Measurement-device-independent verification of channel steering

Extending the concept of steerability for quantum states, channel steerability is an ability to remotely control the given channel from a coherently extended party. Verification of channel steering can be understood as certifying coherence of the channel in a one-sided device-independent manner with respect to a bystander. Here we propose a method to verify channel steering in a measurement-device-independent way. To do this, we first obtain Choi matrices from given channels and use the canonical method of measurement-device-independent verification of quantum steering. As a consequence, exploiting channel-state duality which interconverts steerability of channels and that of states, channel steering is verified. We further analyze the effect of imperfect preparation of entangled states used in the verification protocol, and find that the threshold of the undesired noise that we can tolerate is bounded from below by steering robustness.

Figure 1

Comparison between state steering and channel steering. Left: Given a local state ${\rho }_A$, an extended state ${\rho }_{AB}$ is such that one can retrieve the original local state ${\rho }_A$ by discarding system $B$. If a set of measurements ${\left\{M_{a|x}\right\}}_{a,x}$ is performed on system $B$, system $A$ collapses to an assemblage ${\left\{{\rho }_{a|x}\right\}}_{a,x}$. By checking whether measurements on system $B$ really steered system $A$, we determine steerability of the state. Right: Channel steering can be defined in an analogous way. Given a channel from system $D$ to system $B$, ${\mathrm{\Lambda }}^{D\to B}$, one can come up with an extended channel with one input $D$ and two output systems $B$ and $A$ such that the original channel ${\mathrm{\Lambda }}^{D\to B}$ is retrieved by discarding an added output system $A$, a bystander. If a set of measurements ${\left\{M_{a|x}\right\}}_{a,x}$ is performed on a bystander's system, the channel reduces to an instrument from $D$ to $B$, ${\left\{{\mathrm{\Lambda }}_{a|x}^{D\to B}\right\}}_{a,x}$. By checking whether measurements on a bystander's system really steered a channel from $D$ to $B$, we determine steerability of the channel.

Figure 2

Schematic of an MDI verification protocol of channel steering. (a) Due to Theorem 1, steerability of the channel assemblage (left) is equivalent to steerability of the Choi assemblage obtained by $|\psi \rangle$ (right). Thus we first obtain Choi assemblage using bipartite pure states $|\psi \rangle$ with the full Schmidt rank, and verify steerability of the resultant Choi assemblage ${\left\{{\rho }_{a|x}^{|\psi \rangle }\right\}}_{a,x}$. (b) Left: 1s-DI verification of state steering can be accomplished by requesting and receiving classical information from the experimenter when the whole process can be trusted. It is depicted as a white box which takes and yields classical information. Right: If we encode the question in quantum states, MDI verification of state steering is possible without any trust in the process. It is depicted as a black box because we do not need to concern ourselves with what is taking place inside. Therefore, by verifying the obtained Choi assemblage in the MDI protocol, MDI verification of the channel steering is accomplished.