How difficult is it to prepare a quantum state?

TL;DR

This paper introduces a geometric cost function to quantify the difficulty of quantum state preparation, linking it to quantum coherence and correlations, and providing bounds on the resources needed.

Contribution

It proposes a new geometric measure for the complexity of quantum state preparation based on classical resources and connects it to quantum coherence and correlations.

Findings

The cost function provides a lower bound on the number of unitary transformations needed.

Quantum character of states relates to the amount of coherence and correlations created.

The approach bridges geometric complexity with quantum resource theories.

Abstract

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.