State protection by quantum control before and after noise processes

We discuss protection of a quantum state that goes through a noise process by measurements and operations before and after the noise process. In our previous work, we showed the nonexistence of “truly quantum” protocols that protect an unknown qubit state against depolarizing noise better than “classical” ones. Toward identifying the class of noise processes that is optimally suppressed by such a “classical” protocol, we extend our previous result in two directions. First, we show that the statement is also true in any finite-dimensional Hilbert spaces, which was previously conjectured; the optimal protocol is either the do nothing protocol or the discriminate and reprepare protocol, depending on the strength of the noise. Second, in the case of a single qubit, we show that essentially the same conclusion holds for any unital noise. Thus, the noise must be nonunital for a control protocol beyond “classical” ones to exist.

Figure 1

Schematic diagram of quantum control (ex-ante control and ex-post control).

Figure 2

Left: The space of unital noise channels is (up to certain equivalence) a tetrahedron $T$ with vertices at ${\mathcal{E}}_0(1,1,1)$, ${\mathcal{E}}_1(1,-1,-1),{\mathcal{E}}_2(-1,1,-1)$, and ${\mathcal{E}}_3(-1,-1,1)$, which are unitary operations. A unital noise channel is a convex combination of ${\mathcal{E}}_{\mu }$. Right: The tetrahedron $T$ is decomposed into an octahedron $O$ and four tetrahedra $T_{\mu }$. The six vertices of $O$ are the midpoints ${\mathcal{E}}_{\mu \nu }$ of the edges, which are $(\pm 1,0,0)$, $(0,\pm 1,0)$, and $(0,0,\pm 1)$. This decomposition will be used in Theorem 2.