Continuous-variable steering and incompatibility via state-channel duality

The term Einstein-Podolsky-Rosen steering refers to a quantum correlation intermediate between entanglement and Bell nonlocality, which has been connected to another fundamental quantum property: measurement incompatibility. In the finite-dimensional case, efficient computational methods to quantify steerability have been developed. In the infinite-dimensional case, however, less theoretical tools are available. Here, we approach the problem of steerability in the continuous variable case via a notion of state-channel correspondence, which generalizes the well-known Choi-Jamiołkowski correspondence. Via our approach we are able to generalize the connection between steering and incompatibility to the continuous variable case and to connect the steerability of a state with the incompatibility breaking property of a quantum channel, with applications to noisy NOON states and amplitude damping channels. Moreover, we apply our methods to the Gaussian steering setting, proving, among other things, that canonical quadratures are sufficient for steering Gaussian states.

Figure 1

Critical noise bound for steering through the 1001-state by a coarse-grained pair of quadrature measurements, as a function of the separation angle $\theta$, with coarse-grainings of different number of intervals $N_{\mathrm{int}}$. The case $N_{\mathrm{int}}=2$ can be reproduced analytically. Below $\eta =2/3$ the setting is unsteerable by joint measurability criterion Eq. (19).

Figure 2

Steerability region for the dynamical setting (shaded area). The parameter $u$ is the coupling strength (in units of the spectral linewidth $\lambda$ of the bath) and $t$ is time (in units of ${\lambda }^{-1}$). The two revival regions reflect the non-Markovian character of the evolution in the strong coupling regime, which allows steerability to re-emerge at later times.