Information Measures Along RG Flows

There is no single quantity called “information along the RG flow.” Entropy, mutual information, relative entropy, negativity, and information metrics answer different operational questions and can use different algebras. A useful flow diagnostic is therefore a fixed family of comparisons with controlled endpoint limits and either a theorem or an explicit information channel that fixes the monotonicity direction.

Required background. UV and IR fixed points supplies the endpoint theory data; separating scales fixes the dimensionless ratios and limit order used below.

Helpful background. Universal scaling functions explains crossover collapse; measure selection matches an information quantity to a physical task.

The chapter overview separates the scale roles, compares the available diagnostics in one comparison table, and states the independent validity gates. This page applies those gates to two observables computed on the same massive lattice ensembles.

A flow is a declared family of comparisons

Near a UV fixed point, a relevant scaling field has eigenvalue $y=d-\Delta>0$. A convenient coordinate is

$$x=g_R(\mu)R^y,$$

with its sign, normalization, and renormalization prescription retained. If one coupling sets the correlation length, then $\xi=A_\pm|g_R|^{-1/y}$ up to corrections to scaling; $x$ and $R/\xi$ are related rather than independent. A family $I(R,g_R)$ becomes an RG diagnostic only after the state, region prescription, regulator removal, and finite counterterms are fixed.

Four comparisons must not be conflated:

1. varying $R$ in one theory, which changes the regional algebra;
2. varying a coupling at fixed region in a common regulator;
3. taking controlled UV and IR endpoint limits;
4. applying a declared channel between specified input and output systems.

Wilsonian momentum-shell integration is not automatically the state restriction associated with a spatial subregion. Data processing applies only after an actual algebra map or state channel has been identified.

Regulated formulas and continuum meanings

On the harmonic-chain regulator, every finite set of sites is type I, so density matrices and von Neumann entropies are literal. In continuum relativistic QFT, sharp local algebras are type III under standard regularity and scaling hypotheses Driessler 1977, pp. 295–297, and the individual $S(A)$ terms then need not exist. The safe interpretation depends on the measure.

Diagnostic	Type-I formula	Continuum qualification	What monotonicity would mean
Raw regional entropy	$S(A)=-\operatorname{tr}\rho_A\log\rho_A$	regulator dependent for a sharp region	none without a subtraction theorem
Mutual information	$I(A:B)=S(A)+S(B)-S(AB)$	for a split pair, Araki relative entropy against the normal product state; it can diverge if the split property fails	contraction under a declared channel, not generic $R$-monotonicity
Relative entropy	$D(\rho\Vert\sigma)=\Delta\langle K_\sigma\rangle-\Delta S$	Araki relative entropy of normal states on one algebra; the value may be $+\infty$	contraction under restriction or a normal channel
Entropic $c$-function	$c_E(R)=3R\,dS/dR$	cutoff-independent interval derivative under the theorem hypotheses	$dc_E/dR\le0$ in a unitary Lorentz-invariant vacuum flow
Disk $\mathcal F$	$\mathcal F(R)=RS'(R)-S(R)$	renormalized disk entropy in $2+1$ dimensions	$\mathcal F'(R)=RS''(R)\le0$ under the circle proof hypotheses

Positive separation alone should not be used as a universal finiteness theorem: the needed algebraic condition is a normal product state, supplied in standard cases by a split inclusion. Likewise, the density-matrix identity for relative entropy is a regulator formula; the algebraic quantity does not require a trace-class density matrix.

In the type-I row, $K_\sigma=-\log\sigma$ and $\Delta\langle X\rangle=\operatorname{tr}(\rho X)-\operatorname{tr}(\sigma X)$. The displayed finite formula assumes $\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma$; otherwise $D(\rho\Vert\sigma)=+\infty$. The relative-modular definition and its density-matrix specialization are given in Araki 1976, § 1, Eqs. (1.1)–(1.2), pp. 809–810. Its contraction under the unital map dual to restriction is Uhlmann 1977, Proposition 18, p. 31; this is a licensed data-processing statement, not a theorem that every change of $R$ is monotone.

Endpoint theorems do not transfer between measures

A candidate $C(R)$ should have controlled endpoints and a proved sign,

$$C(0)=C_{\rm UV}, \qquad C(\infty)=C_{\rm IR}, \qquad R\frac{dC}{dR}\le0.$$

Endpoint agreement alone does not imply the last inequality: a crossover can overshoot. Conversely, a decreasing regulated curve can depend on a moving region or subtraction convention and fail to define a universal quantity.

