Coarse-Graining Channels and Recoverability

A coarse graining is an information channel only after its input algebra, retained algebra, state map, and operational task have been specified. Then data processing measures lost distinguishability, while recovery theorems test whether that loss can be reversed on a chosen state family. A Wilsonian change of action or coupling coordinates does not by itself provide these ingredients.

Required background. Positivity, monotonicity, and data processing supplies the relative-entropy inequality; information measures along RG flows supplies the observable and scale direction used here.

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

The chapter overview first separates cutoff, observation, and deformation scales, then gives a comparison table of the resulting claims and a set of independent validity gates. Those distinctions are essential here: “integrate out a shell” and “apply a quantum channel” are different mathematical statements.

A channel needs declared algebras

At a finite regulator, let the full and retained systems have density matrices on Hilbert spaces ${\cal H}_{\rm full}$ and ${\cal H}_{\rm keep}$. A Schrödinger-picture channel

$${\cal N}:{\cal T}({\cal H}_{\rm full})\longrightarrow {\cal T}({\cal H}_{\rm keep})$$

is linear, completely positive, and trace preserving. If the regulator factors as ${\cal H}_{\rm full}={\cal H}_{\rm keep}\otimes{\cal H}_{\rm discard}$, partial trace is one such channel. It is not the only one: finite-resolution measurement, noise, and a declared mode transform followed by partial trace define different operational coarse grainings.

In an algebraic continuum formulation the clean direction is often Heisenberg: a normal unital completely positive map sends retained observables into the full algebra, and its predual sends normal states in the opposite direction. A subalgebra restriction need not come from a tensor factor. This matters in gauge theory and in local QFT, where Gauss constraints, centers, and type-III local algebras obstruct the naive formula “trace out the ultraviolet.”

For normal states $\rho$ and $\sigma$ in a finite-dimensional representative, define

$$\delta_{\cal N}(\rho,\sigma) =D(\rho\Vert\sigma)-D({\cal N}\rho\Vert{\cal N}\sigma)\ge 0.$$

The support condition $\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma$ makes the first relative entropy finite. The number $\delta_{\cal N}$ belongs to the pair and the channel; it is not a state-independent count of degrees of freedom.

Exact and approximate recovery

For a finite-dimensional reference state $\sigma$, the Petz map on the support of ${\cal N}\sigma$ is

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

The inverse is the support inverse; a trace-preserving extension on the orthogonal complement can be chosen without changing the states under discussion. Equality in data processing holds precisely when the relevant states are sufficient for the channel, equivalently when this recovery recovers both $\sigma$ and $\rho$ under the standard support hypotheses Petz 1988, pp. 99–105.

Approximate equality has a stronger, state-independent form. If

$$f(\omega,\eta)=\operatorname{Tr}\sqrt{\sqrt{\omega}\eta\sqrt{\omega}}$$

denotes root fidelity, there is a recovery channel $\widetilde{\cal R}_{\sigma,{\cal N}}$, depending on $\sigma$ and ${\cal N}$ but not on $\rho$, such that

$$\delta_{\cal N}(\rho,\sigma) \ge -2\log f\!\left( \rho,(\widetilde{\cal R}_{\sigma,{\cal N}}\circ{\cal N})(\rho) \right).$$

Theorem 2.1 first proves an averaged rotated-map remainder; the single universal-map form displayed here is Junge et al. 2018, Remark 2.2, Eq. (20). The Fawzi–Renner bound is the important conditional-mutual-information/partial-trace case, not a theorem that every RG transformation has an inverse Fawzi and Renner 2015, Theorem 5.1.

