# Distinguishing classically indistinguishable states and channels

## TL;DR

This paper explores quantum distinguishability problems where states and channels become indistinguishable under decoherence, providing conditions for perfect distinguishability in such scenarios.

## Contribution

It introduces new quantum distinguishability problems involving decoherence and derives conditions for perfect distinguishability of states and channels.

## Key findings

- Conditions for perfect distinguishability of quantum states under decoherence
- Conditions for perfect distinguishability of quantum channels under decoherence
- Relevance to superselection rules and quantum information encoding

## Abstract

We investigate an original family of quantum distinguishability problems, where the goal is to perfectly distinguish between $M$ quantum states that become identical under a completely decohering map. Similarly, we study distinguishability of $M$ quantum channels that cannot be distinguished when one is restricted to decohered input and output states. The studied problems arise naturally in the presence of a superselection rule, allow one to quantify the amount of information that can be encoded in phase degrees of freedom (coherences), and are related to time-energy uncertainty relation. We present a collection of results on both necessary and sufficient conditions for the existence of $M$ perfectly distinguishable states (channels) that are classically indistinguishable.