Relative Entropy and RG Irreversibility

Relative entropy decreases when observables are discarded, but that theorem compares two states through one declared restriction or channel. In QFT, the states must be normal functionals on a common algebra, or they must be represented by a common regulator whose continuum interpretation is controlled. There is no canonical relative entropy between two abstract Lagrangians.

Required background. Relative entropy in QFT supplies the algebraic definition and finiteness conditions; separating scales fixes the regional and regulator limits.

Helpful background. Information measures along RG flows distinguishes region-size comparisons from RG trajectories.

The chapter’s separation of scale roles prevents a region inclusion from being mislabeled as inverse RG flow. Use the comparison table to identify the comparison and the independent validity gates before assigning an irreversibility interpretation.

Relative entropy on one algebra

For regulated density matrices,

$$D(\rho\Vert\sigma)= \begin{cases} \operatorname{Tr}\rho(\log\rho-\log\sigma), &\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma,\\ +\infty,&\text{otherwise}. \end{cases}$$

When the first line is finite,

$$D(\rho\Vert\sigma) =\Delta\langle K_\sigma\rangle-\Delta S, \qquad K_\sigma=-\log\sigma,$$

where $\Delta$ means the value in $\rho$ minus the value in $\sigma$. Faithfulness of $\sigma$ is a convenient sufficient hypothesis, not a necessary one; support restriction is the exact finite-dimensional condition.

For a continuum local algebra, Araki relative entropy is defined intrinsically by the relative modular operator for normal positive functionals Araki 1976, §1, Eqs. (1.1)–(1.2), pp. 809–810. The separate quantities $S$ and $\langle K_\sigma\rangle$ are generally not intrinsic finite numbers for a type-III algebra. Their difference formula should therefore be read in a split/type-I representative, a common regulator, or a setting in which the subtracted terms have been proved meaningful.

The reference state is part of the question. For a deformed vacuum $\rho_g$ and UV-CFT vacuum $\sigma_0$ restricted to the same ball algebra, $D(\rho_{g,R}\Vert\sigma_{0,R})$ measures distinguishability at resolution $R$. It is asymmetric and is not a metric on actions.

Region inclusion fixes the monotonicity direction

If $A\subset B$, restriction of both normal states from $\mathcal A(B)$ to $\mathcal A(A)$ gives

$$D(\rho_A\Vert\sigma_A) \le D(\rho_B\Vert\sigma_B).$$

Thus distinguishability is nondecreasing as the accessible region grows. Identifying larger $R$ with lower energy and then saying that “information decreases toward the IR” reverses this theorem. An RG monotone needs an additional subtraction, geometry argument, or channel with a separately declared direction.

For the vacuum of a CFT restricted to the ball $B_R$ on $t=0$, the modular Hamiltonian is local:

$$K_{B_R}=2\pi \int_{\lvert\mathbf x\rvert<R}d^{d-1}x\, \frac{R^2-\lvert\mathbf x\rvert^2}{2R}\, T_{00}(0,\mathbf x)+c.$$

The additive constant cancels from $\Delta\langle K\rangle$. Positivity gives the precise vacuum-subtracted bound

$$\Delta S_{B_R} \le 2\pi \int_{\lvert\mathbf x\rvert<R}d^{d-1}x\, \frac{R^2-\lvert\mathbf x\rvert^2}{2R}\, \Delta\langle T_{00}(0,\mathbf x)\rangle.$$

The CFT-vacuum and ball hypotheses are essential; a generic modular Hamiltonian is nonlocal. The local ball kernel is Casini 2008, Eq. (9), and the relative-entropy formulation of the Bekenstein bound is Casini 2008, Eqs. (17)–(20).

Common-regulator Wilson-fermion calculation

Here is a finite application in which the embedding is completely explicit. Use a one-dimensional ring of $N$ cells, two fermionic orbitals per cell, Wilson parameter $r=1$, and antiperiodic momenta. In lattice units,

$$h(k)=\sin k\,\sigma_y+ \left[ma+1-\cos k\right]\sigma_z.$$

Fill the negative-energy band. The critical reference has $m_0=0$ on the same orbitals, ring, momentum grid, and interval. Antiperiodic boundary conditions remove an exact $k=0$ mode at finite $N$. Restrict the two correlation matrices to an interval of $n_R$ complete cells:

$$C_{m,ij}=\left.\langle c_j^\dagger c_i\rangle_m\right|_{B_R}, \qquad C_{0,ij}=\left.\langle c_j^\dagger c_i\rangle_0\right|_{B_R}.$$

For number-conserving fermionic Gaussian states,

$$\begin{aligned} D(\rho_{C_m}\Vert\rho_{C_0}) =\operatorname{tr}\big[&C_m(\log C_m-\log C_0)\\ &+(1-C_m)(\log(1-C_m)-\log(1-C_0))\big]. \end{aligned}$$

This follows by writing each reduced state as the exponential of a quadratic entanglement Hamiltonian; the correlation-matrix construction is derived in Peschel 2003, pp. L205–L208.

