubit Greenberger–Horne–Zeilinger (GHZ) states are the maximally entangled states of [Formula: see text] qubits, which have had many important applications in quantum information processing, such as quantum key distribution and quantum secret sharing. Thus how to distinguish the GHZ states from other quantum states becomes a significant problem. In this work, by presenting a family of the generalized Clauser–Horne–Shimony–Holt (CHSH) inequality, we show that the [Formula: see text]-qubit GHZ states can be indeed identified by the maximal violations of the generalized CHSH inequality under some specific measurement settings. The generalized CHSH inequality is simple and contains only four correlation functions for any [Formula: see text]-qubit system, thus has the merit of facilitating experimental verification. Furthermore, we present a quantum phenomenon of robust violations of the generalized CHSH inequality in which the maximal violation of Bell’s inequality can be robust under some specific noises adding to the [Formula: see text]-qubit GHZ states.

From a single-system game to robust violations of the Bell's inequality

“…In this section, we show that the above mappings provide an explanation for the quantum phenomenon of robust violations of the generalized CHSH inequality [20]. And, based on the explanation, one can exactly demonstrate the degeneracy of the Bell function I N CHSH , which corresponds to the dimension of noises for robust violations.…”

“…(i) The maximal violations of the GHZ state cannot exceed the Tsirelson's bound ±2 √ 2, which has been found in Ref. [20]. (ii) When the expected value for the GHZ state…”

“…where σ = (σ x , σ y , σ z ) is the vector of Pauli matrices, n j = (sin α j cos ϕ j , sin α j sin ϕ j , cos α j ) and n j = (sin α j cos ϕ j , sin α j sin ϕ j , cos α j ) denote the measurement direction of the jth qubit. Fan et al [20] defined the measurement operators of Alice and Bob in (5) as…”

“…Multiplying by weighting coefficients (−1) ab and summing up the terms in (19), according to the relations (20) and (21) one can find…”

“…On the other hand, to distinguish N -qubit Greenberger-Horne-Zeilinger (GHZ) states, Fan et al presented a generalized CHSH inequality in their very recent work [20] . It is expressed as…”