Araki Relative Entropy and Regulated Limits

A cutoff calculation approaches continuum Araki relative entropy only when it approximates the same observable comparison. Increasing matrix size is not enough: the algebras must be related by genuine inclusions or compatible embeddings, the two states must restrict consistently, and the limiting family must generate the target algebra. This page states the exact convergence theorem, realizes its monotonic part on one harmonic-chain regulator, and then shows why a sequence of unrelated truncations can move in either direction.

Required background. Start from Relative Entropy for QFT States, especially its state order and support convention. Helpful background. Lattice-to-continuum entropy explains why cutoff stability is weaker than a continuum theorem.

The chapter’s structural map distinguishes an algebraic state comparison from a chosen matrix representation. Its validity map and claim-comparison table summarize the hypotheses checked below.

The increasing-algebra theorem

Let $(\mathfrak N_i)_{i\in I}$ be a directed increasing net of unital von Neumann subalgebras of $\mathfrak M$, with a common identity and

$$\left(\bigcup_{i\in I}\mathfrak N_i\right)''=\mathfrak M.$$

For fixed normal states $\omega$ and $\varphi$ on $\mathfrak M$, write $\omega_i=\omega|_{\mathfrak N_i}$ and $\varphi_i=\varphi|_{\mathfrak N_i}$. Then

$$S_{\mathfrak N_i}(\omega_i\Vert\varphi_i) \nearrow S_{\mathfrak M}(\omega\Vert\varphi)$$

in the extended interval $[0,+\infty]$. Thus a finite target is approached from below, whereas an infinite target means that the restricted relative entropies eventually exceed every finite bound. Faithfulness and conditional expectations are not required for this statement; support failures are included through the $+\infty$ convention.

There are two ideas behind the theorem. First, restriction from a larger algebra to a smaller one is a data-processing map, so the net is nondecreasing and bounded above by the full relative entropy. Second, Kosaki’s variational formula expresses relative entropy as a supremum over finitely many algebra elements. Directedness places every finite test inside some $\mathfrak N_i$, while the bicommutant condition supplies density. This gives the missing lower bound. A modern statement and proof are Hiai 2018, Theorem 4.1(v), pp. 102202-11–102202-12; the variational route is Kosaki 1986, Theorem 3.2 and Remark 3.3, pp. 344–345.

This result should not be attributed to Araki’s 1976 definition paper alone. Araki 1977, Theorems 3.8–3.9, pp. 181–182 and 188–189 establishes the key approximation bounds under the hypotheses available there; later monotonicity and variational formulations give the unrestricted modern statement above.

When density matrices compute the theorem

If an approximating algebra is finite-dimensional and type I, its restricted states are represented by density matrices $\rho_i$ and $\sigma_i$, and

$$S_{\mathfrak N_i}(\omega_i\Vert\varphi_i) =D(\rho_i\Vert\sigma_i) =\operatorname{Tr}\rho_i(\log\rho_i-\log\sigma_i)$$

whenever $\operatorname{supp}\rho_i\leq\operatorname{supp}\sigma_i$. The trace formula is then a representation of the algebraic term on the left of the convergence theorem. It is not a new definition of continuum entropy. For trace-class density operators on an infinite-dimensional type-I algebra, the first branch is the extended Umegaki functional and may still be $+\infty$ despite support inclusion.

Before comparing successive cutoff values, verify all of the following:

1. The arrows are literal unital inclusions in one ambient algebra, or specified normal faithful embeddings that compose.
2. Both state pairs are compatible restrictions through those same arrows.
3. The directed union generates the stated target algebra.
4. Support and relative-modular form-domain conditions are checked rather than repaired by deleting small eigenvalues.
5. Numerical conditioning, finite-volume effects, discretization, and continuum extrapolation are reported separately.

Ordinary lattice refinements usually change the Hilbert space, Hamiltonian, counterterms, and state. They do not satisfy items 1–2 merely because $h$ decreases. Likewise, a split inclusion supplies an intermediate type-I factor, but factors selected for different collar widths are not automatically nested. Collar shrinkage therefore needs an additional construction before the theorem applies; lower semicontinuity by itself gives no equality claim.