Set the physical interval length to $R=1$, take $a=1/n_R$, fix $L_{\rm box}/R=16$, and compare $mR\in\{0.25,0.5,1\}$ at $n_R=4,6,8,10,12,16$. The dedicated JSON dataset is generated by scripts/generate-information-scales-relative-entropy-benchmark.mjs. Its finest results are

$mR$	$R/a$	$D_R$ (nats)	Refinement change	Spectral-floor sensitivity
0.25	16	0.111121	0.003156	0.000152
0.50	16	0.308064	0.012849	0.000597
1.00	16	0.813090	0.051752	0.002315

The refinement column is the absolute change from $R/a=12$ to $16$. The floor column is the maximum change when correlation eigenvalues are clipped at $10^{-12}$, $10^{-13}$, and $10^{-14}$ before taking logarithms. Their sum defines an observed resolution-sensitivity envelope: $0.003307$, $0.013446$, and $0.054067$ nats in the three rows. These deterministic envelopes are diagnostics, not statistical standard deviations or proved upper bounds on the discretization error. Across all 18 rows, the maximum correlation-matrix reconstruction residual is $1.98\times10^{-14}$, the discarded imaginary part is below $1.70\times10^{-16}$, and the momentum-space projector idempotency residual is below $6.67\times10^{-16}$. The growing envelope at $mR=1$ warns that these values are regulated finite-window estimates, not a claimed continuum extrapolation.

As a separate monotonicity check, hold $N=256$ and $ma=0.03125$ fixed and grow the interval. Selected rows are

Interval cells	$mR$	$D_R$ (nats)
2	0.0625	0.011385
4	0.1250	0.034987
8	0.2500	0.104198
12	0.3750	0.196489
16	0.5000	0.308064

Every successive increment in the full eight-row scan is positive; the smallest is $0.023602$ nats. This verifies regional data processing for this regulated family. It does not turn $D_R$ into a decreasing RG charge.

Embedding and reference adversary

At $mR=0.5$, $R/a=8$, and $L_{\rm box}/R=16$, rotate the two orbitals in every retained cell by the real angle $\theta=0.35$. The calculation gives

Comparison	Relative entropy (nats)	Change from base
Original common embedding	0.2746764131	—
Rotate both $C_m$ and $C_0$	0.2746764128	$-3.07\times10^{-10}$
Rotate only $C_m$	14.5352341149	$+14.2605577018$
Keep embedding, change reference to $m_0R=0.1$	0.1445220661	$-0.1301543470$

The simultaneous residual is below the predeclared $10^{-9}$ numerical tolerance, as required by unitary invariance. Rotating only one state changes the field identification and exposes UV-sensitive disagreement with the fixed reference. Changing the reference asks a different discrimination question. The strongest invariant statement is therefore: relative entropy is invariant under a common algebra isomorphism applied to both states; it is not invariant under an independent embedding change or a change of reference state.

Casini, Testé, and Torroba construct a continuum comparison between a CFT vacuum and a relevantly deformed vacuum using a conformal interaction-picture embedding. Their null-Cauchy-surface limit can remove the modular-energy term only in a dimension-dependent convergence window; it is not a license to ignore the embedding or cutoff in an arbitrary theory comparison Casini, Testé, and Torroba 2017, §§2.1–2.4, §3, and Appendix A.

Channel loss and recoverability

For an explicit channel $\mathcal N$ acting on both states,

$$\delta_{\mathcal N}(\rho,\sigma) =D(\rho\Vert\sigma) -D(\mathcal N\rho\Vert\mathcal N\sigma) \ge0.$$

Equality, with the usual support conditions, is equivalent to sufficiency of the channel for the pair and exact recovery by a Petz map. Quantitatively, there is a recovery map depending on $\sigma$ and $\mathcal N$, but not on $\rho$, for which

$$\delta_{\mathcal N}(\rho,\sigma) \ge-2\log f\!\left( \rho,(\mathcal R_{\sigma,\mathcal N}\circ\mathcal N)(\rho) \right),$$

where $f(\rho,\tau)=\operatorname{Tr}\sqrt{\sqrt\rho\tau\sqrt\rho}$ is root fidelity. 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). This is a statement about one pair and one map. It does not by itself produce an $a$-theorem, a local physical inverse, or an energy-feasible recovery protocol.

