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

Abstract

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.

The paper asks whether mixed-state phase equivalence defined by locally reversible channel circuits implies pure-state phase equivalence of canonical purifications, $|\psi_\rho\rangle=|\sqrt\rho\rangle\!\rangle$. It establishes this implication by lifting each locally recoverable channel gate to an isometry on the corresponding doubled region, then composing the lifts into a local unitary circuit, allowing ancillas. For $\sigma_{AC}=\mathcal N_{B\to C}(\rho_{AB})$ and mutual-information loss $\epsilon=I(A:B)_\rho-I(A:C)_\sigma$, a local isometry achieves purification overlap at least $e^{-\epsilon/2}$. The proof uses a Hilbert–Schmidt version of universal rotated-Petz recovery. A separate explicit construction using “half” the unrotated Petz map gives, for $\eta=I(A:C|B)\leq1$, the amplitude bound $\|\sqrt\rho-\rho_{AB}^{1/2}\rho_B^{-1/2}\rho_{BC}^{1/2}\|_2\leq\frac{\sqrt\eta}{\pi}[2+\sqrt{2\log(16d_*/\eta)}]$, where $d_*=\min(d_A,d_C)$. Exact recoverability gives an exact lift; circuit-level equivalence requires superpolynomially small gate-recovery errors. Two numerical examples illustrate the connection. For bit-flip-dephased toric code, random-bond Ising calculations up to $L=16$ locate the purification transition near the decoding threshold $p_c\approx0.109$, separating doubled from single-copy toric-code order. For a Chern insulator with retained dephasing flags, trajectory simulations find exponentially decaying conditional mutual information and purification spin-Hall index approaching $C=-1$ for sampled $0.1\leq p\leq0.9$; the fully measured endpoint shows algebraic behavior. The phase implication is only one-way: a counterexample shows that equivalent canonical-purification phases need not imply equivalent mixed-state phases. The Chern-insulator example concerns flagged, not unflagged, noise, and the numerical critical behavior is subject to finite-size and sampling limitations.