Strong violations of Bell-type inequalities for Werner-like states

We investigate the violation of Bell-type inequalities for two-qubit Werner-like states parametrized by the positive parameter $0⩽p⩽1$. We use an unbalanced homodyne detection scheme to obtain the quantum mechanical probabilities. A violation of the Bell-Wigner and Janssens inequalities is obtained for a large range of the parameter $p$. The range given by these inequalities is greater than the one given by the Clauser-Horne inequality. The range in which a violation is attained actually coincides with the range where the Werner-like states are known to be nonseparable, i.e., for $p>1∕3$. However, the improvement over the Clauser-Horne inequality is achieved at the price of restricting the class of possible local hidden variable theories.

Figure 1

Unbalanced homodyne detection scheme for a Bell experiment with Werner-like states. Here ${\rho }_p^{\mathrm{W}}=p\mid \Psi ⟩⟨\Psi \mid +\frac{(1-p)}{4}\mathbb{I}$ with $\mid \Psi ⟩=\frac{1}{\sqrt{2}}({\mid 1⟩}_a{\mid 0⟩}_b-{\mid 0⟩}_a{\mid 1⟩}_b)$ and $a$ and $b$ label the modes. Alice and Bob mix mode matched local oscillators of amplitude ${\gamma }_a$ and ${\gamma }_b$, respectively, with their signal modes, where $\alpha ={\gamma }_a\sqrt{1-T}$ and $\beta ={\gamma }_b\sqrt{1-T}$ are the arguments of the displacement operator $\widehat{D}$ defined in the text. The symbol $T$ denotes the transmittivity of the beam splitters.

Figure 2

(Color online) Violation of the CH inequality $-1⩽{\mathcal{I}}_{\mathit{CH}}⩽0$ with ${\mathcal{I}}_{\mathit{CH}}$ defined in Eq. (9) as a function of the mixing parameter $p$. The horizontal dashed line marks the classical constraint. We obtain a violation for $p>0.75$.

Figure 3

(Color online) Violation of the Bell-Wigner inequality $W_1⩽1$ with $W_1$ defined in Eq. (10) as a function of the mixing parameter $p$. The horizontal dashed line marks the classical constraint. We obtain a violation for $p>1∕3$.

Figure 4

(Color online) Violation of the Bell-type inequality $0⩽J_1$, where $J_1$ is defined in Eq. (16), as a function of $p$. The horizontal dashed line marks the classical constraint. We obtain a violation for $p>0.4$.

Figure 5

(Color online) Violation of the Bell-type inequality $J_3⩽3$, where $J_3$ is defined in Eq. (17), as a function of $p$. The horizontal dashed line marks the classical constraint. We obtain a violation for $p>1∕3$.

Figure 6

Overview of the current status on the violation of Bell-type inequalities in different models by two-qubit Werner states ${\rho }_p^{\mathrm{W}}$ as a function of $p$. The models refer to the following publications: Barrett [11], Werner [8], Acín, Gisin, and Toner (AGT) [10].