Bell's Inequality, Generalized Concurrence and Entanglement in Qubits.

We demonstrate an alternative evaluation of quantum entanglement by measuring maximum violation of Bell's inequality without performing a partial trace operation in an $n$-qubit system by bridging maximum violation of Bell's inequality and a generalized concurrence of a pure state. We show that this proposal is realized when one subsystem only contains one qubit and a quantum state is a linear combination of two product states. We also use a toric code model with smooth and rough boundary conditions on a cylinder manifold and a disk manifold with holes to show that a choice of a generalized concurrence of a pure state depends on boundary degrees of freedom of a Hilbert space. A relation of a generalized concurrence of a pure state and maximum violation of Bell's inequality is also demonstrated in a $2n$-qubit state. Finally, we apply our theoretical studies to a two-qubit system with a non-uniform magnetic field at a finite temperature as well as a Wen-Plaquette model with four and six qubits at zero temperature.