Coarse-Graining Channels and Recoverability

A coarse graining is an information channel only when it maps states on a declared input algebra to states on a declared retained algebra by a completely positive trace-preserving map. Under that condition, data processing quantifies lost distinguishability and recovery theorems quantify approximate sufficiency. A Wilsonian change of action or coupling coordinates does not, by itself, supply such a map.

Required background. Positivity, monotonicity, and data processing supplies the channel inequality; information measures along RG flows fixes the observable and scale direction.

Helpful background. Recovery and approximate Markovianity supplies Petz and universal recovery maps.

Channelized coarse graining

Let $\mathcal A_{\rm full}$ contain the observables of a regulated theory and $\mathcal A_{\rm keep}$ the observables retained at lower resolution. In a tensor-factor regulator, tracing discarded modes gives

$$\mathcal N(\rho_{\rm full}) =\operatorname{Tr}_{\rm discard}\rho_{\rm full}.$$

More generally, a unital inclusion of retained observables defines a dual state restriction. Smearing, finite-resolution measurement, or noise can also define CPTP maps. For each construction one must state the input/output systems, the state family, and which observables represent the low-energy task.

The structure diagram marks the channel branch as conditional because many useful RG transformations act on actions or correlation functions without defining one canonical state channel.

Channelized coarse graining is the operational branch. It requires an explicit CPTP map and retained algebra; only then do distinguishability loss and recovery have their standard meanings. Schematic and not to scale.

Distinguishability loss

For any pair $\rho,\sigma$,

$$D(\rho\Vert\sigma) \ge D(\mathcal N\rho\Vert\mathcal N\sigma).$$

The loss

$$\delta_{\mathcal N}(\rho,\sigma) =D(\rho\Vert\sigma) -D(\mathcal N\rho\Vert\mathcal N\sigma)$$

depends on the pair and the map. It is not a state-independent count of degrees of freedom. A high-momentum mode can be nearly irrelevant for one low-energy state family and crucial for another containing hard excitations.

Equality under suitable support conditions is characterized by Petz recovery. With reference state $\sigma$, the finite-dimensional Petz map is

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

When $\delta_{\mathcal N}=0$, it recovers both $\sigma$ and $\rho$. Strengthened data-processing results bound a fidelity to a rotated or universal recovery map when the loss is small. These are pairwise guarantees, not formal inverses of an RG semigroup.

Exact sufficiency is characterized in Petz 1988, §§ 3–4. The quantitative fidelity bound is Fawzi and Renner 2015, Theorem 5.1, and the state-independent recovery construction is Junge et al. 2018, Theorem 2.1.

Gaussian mode elimination

Take a regulated free field and perform a symplectic change of variables that separates a retained low-frequency band from discarded modes. Partial trace over the discarded factor is a Gaussian channel: first moments transform linearly and the retained covariance matrix is the corresponding principal block after the chosen mode map.

A reproducible test selects two Gaussian input states, computes the full and retained relative entropies, and verifies $\delta_{\mathcal N}\ge0$. It then applies the Gaussian form of the Petz or optimized recovery map and evaluates fidelity on the same state family. Varying the band edge shows that “low energy” is a task-dependent tolerance, not an exact property of the Hilbert-space factorization.

Gauge constraints complicate this example because physical modes need not factorize. One must instead identify a retained gauge-invariant algebra and its center; a naive tensor trace can introduce unphysical edge variables.

Wilsonian RG versus a state channel

Integrating a momentum shell in a Euclidean path integral produces an effective action for remaining fields. Field rescaling and coupling redefinitions then compare actions at different cutoffs. This is a powerful calculational transformation, but it does not uniquely fix:

• a Schrödinger-picture CPTP map on all states;
• a common Hilbert-space factorization across cutoffs;
• or an operational recovery task.

Special tensor-network, open-system, or algebraic constructions can add those data. Their recovery claims apply to that construction and should not be promoted to all RG flows.

Validity map

The operational branch is licensed only after the map and retained algebra are explicit. If the calculation only changes an effective action, recovery language remains a proposal until a state channel is constructed. Schematic and not to scale.

Common pitfalls

Calling an effective action a channel. The effective action reproduces selected correlators; a CPTP state map is additional structure.

Treating a recovery map as a physical inverse. Recovery is optimized for a state family and reference. It need not reconstruct arbitrary discarded excitations.

Ignoring gauge or continuum factorization. A mode trace is straightforward only in a regulator with a declared physical tensor product or algebraic restriction.

References

• Fawzi, Omar, and Renato Renner. “Quantum Conditional Mutual Information and Approximate Markov Chains.” Communications in Mathematical Physics 340 (2015): 575–611. DOI.
• 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.
• Petz, Dénes. “Sufficiency of Channels over von Neumann Algebras.” Quarterly Journal of Mathematics 39 (1988): 97–108. DOI.