# Quantum algorithm for Petz recovery channels and pretty good measurements

arXiv: https://arxiv.org/abs/2006.16924 · HTML: https://mlsanity.com/paper/2006.16924/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jun 2020.

## Built on by (most cited first, 3 of 3)

- [Statistical Complexity of Quantum Learning](https://mlsanity.com/paper/2309.11617/) (2023)
- [An invitation to the sample complexity of quantum hypothesis testing](https://mlsanity.com/paper/2403.17868/) (2024)
- [Density Operator Expectation Maximization](https://mlsanity.com/paper/2507.22786/) (2025)

## Abstract (arXiv)

The Petz recovery channel plays an important role in quantum information science as an operation that approximately reverses the effect of a quantum channel. The pretty good measurement is a special case of the Petz recovery channel, and it allows for near-optimal state discrimination. A hurdle to the experimental realization of these vaunted theoretical tools is the lack of a systematic and efficient method to implement them. This paper sets out to rectify this lack: using the recently developed tools of quantum singular value transformation and oblivious amplitude amplification, we provide a quantum algorithm to implement the Petz recovery channel when given the ability to perform the channel that one wishes to reverse. Moreover, we prove that, in some sense, our quantum algorithm's usage of the channel implementation cannot be improved by more than a quadratic factor. Our quantum algorithm also provides a procedure to perform pretty good measurements when given multiple copies of the states that one is trying to distinguish.