Petz, Rotated, and Universal Recovery Maps

Petz, rotated Petz, and universal recovery maps turn data-processing loss into explicit reconstruction procedures. They share a reference state and channel but differ in modular rotation and averaging. Their formulas involve supports and sometimes unbounded inverses, so a regulated implementation must not be confused with an automatically local continuum operation.

Required background. Use Recovery Maps and Approximate Markovianity. Helpful background. Araki Relative Entropy and Regulated Limits supplies the modular and support setting.

Petz recovery

For a channel $\mathcal N$ and faithful reference state $\sigma$, the finite-dimensional Petz map is

$$\mathcal P_{\sigma,\mathcal N}(X) =\sigma^{1/2}\, \mathcal N^\dagger\!\left[ \mathcal N(\sigma)^{-1/2} X \mathcal N(\sigma)^{-1/2} \right]\sigma^{1/2},$$

with inverses taken on the appropriate support. It recovers $\sigma$ exactly and characterizes equality in data processing: when relative entropy is preserved for $\rho$ and $\sigma$, it also recovers $\rho$. This sufficiency theorem is due to Petz 1986, pp. 123–131.

The diagram places Petz recovery on the channel branch attached to the central relative-entropy comparison.

Recovery maps are built from a channel, its adjoint, and reference modular data. Correlation bounds can guarantee performance, but the algebra, support, rotation, and resource domain remain part of the construction. Schematic.

Rotations and universal averaging

Modularly rotated versions conjugate the input and output factors by imaginary powers such as $\sigma^{it}$ and $\mathcal N(\sigma)^{-it}$. Averaging these rotated maps with a fixed probability density yields a recovery map that depends on $\sigma$ and $\mathcal N$ but not on the particular recovered state. Junge et al. 2018, pp. 2955–2978 establish universal remainder bounds for data processing in von Neumann algebras.

“Universal” has this limited meaning: one map works for a class of input states relative to the fixed reference and channel. It does not mean independent of the algebra, reference, representation, or physical resource limits.

For three finite Gaussian regions, regularize a full-rank reference covariance matrix, construct the Petz map, and compare it with a rotated average. Then vary the rank regularization and mode cutoff. Stability of the recovered observables is necessary before assigning a continuum meaning.

Domain and implementation

The failure map highlights the inverse-support and operation-domain issues.

Support restrictions and modular domains are mathematical inputs, not numerical details. A regulated inverse that diverges as the cutoff is removed may prevent a bounded or energy-feasible continuum recovery operation. Schematic.

State the Schrödinger or Heisenberg convention, the channel adjoint pairing, the reference state, supports, rotation measure, regulator, and any locality or energy requirement. These data distinguish an equality theorem from a physical recovery protocol.

References

• Junge, Marius, Renato Renner, David Sutter, Mark M. Wilde, and Andreas Winter. “Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy.” Annales Henri Poincaré 19 (2018): 2955–2978. DOI. Open preprint.
• Petz, Dénes. “Sufficient Subalgebras and the Relative Entropy of States of a von Neumann Algebra.” Communications in Mathematical Physics 105 (1986): 123–131. DOI.

Further reading

• Sutter, David, Mario Berta, and Marco Tomamichel. “Multivariate Trace Inequalities.” Communications in Mathematical Physics 352 (2017): 37–58. DOI. Open preprint.