# Completing the Grand Tour of asymptotic quantum coherence manipulation

## TL;DR

This paper provides a comprehensive analysis of asymptotic quantum coherence manipulation, deriving explicit formulas for distillable coherence and coherence cost under physically and strictly incoherent operations, with implications for practical quantum technologies.

## Contribution

It offers the first closed-form expression for distillable coherence under SIO, shows equivalence with PIO, and presents a single-letter formula for coherence cost, completing the understanding of coherence manipulation.

## Key findings

- Distillable coherence under SIO equals that under PIO.
- Coherence cost under PIO is finite for a broad class of states.
- Identifies 'abyssally bound' states with zero distillable coherence but infinite coherence cost.

## Abstract

We compute on all quantum states several measures that characterise asymptotic quantum coherence manipulation under restricted classes of operations. We focus on the distillable coherence, i.e. the maximum rate of production of approximate pure bits of coherence starting from independent copies of an input state $\rho$, and on the coherence cost, i.e. the minimum rate of consumption of pure coherence bits that is needed to generate many copies of $\rho$ with vanishing error. We obtain the first closed-form expression for the distillable coherence under strictly incoherent operations (SIO), proving that it coincides with that obtained via physically incoherent operations (PIO). This shows that SIO and PIO are equally weak at distilling coherence, sheds light on the recently discovered phenomenon of generic bound coherence, and provides us with an explicit optimal distillation protocol that is amenable to practical implementations. We give a single-letter formula for the coherence cost under PIO, showing that it is finite on a set of states with nonzero volume. Since PIO can be realised in a laboratory with incoherent ancillae, unitaries, and measurements, our result puts fundamental limitations on coherence manipulation in an experimentally relevant setting. We find examples of 'abyssally bound' states with vanishing PIO distillable coherence yet infinite PIO coherence cost. Our findings complete the picture of asymptotic coherence manipulation under all the main classes of incoherent operations.