A controlled coherent-state benchmark

Consider the massless real scalar in $1+1$ dimensions at $t=0$. The continuum interval is $I_R=(-R,R)$, and the time-symmetric coherent data are

$$F(x)= \begin{cases} A\exp\!\left[-\dfrac{x^2}{r^2-x^2}\right],&|x|<r,\\[4pt] 0,&|x|\geq r, \end{cases} \qquad G(x)=0,$$

with $R=1$, $r=1/2$, and $A=1$. The support lies strictly inside the interval. For the vacuum as the second argument, the exact continuum formula is

$$S_{I_R}(\omega_F\Vert\omega_0) =2\pi\int_{-R}^{R}\frac{R^2-x^2}{2R} \frac{[F'(x)]^2+[G(x)]^2}{2}\,dx \approx8.415964488746008\ \text{nats}.$$

The energy density and vacuum bounded-interval relative entropy are Garbarz and Palau 2023, Eq. (91), p. 125016-9, and § IV.B.2, Eq. (95), p. 125016-10. A 90-digit quadrature begins $8.4159644887460076\ldots$; the displayed decimal is its binary64 rendering. Two independent JavaScript quadratures agree with it within $1.7\times10^{-13}$ nats.

For a finite regulator, place the theory in a Dirichlet box $[-B,B]$ with spacing $h$ and canonical variables

$$q_j=\sqrt h\,\Phi(x_j), \qquad p_j=\sqrt h\,\Pi(x_j), \qquad [q_j,p_k]=i\delta_{jk},$$

so that

$$H_h=\frac12(p^Tp+q^TKq), \qquad K_{jk}=\frac{2\delta_{jk}-\delta_{j,k+1}-\delta_{j,k-1}}{h^2}.$$

The vacuum covariance is $V=\tfrac12\operatorname{diag}(K^{-1/2},K^{1/2})$. Restrict one global vacuum and one global displacement $d_q=\sqrt h F$, $d_p=0$ to nested centered site sets. Let $\sigma_L$ be the restricted vacuum and $K_{\sigma,L}=-\log\sigma_L$. Because a Weyl displacement changes first moments but not covariance, $\Delta S_L=0$ exactly, and the Gaussian modular kernel $\mathcal G_L$ gives

$$D(\rho_L\Vert\sigma_L) =\Delta\langle K_{\sigma,L}\rangle =\frac12 d_L^T\mathcal G_Ld_L.$$

This equal-covariance specialization follows from the Gaussian relative-entropy formula in Wilde et al. 2017, Eqs. (3)–(6) and (9), pp. 120501-2–120501-3.

At fixed $B=8$, $h=1/16$, and grid phase zero, the half-lengths $L=0.25,0.50,0.75,1.00$ give respectively

$$D_L=0.745892, 2.714598, 5.759384, 8.143612\ \text{nats}.$$

All seven recorded nested intervals are nondecreasing; the smallest step is $0.433881$ nats. This conclusion survives symplectic-gap conditioning floors $10^{-8}$, $10^{-10}$, and $10^{-12}$. It is a genuine finite-dimensional DPI test because every row is a principal restriction of the same 255-site covariance and displacement.

What the regulator data do—and do not—show

The endpoint comparison is less precise than the monotonicity check. At the preferred $B=8$, $h=1/16$ regulator, changing the symplectic-gap conditioner moves $D_{L=1}$ from $7.852675$ to $8.277415$ nats, a spread of $0.424740$ nats. Shifting the grid by half a lattice cell gives $8.378718$ nats, while changing $B$ also produces visible drift. Near-pure entanglement modes make the binary64 evaluation of $\operatorname{arccoth}(2\nu)$ ill-conditioned when $\nu$ is extremely close to $1/2$.

The strongest supported statement is therefore deliberately split:

• Exact continuum formula: the analytic coherent-state integral has 90-digit numerical value beginning $8.4159644887460076\ldots$ nats; $8.415964488746008$ is its binary64 rendering.
• Theorem-backed finite-regulator result: nested site algebras on one master chain give nondecreasing $D_L$ for every tested conditioner.
• Empirical diagnostic only: the $h$, $B$, grid-phase, and conditioning sweeps are compatible with the continuum scale, but they do not constitute precision evidence for a continuum limit.

The complete model, site ranges, covariance conventions, raw symplectic diagnostics, quadratures, and all control sweeps are in the reproducible interval benchmark. This separation of claims is more informative than quoting the closest lattice number as an error estimate.