Exact displaced-thermal-mode benchmark

Here is a complete channel calculation with no continuum or fitting ambiguity. Take two oscillator modes $L,H$ with $[a_j,a_k^\dagger]=\delta_{jk}$ and dimensionless thermal parameters $\beta\omega_L=1$, $\beta\omega_H=4$. Set

$$\sigma=\tau_L\otimes\tau_H, \qquad \tau_j=(1-e^{-\beta\omega_j})e^{-\beta\omega_j a_j^\dagger a_j},$$

and displace both modes,

$$\rho=D_L(\alpha_L)D_H(\alpha_H)\,\sigma\, D_H(\alpha_H)^\dagger D_L(\alpha_L)^\dagger.$$

The channel is ${\cal N}=\operatorname{Tr}_H$, so its input is the two-mode trace class and its output is the low-mode trace class. Because a displacement preserves entropy and increases $\langle a_j^\dagger a_j\rangle$ by $\lvert\alpha_j\rvert^2$,

$$\begin{aligned} D(\rho\Vert\sigma) &=\beta\omega_L\lvert\alpha_L\rvert^2 +\beta\omega_H\lvert\alpha_H\rvert^2,\\ D({\cal N}\rho\Vert{\cal N}\sigma) &=\beta\omega_L\lvert\alpha_L\rvert^2,\\ \delta_{\cal N} &=\beta\omega_H\lvert\alpha_H\rvert^2. \end{aligned}$$

For this product reference the Petz map is especially transparent:

$${\cal R}_{\sigma,{\cal N}}(X_L)=X_L\otimes\tau_H.$$

Thus recovery is exact exactly on the subfamily with no discarded-mode displacement. Bosonic Gaussian states and channels remain Gaussian under Petz recovery; the general covariance-and-first-moment construction is given in Lami, Das, and Wilde 2018, Theorem 1 and § IV.

For equal thermal covariances, the recovered-state root fidelity depends only on the missed high-mode displacement:

$$f=\exp\!\left[-\frac{\lvert\alpha_H\rvert^2}{2\nu_H}\right], \qquad \nu_H=2\bar n_H+1=\coth(\beta\omega_H/2).$$

With $\alpha_L=0.2$, $\alpha_H=0.1$, one has $\nu_H=1.0373147207$. All logarithms below are natural.

Input family	$D(\rho\Vert\sigma)$	$D({\cal N}\rho\Vert{\cal N}\sigma)$	$\delta_{\cal N}$	$f$	$-2\log f$
$\alpha_L=0.2,\ \alpha_H=0$	$0.040000$	$0.040000$	$0$	$1$	$0$
$\alpha_L=0.2,\ \alpha_H=0.1$	$0.080000$	$0.040000$	$0.040000$	$0.99519146$	$0.00964028$
$\alpha_L=0.2,\ \alpha_H=0.2$	$0.200000$	$0.040000$	$0.160000$	$0.98090413$	$0.0385611$

The recovery inequality is comfortably satisfied in the last two rows. These values follow from closed formulas; double-precision evaluation changes the displayed residuals by less than $10^{-14}$. A Fock-space reproduction should report its cutoff. For the undisplaced thermal reference, truncating after $n=32$ leaves probability $e^{-33}=4.66\times10^{-15}$ in the least-suppressed low mode, but the displaced-state tail must be checked separately rather than inferred from that number.

The benchmark also states its operational scope. Low-mode quadratures are retained exactly. The unknown high-mode first moment is erased. A fidelity guarantee controls bounded observables through the Fuchs–van de Graaf inequalities; an energy or moment bound is additionally required before using it for unbounded field quadratures.

Changing the retained information changes the answer

The promised adversarial test modifies the retained algebra rather than the state. For the middle row above:

Retained information	Allowed recovery	Loss	Root fidelity after recovery	Conclusion
Low mode only	One fixed channel depending on $\sigma,{\cal N}$	$0.040000$	$0.99519146$ for the Petz map	The unknown high displacement is lost.
Both modes	Identity channel	$0$	$1$	No coarse graining occurred.
Low mode plus a classical label for $\alpha_H$	Reprepare $D_H(\alpha_H)\tau_HD_H^\dagger$ conditionally	$0$ on the labelled family	$1$	Side information changed the task.
Low mode plus an uncorrelated fixed ancilla	Append any state independent of $\alpha_H$	$0.040000$ before recovery	Cannot be $1$ for two distinct unknown $\alpha_H$ values	A UV ancilla is not the missing information.

The third row is not a better inverse to the original channel: its classical side register was absent from the original output. The fourth row is a useful no-go control. Two inputs with identical low marginal but different high displacements give the same channel output, so no single recovery map acting only on that output can return both inputs exactly.

