Generalising the Horodecki criterion to nonprojective qubit observables

Abstract

The Horodecki criterion provides a necessary and sufficient condition for a two-qubit state to be able to manifest Bell nonlocality via violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. It requires, however, the assumption that suitable projective measurements can be made on each qubit, and is not sufficient for scenarios in which noisy or weak measurements are either desirable or unavoidable. By characterising two-valued qubit observables in terms of strength, bias, and directional parameters, we address such scenarios by providing necessary and sufficient conditions for arbitrary qubit measurements having fixed strengths and relative angles for each observer. In particular, we find the achievable maximal values of the CHSH parameter for unbiased measurements on arbitrary states, and, alternatively, for arbitrary measurements on states with maximally-mixed marginals, and determine the optimal angles in some cases. We also show that for certain ranges of measurement strengths it is only possible to violate the CHSH inequality via biased measurements. Finally, we use the CHSH inequality to obtain a simple necessary condition for the compatibility of two qubit observables.

Recycling qubits for the generation of Bell nonlocality between independent sequential observers

“…This upper bound is proved in Appendix A, and will be generalised elsewhere [43]. Note that it simplifies to the maximum quantum value of 2 √ 2 for the case of unit strengths and orthogonal measurement directions.…”

“…It will be shown elsewhere that the bound in Eq. ( 57) can be extended to a generalised Horodecki criterion for nonprojective observables [43]. Finally, we note that since our approach is based on an instrumental formalism which can incorporate the most general measurements, our analysis can be applied to similar problems in the sequential sharing of other quantum properties, such as EPR-steering and entanglement, including for cases in which more measurement settings as per observer are allowed.…”

Quantum Bell nonlocality cannot be shared under a special kind of bilateral measurements for high-dimensional quantum states

“…For multipartite quantum states, the sharing ability of nonlocality in unilateral POVM measurement is already very weak [30,34], so it should be weaker than bipartite quantum state in bilateral measurement, and we can continue to study it. In the latest literature [35], by characterising two-valued qubit observables in terms of strength, bias, and directional parameters, the authors investigated generalising the Horodecki criterion to nonprojective qubit observables. Therefore, we may continue to think about a series of problems such as the sharing of network nonlocality [36] or other quantum resources under nonprojective measurement.…”

Mermin and Svetlichny inequalities for non-projective measurement observables

“…quantum entanglement, nonlocality, steering, etc [10,11]. The framework provided by Hall and Cheng in [6], has inspired us to do the similar analysis for three qubit Bell inequalities.…”

“…most importantly, nonorthogonal state discrimination and its crucial role in randomness extraction plus quantum cryptography [5]. To deal with such situations, Hall and Cheng have provided the necessary and sufficient conditions to violate the Bell-CHSH inequalities for non-projective measurements on arbitrary two qubits [6], by improving the Horodecki bound (of sharp observables) [7]. The generalization of the bound is also useful for the task of 'resource recycling' where bonafide parties use the noisy detector to implement the task [8,9].…”