Semi-device-independent bounds on entanglement

Detection and quantification of entanglement in quantum resources are two key steps in the implementation of various quantum-information processing tasks. Here, we show that Bell-type inequalities are not only useful in verifying the presence of entanglement but can also be used to bound the entanglement of the underlying physical system. Our main tool consists of a family of Clauser-Horne-like Bell inequalities that cannot be violated maximally by any finite-dimensional maximally entangled state. Using these inequalities, we demonstrate the explicit construction of both lower and upper bounds on the concurrence for two-qubit states. The fact that these bounds arise from Bell-type inequalities also allows them to be obtained in a semi-device-independent manner, that is, with assumption of the dimension of the Hilbert space but without resorting to any knowledge of the actual measurements being performed on the individual subsystems.

Figure 1

Plot of the concurrence $C$ of the two-qubit state $|{\psi }_{\tau }^{*}\rangle$ which violates maximally the inequality $I^{(\tau )}$ as a function of $\tau$ (blue solid line). Also plotted is the concurrence $C_{\mathrm{cr}}$ of the critical two-qubit state $|{\psi }_c^{\tau }\rangle$ (red dashed line). All states with $C>C_{\mathrm{cr}}$ do not violate $I^{(\tau )}$ (see text at Sec. 4b for details); hence, the violation of $I^{(\tau )}$ allows one to derive upper bounds on the entanglement.

Figure 2

Maximal quantum violation of $I^{(\tau )}$ as a function of $\tau$ (blue solid line). Also shown is the analytic upper bound on ${\mathcal{S}}_{\mathrm{qm}}^{(\tau )}(|{\psi }_{\tau }^{*}\rangle )$ computed from Eq. (11) (red dashed line), which provides an upper bound on ${\mathcal{S}}_{\mathrm{qm}}^{(\tau )}$.