Relative Entropy for QFT States

Relative entropy compares two states only through observables in a declared algebra. In continuum QFT that algebraic formulation is the intrinsic one: it remains meaningful for type-III local algebras, is ultraviolet finite in many local comparisons, and does not require a reduced density matrix.

Required background. Use operator algebras and normal positive functionals and restricted states on subregions. Helpful background. Type-III local algebras explain why the density-matrix formula is not the continuum definition.

Relative modular definition

Let $\omega$ and $\varphi$ be normal positive functionals on a von Neumann algebra $\mathfrak M$, represented in standard form. Their relative modular operator $\Delta_{\varphi,\omega}$ acts on the support selected by the two states. With the usual support convention, Araki relative entropy is

$$S(\omega\Vert\varphi) =-\langle \xi_\omega,\log\Delta_{\varphi,\omega}\,\xi_\omega\rangle,$$

and is $+\infty$ when the required absolute-continuity condition fails. For $\mathfrak M=\mathcal B(\mathcal H)$ this reduces to the Umegaki expression

$$D(\rho\Vert\sigma) =\operatorname{Tr}\rho(\log\rho-\log\sigma)$$

provided $\operatorname{supp}\rho\leq\operatorname{supp}\sigma$. The algebraic definition and its positivity, lower semicontinuity, and monotonicity were established in Araki 1976, pp. 809–833.

The diagram locates this quantity among the operational comparisons developed later. Its arrows are conditional: the algebra, support, allowed tests, and resources must be fixed before an interpretation is chosen.

Relative entropy is the algebraic state-comparison core. Many-copy tests, fidelity bounds, mutual information, and recovery or channel statements require distinct support, copy, algebra, and energy hypotheses. Schematic.

Local comparison in QFT

For a vacuum state $\omega_0$ and a coherent excitation $\omega_f$ restricted to an interval algebra $\mathfrak A(O)$, $S(\omega_f\Vert\omega_0)$ compares every admissible local measurement at once. In situations with a known vacuum modular Hamiltonian $K_O$, a regulated computation often takes the form

$$D(\rho_f^O\Vert\rho_0^O)=\Delta\langle K_O\rangle-\Delta S_O.$$

The separate terms may depend on the cutoff, while their matched difference has a continuum limit. This identity is a regulated representation of the algebraic quantity, not evidence for an intrinsic trace on $\mathfrak A(O)$.

Relative entropy is asymmetric and does not itself equal an error probability. Its usefulness comes from inequalities and asymptotic theorems whose hypotheses are stated on the following pages. It also depends on the observable algebra: enlarging the algebra can only increase distinguishability.

Support and comparison discipline

The lower figure summarizes the conditions that prevent a finite formula from being used outside its domain. In particular, a state with support outside that of the reference has infinite relative entropy; deleting the singular directions changes the question.

The algebra and reference support are part of the relative-entropy problem. Channel monotonicity and operational bounds additionally require a positive physical map and fixed resource constraints. The lower row shows characteristic failures when those hypotheses are changed. Schematic.

In the coherent-state example, first verify normality on the chosen interval algebra and finite modular energy relative to the vacuum. A mismatched reference representation, a projection that discards unsupported components, or a different algebra on the two sides invalidates the comparison.

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

• Umegaki, Hisaharu. “Conditional Expectation in an Operator Algebra, IV: Entropy and Information.” Kodai Mathematical Seminar Reports 14 (1962): 59–85. DOI.