Information Across Scales and Renormalization

Information measures can track the loss of distinguishable structure from short to long distances. Their relation to renormalization depends on which scale is varied and which theorem or coarse-graining map controls the comparison.

Enter this chapter

Begin with Separating Cutoff, Region, Correlation, and RG Scales to separate region size, regulator, coupling, and renormalization scales.

Helpful background. Universal terms and geometry identifies the finite and logarithmic data used at conformal fixed points. Data processing supplies the information inequality behind genuine coarse-graining channels. UV and IR fixed points, flow constraints, and anomaly coefficients provide the RG and CFT endpoints.

Topics

1. Separating Cutoff, Region, Correlation, and RG Scales — Distinguishes the physical roles of the scales in an information calculation.
2. Information Measures Along RG Flows — Defines a family of states or regions before comparing information.
3. Entropic Monotones in Two Dimensions — Builds an interval-based monotone in two dimensions.
4. Sphere Entanglement and Three-Dimensional F Monotonicity — Derives a disk-based monotonicity statement in three dimensions.
5. Four-Dimensional a-Type Flow Constraints — Relates four-dimensional irreversibility to the Euler anomaly.
6. Renormalized Entanglement and Scheme Dependence — Tracks the finite freedom in subtracted entropy quantities.
7. Relevant Deformations and Entanglement Crossover Functions — Separates universal endpoints from crossover functions.
8. Relative Entropy and RG Irreversibility — Tests irreversibility through comparison on a fixed algebra.
9. Coarse-Graining Channels and Recoverability — Defines coarse graining as a channel before asking about recovery.
10. Scale-Space Entanglement and Factorization Caveats — Explains why a scale decomposition need not define tensor factors.
11. Information Geometry on Theory Space — Compares metrics on specified families of theories.

How the ideas fit together

Entropic monotonicity combines information inequalities with geometric and symmetry assumptions. In two dimensions, a suitable interval derivative cancels the cutoff-dependent constant and yields a quantity constrained along relativistic vacuum flows Casini and Huerta 2007, pp. 7032–7035.

Higher-dimensional results use different constructions and hypotheses. A subtraction prescription or a finite crossover curve is not automatically another monotone. Likewise, integrating out momentum modes becomes a data-processing statement only after a common state space and an actual channel have been specified.

The comparison below separates the assumptions and tests needed for these different conclusions.

Comparison table

Information diagnostics compared by system, definition, hypotheses, finite-flow evidence, operational consequence, and characteristic failure.

Diagnostic	System	Definition or prescription	Hypotheses and endpoint anchor	Finite-flow evidence	Operational consequence	Adversarial failure
Entropic c-function	Vacuum interval in 1+1 dimensions	cE(R) = 3R S′(R)	Unitary Lorentz-invariant QFT and strong subadditivity; equals the CFT central charge at an endpoint	Correlated derivative fit with cutoff and box refinement	No recovery map is required for the theorem	A thermal state adds β and gives an increasing exact control curve
Renormalized F-function	Vacuum disk in 2+1 dimensions	𝓕(R) = (R∂R − 1)S(R)	Relativistic vacuum flow; equals the sphere free energy at a CFT	The differential operator itself removes the area term; fit-window and radial-cutoff stability remain to be checked	No recovery map is required for the entropic inequality	Raw disk entropy, a narrow fit window, or an untracked boundary contribution gives a false plateau
Four-dimensional a constraint	Smooth spheres or null-cone regions in 3+1 dimensions	Euler-anomaly endpoint difference; at a CFT the spherical logarithm is −4a log(R/ε)	Unitary relativistic flow between bulk CFTs; aUV ≥ aIR	Endpoint anomaly match or a controlled dilaton or null-cone calculation	No canonical finite a(R) and no generic recovery map follow	A finite counterterm, boundary anomaly, or non-CFT endpoint cannot be relabeled as bulk a
Renormalized entropy prescription	One scalable region family in a declared dimension	Dimension-dependent differential polynomial, fixed local counterterms, or symmetrically split mutual information	Surface geometry, framing, and subtraction convention fixed globally; fixed-point normalization checked	Agreement of universal endpoints and quantified prescription differences through the crossover	None by itself	Changing the split, fit, or counterterm between radii manufactures running
Relevant-deformation crossover	A fixed state and region family near an RG fixed point	Normalized scaling field gRRy, equivalently R/ℓg, plus irrelevant corrections	The scaling field, branch, nonuniversal metric factor, regulator path, and fit window declared	Data collapse, endpoint asymptotics, exponent stability, and correction-to-scaling residuals	A controlled crossover, not a theorem merely from collapse	Bare-coupling fits or coarse lattices can shift the apparent exponent
Relative-entropy diagnostic	Two normal states on one regional algebra, or one common type-I regulator	Araki relative entropy; on a regulated factor, D = Δ⟨K⟩ − ΔS	Support condition and reference state fixed; a cross-theory comparison needs an explicit embedding	Region-size dependence, conditioning, continuum refinement, and endpoint asymptotics	Recovery follows only after a restriction or channel is specified and its loss is controlled	Changing only one state’s embedding changes the answer
Coarse-graining channel	Retained regulated subsystem, or a continuum subalgebra	CPTP state map in type I; predual of a normal unital completely positive algebra map in the continuum	Input, output, reference state, side information, and operational task fixed	Finite distinguishability loss under the declared map	Exact sufficiency at zero loss; quantitative recovery under a stated fidelity bound	A Wilsonian change of action without a state map licenses neither data processing nor recovery
Scale-space factorization	Independent real Fourier modes, wavelet factors, or another declared canonical split	Entropy or mutual information after a canonical transformation of both coordinates and momenta	Mode reality constraints, gauge algebra, basis, retained factors, and reconstruction accuracy fixed	Symplectic-spectrum and reconstruction checks under Fourier, Haar, and rotated bases	Only a basis-defined resource statement	A non-block-diagonal basis rotation changes the factorization and can create mode entanglement
Theory-space metric	A smooth regulated source or state family at fixed region	BKM relative-entropy Hessian, fidelity metric, normalized Fisher metric, or partition-function Hessian—named explicitly	Operator basis, contact prescription, boundaries, and directions that leave the chosen physics unchanged are fixed	Connected correlators, coordinate-covariant line elements, null-direction and refinement tests	Contractivity only when an explicit channel acts on the state family	A field-independent contact counterterm can shift ∂i∂j log Z without shifting normalized Fisher information

Where to go next

Continue to Symmetry, Superselection, and Information Resources to specify the symmetry constraints on states, operations, and ancillary systems.

References

• Casini, Horacio, and Marina Huerta. “A c-Theorem for the Entanglement Entropy.” Journal of Physics A: Mathematical and Theoretical 40 (2007): 7031–7036. DOI.