Araki Relative Entropy and Regulated Limits

A regulated density-matrix relative entropy approximates Araki relative entropy only when the regulated algebras and states converge in a controlled way. Agreement at a few cutoff values is insufficient: the inclusions, representation, support, and topology of convergence must be part of the limiting statement.

Required background. Start from Relative Entropy for QFT States. Helpful background. Lattice-to-continuum entropy supplies the distinction between a regulated answer and its continuum target.

Algebraic target and approximants

Let $\mathfrak M_1\subset\mathfrak M_2\subset\cdots\subset\mathfrak M$ be an increasing family whose union is weakly dense in $\mathfrak M$. Restrict normal states $\omega,\varphi$ to $\mathfrak M_n$. Monotonicity gives

$$S(\omega|_{\mathfrak M_n}\Vert\varphi|_{\mathfrak M_n}) \leq S(\omega|_{\mathfrak M_{n+1}}\Vert\varphi|_{\mathfrak M_{n+1}}) \leq S(\omega\Vert\varphi).$$

Under the standard normality and density hypotheses, the increasing limit reaches the Araki relative entropy; see Araki 1976, pp. 809–833. A lattice or split approximation may realize each $\mathfrak M_n$ as type I and thereby replace its restriction by density matrices $\rho_n,\sigma_n$. The trace formula then computes the left-hand side, not a separate continuum entropy.

The chapter diagram makes the relation explicit: regulated matrices are one representation of the central algebraic comparison, while operational branches require further resources.

A monotone, representation-compatible sequence of type-I approximants can converge to Araki relative entropy. The other branches are not automatic consequences of having a cutoff. Schematic.

A controlled interval limit

Take a scalar field on lattices of spacing $a_n\downarrow0$ and fix a physical interval $O$. Choose site algebras $\mathfrak M_n$ nested under an explicit embedding and states whose local $n$-point functions converge normally. If the reference is faithful on each support, compute

$$D_n=\operatorname{Tr}\rho_n(\log\rho_n-\log\sigma_n).$$

The claim $\lim_nD_n=S(\omega_O\Vert\varphi_O)$ is licensed only after showing that the embeddings approximate $\mathfrak A(O)$ and that the state restrictions correspond under them. A sequence of matrices whose dimensions merely grow need not define any common algebraic limit.

The same caution applies to split inclusions. Sending the collar width to zero can recover a sharp local comparison, but the interpolating type-I factor depends on the collar and need not be unique. Relative entropy is stable when the induced restrictions converge to the same normal functionals; the intermediate density matrices themselves need not converge in trace norm.

Failure modes

The diagram below highlights the data that must remain fixed. A particularly deceptive failure is nonmonotone truncation: discarding high-energy modes at each step may lower the computed relative entropy without producing a channel or inclusion relating successive approximants.

Convergence to Araki relative entropy requires compatible algebras, normal states, support, and embeddings. Projecting away unsupported directions, changing representation, or varying the cutoff prescription without a connecting channel can produce a finite sequence with the wrong limit. Schematic.

For every numerical continuum extrapolation, report the physical region, regulator family, embeddings, reference state, support test, and residual cutoff dependence. These data distinguish a theorem about a local algebra from an uncontrolled extrapolation of matrices.

References

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11 (1976): 809–833. DOI.

Further reading

• Petz, Dénes. “Sufficient Subalgebras and the Relative Entropy of States of a von Neumann Algebra.” Communications in Mathematical Physics 105 (1986): 123–131. DOI.