Type-III Information Tasks, Split-Regulated Entropy, and Energy Constraints

Sharp local algebras in relativistic QFT are generally type III, so they have no normal trace, no density matrix intrinsic to the region, and no finite von Neumann entropy obtained by tracing out its complement. Well-posed substitutes include Araki relative entropy, mutual information defined as relative entropy, split-collar approximations, and channel distances restricted to an energy-bounded state class. These objects answer different questions and must not be interchanged.

Required background. Type-III local algebras and entropy limits establish the obstruction, while Araki relative entropy supplies an intrinsic quantity. Helpful background. Split inclusions introduce a collar, and noncommutative divergences supply energy-truncated bounds. See species and regulator dependence, continuum type-III obstacles, fine- and coarse-grained entropy, horizon entanglement, and species, gauge, and edge terms.

A domain table for type-III tasks

Let $\mathcal A(O)$ be a type-III local factor. The following distinctions determine which claim exists.

• $S_{\mathcal A(O)}(\omega\Vert\sigma)$ is intrinsic for normal positive functionals, extended to $+\infty$ when supports or modular integrability fail.

• For separated commuting algebras $\mathcal A$ and $\mathcal B$ with the split property, the mutual information is

  $$I_\omega(\mathcal A:\mathcal B) =S_{\mathcal A\vee\mathcal B} (\omega\Vert\omega_\mathcal A\otimes\omega_\mathcal B).$$

  The product functional must be normal on the joined algebra; split supplies that fact.

• A split entropy chooses an intermediate type-I factor $\mathcal F$ and computes a density-matrix entropy there. It depends on $\mathcal F$ and the collar and is not an intrinsic entropy of $\mathcal A(O)$.

• An energy-constrained channel norm declares a Hamiltonian $H$, bound $E$, and state set $\mathfrak S_E=\{\rho:\rho(H)\leq E\}$. Finiteness on $\mathfrak S_E$ does not imply an unconstrained diamond-norm statement.

For two Schrödinger channels $\Phi_*,\Psi_*$, a typical precise choice is

$$\lVert\Phi_*-\Psi_*\rVert_{\diamond,E} =\sup_{\substack{\rho_{RA}\ \mathrm{normal}\\ \rho_A(H)\leq E}} \left\lVert (\operatorname{id}_R\otimes\Phi_*)(\rho_{RA}) -(\operatorname{id}_R\otimes\Psi_*)(\rho_{RA}) \right\rVert_1.$$

The reference system is retained so that ancillary distinguishability is tested. Changing $H$, its ground-energy normalization, or the allowed state class changes the quantity. A finite bound at each $E$ says nothing about the limit $E\to\infty$ without a uniform estimate.

The absence of a trace is not an inconvenience cured by picking a basis. It is invariant under isomorphism of the factor. Araki’s modular definition is designed precisely to avoid that missing structure Araki 1976, §§1–4, pp. 810–824.

This classification also fixes the order of comparison. First choose the sharp algebras and normal states, then decide whether the task is intrinsic, split regulated, or energy constrained. Only in the second case is an intermediate type-I density operator part of the object. Agreement of two regulators in one limit must be proved; it is not implied by the common sharp algebra they are meant to approximate.

Two intervals in the massless Dirac net

Consider the vacuum of the full massless Dirac field in $1+1$ dimensions and two equal intervals

$$A=(0,L),\qquad B=(L+\delta,2L+\delta), \qquad L,\delta>0.$$

Positive separation gives a split pair. The vacuum mutual information, understood as the relative entropy of the joint state against the normal product of its restrictions, is

$$I_0(A:B)=\frac13\log \frac{(L+\delta)^2}{\delta(2L+\delta)}.$$

This is the two-interval specialization of the exact extensive entropy formula for a free massless Dirac field Casini and Huerta 2009, §2, equations (2)–(4), article 048. It is finite for every collar $\delta>0$, positive because $(L+\delta)^2-\delta(2L+\delta)=L^2$, and tends to zero as $L/\delta\to0$. These are independent checks of positivity and clustering.

For a finite-energy family $\mathfrak S_E$, one may ask for a uniform bound on

$$\sup_{\omega\in\mathfrak S_E} S(\omega_{AB}\Vert\omega_A\otimes\omega_B)$$

or compare channels using a supremum restricted to $\mathfrak S_E$. Such a bound requires an explicit Hamiltonian and a phase-space or nuclearity estimate; the vacuum formula alone proves no uniform statement for excited states. The construction illustrates the role of the split property: it makes the product reference normal at positive separation while leaving the sharp local algebras type III.

Removing the collar

At fixed $L$,

$$I_0(A:B)=\frac13\log\frac{L}{2\delta}+O(\delta/L) \qquad(\delta\downarrow0).$$

Thus the mutual information diverges as the collar closes. A density-matrix entropy computed in an intermediate type-I factor likewise acquires regulator and collar dependence. This divergence does not imply that every algebraic relative entropy diverges: for example, $S(\omega\Vert\omega)=0$ on the sharp factor, and suitably matched state pairs may have finite relative entropy.

The adversarial failure is therefore twofold. First, hold a particular ultraviolet prescription fixed and send $\delta$ to zero; the split-regulated entropy need not converge. Second, infer from that divergence that the intrinsic Araki relative entropy of any two normal states is infinite; the identical-state counterexample refutes the converse immediately. Energy constraints control only the declared state class and do not manufacture a local trace.

Exercises

1. Check the limiting regimes. Expand the exact mutual information for $\delta\gg L$ and for $\delta\ll L$.

Solution

For $x=L/\delta\ll1$, the logarithm is $\log((1+x)^2/(1+2x))=x^2+O(x^3)$, so $I_0=L^2/(3\delta^2)+O(L^3/\delta^3)$. For $\delta/L\ll1$, it is $\log(L/(2\delta))+O(\delta/L)$, giving the stated logarithmic divergence.

2. Why split is needed. Explain why the formal symbol $\omega_A\otimes\omega_B$ is insufficient without a split or another normal-product-state theorem.

Solution

The algebraic tensor product always admits a product functional, but relative entropy is being taken on the von Neumann algebra $\mathcal A\vee\mathcal B$ in a specified representation. Without split, the product functional need not extend normally to that join, so the required normal reference state—and hence the displayed Araki relative entropy—may not exist there.

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, and Marina Huerta. “Remarks on the Entanglement Entropy for Disconnected Regions.” Journal of High Energy Physics 2009, no. 3 (2009): 048. DOI.