A dimension-growing counterexample

Matrix dimension does not order information. Let $\mathfrak M=\mathbb C^4$ with

$$p=(0.6,0.2,0.1,0.1), \qquad q=(0.1,0.2,0.3,0.4).$$

Coarse-grain first by the partition $\{1\}|\{2,3,4\}$, then by the three-cell partition $\{1,2\}|\{3\}|\{4\}$, and finally use the full four-cell algebra. The corresponding classical relative entropies are

$$D_1=0.750684, \qquad D_2=0.536173, \qquad D_3=0.826565\ \text{nats}.$$

Although the dimensions grow $2\to3\to4$, the first two algebras are incomparable: neither partition refines the other. Each $D_i$ is at most the full $D_3$, but there is no theorem ordering $D_1$ and $D_2$. The observed drop is therefore not a violation of data processing; it diagnoses the missing inclusion.

For the operator-algebraic domain and approximation theorems behind these comparisons, continue to the rigorous noncommutative treatment.

Common pitfalls

Equating larger matrices with larger algebras. A dimension count supplies no embedding and no compatible restriction of states. Write the arrows and test them.

Using one stable digit as a continuum error bar. Cutoff, box, grid, and numerical-conditioning errors are different controls. Vary them independently and report the least favorable relevant sensitivity.

Assuming a split collar gives a canonical nested sequence. Each split inclusion may admit a type-I interpolation without those interpolations forming a directed family. Prove nesting or downgrade the claim to an approximation proposal.

Confusing convergence of correlators with convergence of relative entropy. Relative entropy is lower semicontinuous, not generally continuous. The theorem uses compatible restrictions and dense generation, not a few converged $n$-point functions.

Exercises

1. Recover the upper bound

For $\mathfrak N_i\subset\mathfrak M$, use data processing to show $S_{\mathfrak N_i}(\omega_i\Vert\varphi_i)\leq S_{\mathfrak M}(\omega\Vert\varphi)$. Why does this alone not prove convergence?

Solution

The inclusion $\iota_i:\mathfrak N_i\hookrightarrow\mathfrak M$ is a normal unital $*$-homomorphism. Pulling both states back through $\iota_i$ is restriction, so data processing gives the inequality. It supplies only an upper bound. To reach the full value, the family must be directed and its union must generate $\mathfrak M$; otherwise observables that distinguish the states can remain absent forever.

2. Check the incomparable partitions

Compute $D_1$, $D_2$, and $D_3$ for the distributions above. Identify exactly which implication fails when $D_2<D_1$.

Solution

Aggregating the cells gives

$$\begin{aligned} D_1&=0.6\log6+0.4\log\frac49=0.750683595\ldots,\\ D_2&=0.8\log\frac83+0.1\log\frac13+0.1\log\frac14 =0.536172737\ldots,\\ D_3&=0.6\log6+0.1\log\frac13+0.1\log\frac14 =0.826565017\ldots. \end{aligned}$$

The two-cell partition is not a subalgebra of the three-cell partition, so no restriction channel connects the first comparison to the second. Only the individual bounds $D_1,D_2\leq D_3$ survive.

3. Separate theorem and extrapolation

Suppose $D_L$ increases with the number of retained sites at fixed $(B,h)$, while values at fixed physical $L$ oscillate as $h$ changes. Which observation is protected by data processing, and what additional information is needed for a continuum claim?

Solution

Only the growing-region result at fixed $(B,h)$ is protected: those regions are nested subalgebras of one finite system and the states are common restrictions. Different $h$ values define different regulated systems. A continuum claim needs an explicit common embedding or another convergence theorem, plus independent discretization, volume, grid, support, and numerical-conditioning controls.

References

• Araki, Huzihiro. “Relative Entropy for States of von Neumann Algebras II.” Publications of the Research Institute for Mathematical Sciences 13, no. 1 (1977): 173–192. DOI.
• Garbarz, Alan, and Gabriel Palau. “Relative Entropy of an Interval for a Massless Boson at Finite Temperature.” Physical Review D 107 (2023): 125016. DOI. Open preprint.
• Hiai, Fumio. “Quantum $f$-Divergences in von Neumann Algebras. I. Standard $f$-Divergences.” Journal of Mathematical Physics 59 (2018): 102202. DOI. Open preprint.
• Kosaki, Hideki. “Relative Entropy of States: A Variational Expression.” Journal of Operator Theory 16 (1986): 335–348. Open PDF.
• Wilde, Mark M., Marco Tomamichel, Seth Lloyd, and Mario Berta. “Gaussian Hypothesis Testing and Quantum Illumination.” Physical Review Letters 119 (2017): 120501. DOI. Open PDF.