For vacuum intervals in a unitary Lorentz-invariant $1+1$-dimensional QFT, strong subadditivity gives $d(3RS')/dR\le0$ Casini and Huerta 2007, pp. 7032–7035. For disks in $2+1$ dimensions, the circle construction gives $S''(R)\le0$, hence $\mathcal F'(R)=RS''(R)\le0$ Casini and Huerta 2012, § 4, especially Eqs. (19), (22), and (23). The differential subtraction and its fixed-point normalization are developed in Liu and Mezei 2013, § 2. Four-dimensional irreversibility is the endpoint statement $a_{\rm UV}\ge a_{\rm IR}$; an arbitrary finite spherical entropy is not automatically an interpolating $a$-function Komargodski and Schwimmer 2011, §§ 3–4.

None of these theorems makes separated-region mutual information monotone in $mR$. Mutual information remains valuable because it is positive, geometrically sensitive, and free of the local endpoint divergences when its continuum definition exists.

One massive deformation, two diagnostics

Use the periodic harmonic-chain ground state and covariance prescription defined on separating scales. Set $R=1$ and choose two equal intervals

$$A=[0,R], \qquad B=[R+L,2R+L], \qquad \frac LR=1.$$

The same ensembles provide both

$$c_E^{(h)}=3N_A\frac{S(N_A+h)-S(N_A-h)}{2h}, \qquad h=1,2,$$

and

$$I(A:B)=2S(N_A)-S(A\cup B).$$

The controlled variables are

$$t=mR\in\{0.25,0.35,0.5,0.75,1,1.5,2,3,4,5,6\},$$ $$R/a\in\{16,24,32,48\}, \quad L/R=1, \quad L_{\rm box}/R\in\{12,16\}.$$

Both observables are extrapolated linearly in $(a/R)^2$. For $c_E$, the reported resolution-sensitivity envelope is defined as the sum of the observed fit-window, box-size, and derivative-stencil changes. For mutual information, it sums the fit-window and box-size changes. These are operational convergence diagnostics rather than statistical errors or rigorous bounds on every omitted effect.

$t=mR$	Continuum $c_E$	$\delta c_E$	Continuum $I(A:B)$	$\delta I$
0.25	0.420732	0.009573	0.105129	0.006941
0.50	0.267819	0.000753	0.0477248	0.0003987
1.00	0.112797	0.000011	0.0133221	0.00000129
2.00	0.0190698	0.0000089	0.00135774	$4.1\times10^{-10}$
4.00	0.00046545	0.00000251	0.0000182398	$4.1\times10^{-10}$
6.00	0.00001012	0.00000030	$2.7483\times10^{-7}$	$8.9\times10^{-11}$

Both decrease across this controlled window, but their crossover widths and logical status differ. The sign of $dc_E/dR$ is backed by the interval theorem after the continuum hypotheses are met. The mutual-information decrease is only a result for this fixed equal-interval geometry and this Gaussian state family.

The infrared limits are $c_E\to0$ and $I(A:B)\to0$. In the opposite direction, $c_E\to1$, the central charge of the noncompact scalar, but the periodic scalar zero mode makes the mutual information grow as $m\to0$ rather than approach a finite universal constant. The small-mass expansion and the $\tfrac12\log[-\log m]$ zero-mode divergence are given in Casini and Huerta 2009, § 3.1.1, Eqs. (131)–(132). Thus the benchmark reaches a UV scaling direction, but it does not license a finite UV mutual-information anchor. That limitation is a property of this noncompact scalar family, not a numerical error.

The complete fixed-$t$ rows, including $t=3,5$ for finite-window decay fits, are available as JSON and CSV.

Adversarial test: change the geometry or subtraction

First keep $R/a=48$ and $L_{\rm box}/R=16$ but change the gap along the trajectory according to

$$\frac{L(t)}R=1+\frac{t}{2(1+t)},$$

rounded to the nearest lattice site. The fixed- and moving-geometry results are:

$t$	Fixed $L/R$	Moving $L/R$	Fixed-geometry $I$	Moving-geometry $I$
0.25	1.0000	1.1042	0.105128	0.0942096
0.50	1.0000	1.1667	0.0477248	0.0366557
1.00	1.0000	1.2500	0.0133230	0.00704494
2.00	1.0000	1.3333	0.00135839	0.000304914
4.00	1.0000	1.3958	$1.82975\times10^{-5}$	$6.51541\times10^{-7}$
6.00	1.0000	1.4375	$2.77633\times10^{-7}$	$1.22052\times10^{-9}$

The moving-geometry curve falls faster because increasing the separation suppresses correlations. It is not a second estimate of the same RG observable. Its strongest licensed interpretation is a two-parameter response along the declared path $(mR,L/R)$.

