How Difficult is it to Prepare a Quantum State?

Consider a quantum system prepared in an input state. One wants to drive it into a target state. Assuming classical states and operations as free resources, I identify a geometric cost function which quantifies the difficulty of the protocol in terms of how different it is from a classical process. The quantity determines a lower bound to the number of commuting unitary transformations required to complete the task. I then discuss the link between the quantum character of a state preparation and the amount of coherence and quantum correlations that are created in the target state.

Figure 1

The optimal path ${\overline{\gamma }}_t$ to drive a system from $\rho \in {\mathcal{M}}_{i_R}$ into $\tau$ is the minimizer of the quantum component of the energy $Q_{\rho }(\tau )=E_q^{{\overline{\gamma }}_t}(\overline{\rho }\to \tau )$ over all the free states. The unitary map ${\overline{\gamma }}_t^u$ generates the energy minimizing path from a free state ${\overline{\rho }}^u$ isospectral to the target, $Q_{\rho }^u(\tau )=E^{{\overline{\gamma }}_t^u}({\overline{\rho }}^u\to \tau )$.