Exercises

1. Derive the ball entropy bound

Starting from positivity of $D(\rho_{B_R}\Vert\sigma_{B_R})$, where $\sigma$ is the CFT vacuum, derive the displayed weighted-energy inequality. Explain why the additive constant in $K_{B_R}$ drops out.

Solution

The relative-entropy identity gives

$$0\le D(\rho_{B_R}\Vert\sigma_{B_R}) =\Delta\langle K_{B_R}\rangle-\Delta S_{B_R}.$$

Therefore $\Delta S_{B_R}\le\Delta\langle K_{B_R}\rangle$. Substitute the local CFT-vacuum modular Hamiltonian. If $K_{B_R}=K_{B_R}^{\rm local}+c\,1$, then

$$\Delta\langle c\,1\rangle =c(\operatorname{Tr}\rho-\operatorname{Tr}\sigma)=c(1-1)=0.$$

Only the stress-tensor integral remains, giving exactly the weighted-energy bound. For a non-CFT reference or a nonball region, this local substitution is unavailable even though relative-entropy positivity still holds.

2. Fermionic Gaussian relative entropy

Let a faithful number-conserving Gaussian state have correlation matrix $C$ and density matrix

$$\rho_C=Z_C^{-1}e^{-c^\dagger h_Cc}, \qquad h_C=\log[(1-C)C^{-1}].$$

Derive the correlation-matrix formula for $D(\rho_C\Vert\rho_Q)$ without assuming that $C$ and $Q$ commute.

Solution

The normalization can be written

$$\log\rho_C =\operatorname{tr}\log(1-C) +c^\dagger[\log C-\log(1-C)]c.$$

For any one-particle matrix $A$,

$$\operatorname{Tr}_{\rm Fock}(\rho_Cc^\dagger Ac) =\operatorname{tr}(CA).$$

Insert the two logarithms into $D=\operatorname{Tr}\rho_C(\log\rho_C-\log\rho_Q)$. The scalar normalization terms combine with the quadratic expectations to give

$$\begin{aligned} D(\rho_C\Vert\rho_Q)=\operatorname{tr}\big[&C\log C+(1-C)\log(1-C)\\ &-C\log Q-(1-C)\log(1-Q)\big]. \end{aligned}$$

This is the displayed formula. No simultaneous diagonalization was used: the matrix logarithms are defined spectrally and only cyclicity and linearity of the trace enter.

3. Common versus one-sided basis changes

Let $U$ be a unitary on the one-particle interval space. Prove that replacing $(C,Q)$ by $(UCU^\dagger,UQU^\dagger)$ leaves the Gaussian relative entropy unchanged. Why does the same proof fail if only $C$ is rotated?

Solution

Functional calculus gives

$$\log(UCU^\dagger)=U(\log C)U^\dagger,$$

and similarly for $1-C,Q,$ and $1-Q$. Every term in the transformed relative entropy is therefore $U(\cdots)U^\dagger$. Trace invariance gives the original value.

If only $C$ changes, the cross term becomes

$$-\operatorname{tr}\!\left[UCU^\dagger\log Q +(1-UCU^\dagger)\log(1-Q)\right],$$

which cannot generally be conjugated back because $Q$ was not transformed. Equality survives only for special $U$ commuting with the reference data or by accidental state-specific cancellation. The one-sided operation changes the embedding relative to which the two theories are compared.

References

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11 (1976): 809–833. DOI.
• Casini, Horacio. “Relative Entropy and the Bekenstein Bound.” Classical and Quantum Gravity 25 (2008): 205021. DOI; Open preprint.
• Casini, Horacio, Eduardo Testé, and Gonzalo Torroba. “Relative Entropy and the RG Flow.” Journal of High Energy Physics 2017, no. 3 (2017): 089. DOI; Open preprint.
• 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.
• Peschel, Ingo. “Calculation of Reduced Density Matrices from Correlation Functions.” Journal of Physics A: Mathematical and General 36 (2003): L205–L208. DOI; Open preprint.