A subtraction test reaches the same conclusion more sharply. Add a finite endpoint term $\alpha\log(m/\mu)$ per boundary, with $\alpha=0.05$ and fixed $\mu=1$. Applied consistently,

$$\begin{aligned} S(A)&\mapsto S(A)+2b,\\ S(B)&\mapsto S(B)+2b,\\ S(AB)&\mapsto S(AB)+4b, \end{aligned} \qquad b=\alpha\log(m/\mu),$$

so mutual information is unchanged. The computed cancellation residual is zero to the twelve reported decimal places at every grid point. If the term is applied to $S(A)$ and $S(B)$ but not $S(AB)$, the apparent mutual information shifts by $4\alpha\log(m/\mu)$; at $t=0.25$ it becomes negative. Positivity itself then diagnoses the inconsistent prescription. A raw entropy trajectory can be conventional, while consistently defined $c_E$ and mutual information survive this local endpoint shift.

Common pitfalls

Treating a density matrix as a continuum local object. Density matrices are literal in the lattice benchmark. In algebraic QFT, state the von Neumann algebra and use Araki relative entropy or a split construction.

Changing geometry while claiming one-variable flow. A curve parameterized by $t$ is not a function of $t$ alone if $L/R$, the shape, or the state prescription also changes.

Promoting a finite-window fit to a theorem. The fixed-geometry mutual information decreases in this dataset, but no general mutual-information RG theorem follows. Report the model, window, observed resolution-sensitivity envelope, and endpoint limitations.

Exercises

1. Prove cancellation of endpoint terms

Suppose each endpoint contributes an arbitrary finite number $b(m/\mu)$. Show that it cancels from the two-interval mutual information and from $c_E$ evaluated by varying $R$ at fixed $m$ and $\mu$.

Solution

Each single interval has two endpoints and the disjoint union has four, so

$$I_b=(S_A+2b)+(S_B+2b)-(S_{AB}+4b)=I.$$

For the interval derivative, $b(m/\mu)$ is independent of $R$ because $m$ and $\mu$ are held fixed. Therefore

$$3R\frac{d}{dR}[S(R)+2b(m/\mu)]=3R\frac{dS}{dR}=c_E(R).$$

Changing $b$ between the three entropy terms or while taking the derivative defines a different, inconsistent observable.

2. Quantify the moving-geometry contamination

At $t=1$, calculate the fractional change produced by moving from $L/R=1$ to $L/R=1.25$. Does it estimate a discretization error?

Solution

The fractional change is

$$\frac{0.01332298-0.00704494}{0.01332298} \approx0.471.$$

The mutual information changes by about $47.1\%$. Both values were computed at the same $R/a=48$ and box ratio, so the difference is not a discretization estimate. It measures a deliberate change of the physical separation.

3. Test monotonicity against the sensitivity envelopes

Use the rows at $t=0.5$ and $t=1$ to check whether the quoted convergence envelopes overlap for either diagnostic. State the strongest conclusion.

Solution

For $c_E$,

$$0.267819-0.000753=0.267066 >0.112797+0.000011=0.112808.$$

For mutual information,

$$0.0477248-0.0003987=0.0473261 >0.0133221+0.00000129=0.0133234.$$

The envelopes are disjoint and both observables decrease between these two controlled points. This supports a benchmark-specific finite-window statement. Only the $c_E$ sign is additionally licensed by the entropic theorem; the mutual-information sign is not promoted beyond this state and geometry family.

References

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11, no. 3 (1976): 809–833. DOI.
• Casini, Horacio, and Marina Huerta. “A c-Theorem for the Entanglement Entropy.” Journal of Physics A: Mathematical and Theoretical 40 (2007): 7031–7036. DOI.
• Casini, Horacio, and Marina Huerta. “Entanglement Entropy in Free Quantum Field Theory.” Journal of Physics A: Mathematical and Theoretical 42 (2009): 504007. DOI.
• Casini, Horacio, and Marina Huerta. “On the RG Running of the Entanglement Entropy of a Circle.” Physical Review D 85 (2012): 125016. DOI.
• Driessler, Werner. “On the Type of Local Algebras in Quantum Field Theory.” Communications in Mathematical Physics 53, no. 3 (1977): 295–297. DOI.
• Komargodski, Zohar, and Adam Schwimmer. “On Renormalization Group Flows in Four Dimensions.” Journal of High Energy Physics 2011, no. 12 (2011): 099. DOI.
• Liu, Hong, and Mark Mezei. “A Refinement of Entanglement Entropy and the Number of Degrees of Freedom.” Journal of High Energy Physics 2013, no. 4 (2013): 162. DOI.
• Uhlmann, Armin. “Relative Entropy and the Wigner–Yanase–Dyson–Lieb Concavity in an Interpolation Theory.” Communications in Mathematical Physics 54, no. 1 (1977): 21–32. DOI.