# When is the Chernoff Exponent for Quantum Operations finite?

## TL;DR

This paper investigates the conditions under which the Chernoff exponent for quantum operations is finite, establishing a clear criterion related to perfect distinguishability and providing bounds on the exponent.

## Contribution

It provides a necessary and sufficient condition for the finiteness of the Chernoff exponent for quantum operations and offers upper bounds, clarifying the asymptotic error decay behavior.

## Key findings

- Chernoff exponent is finite iff quantum operations cannot be perfectly distinguished with finite uses.
- Error probability decays exponentially when operations are not perfectly distinguishable.
- Super-exponential decay of error probability is ruled out.

## Abstract

We consider the problem of testing two hypotheses of quantum operations in a setting of many uses where an arbitrary prior probability distribution is given. The Chernoff exponent for quantum operations is investigated to track the minimal average error probability of discriminating two quantum operations asymptotically. We answer the question, "When is the Chernoff exponent for quantum operations finite?" We show that either two quantum operations can be perfectly distinguished with finite uses, or the minimal discrimination error decays exponentially with respect to the number of uses asymptotically. That is, the Chernoff exponent is finite if and only if the quantum operations can not be perfectly distinguished with finite uses. This rules out the possibility of super-exponential decay of error probability. Upper bounds of the Chernoff exponent for quantum operations are provided.