# Quantum algorithms for the Petz recovery channel, pretty-good measurements and polar decomposition

### APA

Quek, Y. (2020). Quantum algorithms for the Petz recovery channel, pretty-good measurements and polar decomposition. Perimeter Institute. https://pirsa.org/20100030

### MLA

Quek, Yihui. Quantum algorithms for the Petz recovery channel, pretty-good measurements and polar decomposition. Perimeter Institute, Oct. 07, 2020, https://pirsa.org/20100030

### BibTex

@misc{ pirsa_PIRSA:20100030, doi = {10.48660/20100030}, url = {https://pirsa.org/20100030}, author = {Quek, Yihui}, keywords = {Quantum Information}, language = {en}, title = {Quantum algorithms for the Petz recovery channel, pretty-good measurements and polar decomposition}, publisher = {Perimeter Institute}, year = {2020}, month = {oct}, note = {PIRSA:20100030 see, \url{https://pirsa.org}} }

Yihui Quek Freie Universität Berlin

## Abstract

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. We rectify this lack using the recently developed tools of quantum singular value transformation and oblivious amplitude amplification, providing a quantum algorithm to implement the Petz recovery channel. 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.

Using the same toolbox, we also develop a quantum algorithm for enacting the polar decomposition, a workhorse in linear algebra. This provides an alternative route to implementing a pretty-good measurements for the special case of pure states, which speeds up the general-purpose algorithm developed above.