Wilsonian transformations are not automatically channels

Integrating a shell in a Euclidean path integral produces an effective action for the remaining integration variables. Rescaling coordinates and fields then compares coupling descriptions at different cutoffs. This is a powerful calculational operation, but it does not uniquely specify a CPTP map on all states, a common Hilbert-space factorization, or a recovery task.

Tensor-network, open-system, lattice, and algebraic constructions can add those data. Their recovery statements apply to the declared construction. In a gauge theory one must specify a gauge-invariant retained algebra and any center or edge-mode convention; tracing an unconstrained tensor factor generally answers a different question.

Common pitfalls

Calling an effective action a channel. Reproducing selected low-momentum correlators does not construct a completely positive state map.

Treating recovery as a physical time reverse. A universal recovery map is universal over $\rho$ only after $\sigma$ and ${\cal N}$ are fixed. It need not reconstruct arbitrary discarded excitations.

Quoting fidelity without its convention. Here $f$ is root fidelity. If a source instead calls $f^2$ “fidelity,” the logarithmic remainder changes appearance by a factor of two.

Exercises

1. Petz recovery for a product reference

Let ${\cal N}=\operatorname{Tr}_H$ and $\sigma=\sigma_L\otimes\sigma_H$, with both factors faithful. Derive ${\cal R}_{\sigma,{\cal N}}(X_L)=X_L\otimes\sigma_H$.

Solution

The Hilbert–Schmidt adjoint of partial trace is ${\cal N}^\dagger(X_L)=X_L\otimes I_H$, and ${\cal N}\sigma=\sigma_L$. Substitution in the Petz formula gives

$$(\sigma_L^{1/2}\otimes\sigma_H^{1/2}) (\sigma_L^{-1/2}X_L\sigma_L^{-1/2}\otimes I_H) (\sigma_L^{1/2}\otimes\sigma_H^{1/2}) =X_L\otimes\sigma_H.$$

Faithfulness avoids writing support projectors. With nonfaithful factors the same calculation holds on the appropriate supports, followed by a trace-preserving extension off the support.

2. Check the quantitative recovery bound

For $\beta\omega_H=4$ and $\alpha_H=0.1$, compute $\delta_{\cal N}$, $\nu_H$, $f$, and $-2\log f$. Verify the universal-recovery inequality numerically.

Solution

The exact information loss is

$$\delta_{\cal N}=4(0.1)^2=0.04.$$

Next,

$$\nu_H=\coth 2=1.0373147207, \qquad f=\exp\!\left[-\frac{0.01}{2(1.0373147207)}\right] =0.99519146.$$

Therefore

$$-2\log f=\frac{0.01}{1.0373147207}=0.00964028<0.04.$$

The gap $0.03035972$ is not a numerical error; the recovery theorem gives a lower bound, not equality for this family.

3. Why a fixed ancilla cannot restore an unknown displacement

Choose two inputs with the same low displacement and distinct high displacements $\alpha_H\ne\alpha_H'$. Show that no channel acting only on the low output can recover both inputs exactly. Explain why supplying the displacement as a classical label evades the argument.

Solution

Partial trace sends both inputs to the same low state. A fixed recovery channel is a function of that output, so it must produce the same recovered two-mode state in both cases. The two targets are different because their high-mode first moments are $\alpha_H$ and $\alpha_H'$. Hence the same recovered state cannot equal both targets.

If the output is enlarged by a classical register containing the displacement, the two outputs are no longer identical. A controlled channel can read the register and prepare the corresponding displaced thermal high mode. Exact recovery is then possible, but the retained algebra and operational task have changed.

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.
• Lami, Ludovico, Siddhartha Das, and Mark M. Wilde. “Approximate Reversal of Quantum Gaussian Dynamics.” Journal of Physics A: Mathematical and Theoretical 51 (2018): 125301. DOI; Open preprint.
• Petz, Dénes. “Sufficiency of Channels over von Neumann Algebras.” Quarterly Journal of Mathematics 39 (1988): 97–108. DOI.