Local vertical measurements and violation of Bell inequality

TL;DR

This paper derives bounds on Bell inequality violations under local vertical measurements for pure and mixed two-qubit states, linking violation potential to entanglement measures like entropy and concurrence.

Contribution

It introduces a method to determine Bell inequality bounds based on local vertical measurements and relates these bounds to entanglement measures for pure and mixed states.

Findings

Bell violation impossible if entanglement < 0.2644 for pure states

Maximal violation occurs only at full entanglement (1)

Violation depends on concurrence, requiring >0.6 for mixed states

Abstract

For two qubits belonging to Alice and Bob, we derive an approach to setup the bound of Bell operator in the condition that Alice and Bob continue to perform local vertical measurements. For pure states we find that if the entanglement of the two qubits is less than 0.2644 (measured with von Neumann entropy) the violation of the Bell inequality will never be realized, and only when the entanglement is equal to 1 the maximal violation ($2\sqrt{2}$) can occur. For specific form of mixed states, we prove that the bound of the Bell inequality depends on the concurrence. Only when the concurrence is greater than 0.6 the violation of the Bell inequality can occur, and the maximal violation can never be achieved. We suggest that the bound of the Bell operator in the condition of local vertical measurements may be used as a measure of the entanglement.