Lifting the mixed to the pure: quantum phases of matter from canonical purification

@misc{huang2026lifting,
  title = {{Lifting the mixed to the pure: quantum phases of matter from canonical purification}},
  author = {Xiaoyang Huang and Shengqi Sang and Chong Wang and Yijian Zou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41926},
  eprint = {2610.41926},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41926},
}

TY  - UNPB
TI  - Lifting the mixed to the pure: quantum phases of matter from canonical purification
AU  - Xiaoyang Huang
AU  - Shengqi Sang
AU  - Chong Wang
AU  - Yijian Zou
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41926
UR  - https://arxiv.org/abs/2610.41926
AB  - Pure-state quantum phases are defined through finite-depth local unitary connections. This concept of phases has recently been generalized to mixed states in terms of locally reversible channels. In this work, we establish a fundamental relation between mixed-state phases and pure-state phases. We show that a locally reversible channel between mixed states can be lifted to a local isometry between their canonical purifications; thus, if two mixed states are in the same phase, then their canonical purifications are in the same pure-state phase. Furthermore, an explicit construction of the lifting isometry is given by ``half'' of the Petz recovery map, suggesting that the Petz recovery map \emph{without rotation} is enough to connect mixed state phases. We illustrate our results with two examples: toric code under bit flip noise and Chern insulator under fermion number noise.
ER  -