Uncertainty principle of quantum processes

Heisenberg's uncertainty principle, which imposes intrinsic restrictions on our ability to predict the outcomes of incompatible quantum measurements to arbitrary precision, demonstrates one of the key differences between classical and quantum mechanics. The physical systems considered in the uncertainty principle are static in nature and described mathematically with a quantum state in a Hilbert space. However, many physical systems are dynamic in nature and described with the formalism of a quantum channel. In this paper, we show that the uncertainty principle can be reformulated to include process measurements that are performed on quantum channels. Since both the preparation of quantum states and the implementation of quantum measurements are themselves special cases of quantum channels, our formalism encapsulates the uncertainty principle in its utmost generality. More specifically, we obtain expressions that generalize the Maassen–Uffink uncertainty relation and the universal uncertainty relations from quantum states to quantum channels.

Figure 1

Schematic illustration of the process positive-operator-valued measures (PPOVMs): (a) PPOVM ${\mathcal{T}}_1$ and (b) PPOVM ${\mathcal{T}}_2$.

Figure 2

Schematic illustration of the lattice structure exhibited in example t17 excluding the GLB. Each point stands for an element, and the red line represents the binary relation “$\prec$” between elements. In this plot, a lower point is majorized by the higher point whenever they are connected with a red line. Obviously, here, ${\mathbf{b}}_S^{\downarrow }\prec {\mathbf{b}}_S$ and ${\mathbf{b}}_S\prec {\mathbf{b}}_S^{\downarrow }$, but ${\mathbf{b}}_S\ne {\mathbf{b}}_S^{\downarrow }$.