Positivity, Monotonicity, and Data Processing

Positivity says that a state is never closer to a distinct reference than to itself; data processing says that discarding accessible information cannot improve distinguishability. In QFT both statements are algebraic and powerful, but their channel, support, and representation hypotheses must be visible.

Required background. Use operator-algebraic positive maps and Relative Entropy for QFT States. Helpful background. Markov generators and semigroups give a dynamical source of channels, while regulated Araki limits explain compatible cutoff approximations.

Positivity and equality

For normal states on a fixed von Neumann algebra,

$$S(\omega\Vert\varphi)\geq0,$$

with equality, under the usual support assumptions, exactly when the states agree. If their restrictions to the chosen algebra coincide, the quantity vanishes even if the global preparations differ. Thus equality always concerns the declared observable algebra.

Let $\Phi:\mathfrak N\to\mathfrak M$ be a normal unital completely positive map in the Heisenberg picture. Pulling states back gives $\omega\circ\Phi$ and $\varphi\circ\Phi$ on $\mathfrak N$, and data processing reads

$$S(\omega\circ\Phi\Vert\varphi\circ\Phi) \leq S(\omega\Vert\varphi).$$

Restriction to a subalgebra is the basic QFT example. The result follows from Araki monotonicity; equality is tied to sufficiency and recovery in the sense developed by Petz 1986, pp. 123–131.

The structural diagram shows where this theorem acts: it controls every branch obtained from a legitimate channel, but it does not choose the operational task.

Data processing transports the central relative-entropy comparison through a restriction or physical channel. Hypothesis-testing, fidelity, and recovery interpretations still require their own conventions and resources. Schematic.

Restriction and Gaussian noise

Suppose two states are compared on an interval $O_2$, then restricted to $O_1\subset O_2$. The inclusion of observable algebras produces

$$S_{\mathfrak A(O_1)}(\omega\Vert\varphi) \leq S_{\mathfrak A(O_2)}(\omega\Vert\varphi).$$

Likewise, a Gaussian attenuation channel applied to both regulated field states cannot increase their relative entropy. The channel parameters and the Hamiltonian domain must be identical for the two inputs.

Nonlinear conditioning on a measurement outcome is not the same map. A postselected branch can appear more distinguishable because its success probability has been removed; only the full instrument, including the outcome register, is trace preserving. Comparing different regulators without an explicit channel between them is another common misuse.

A hypothesis checklist

The lower diagram separates four theorem inputs. Verify each one before invoking monotonicity.

Changing the algebra or support changes the states being compared; replacing a channel by postselection changes normalization; changing energy or cutoff changes the admissible resource set. None is covered by the original data-processing inequality. Schematic.

For a semigroup $\Phi_t$, monotonicity holds at each $t$ when $\Phi_t$ is normal, unital, and completely positive. Differentiating it may yield entropy-production inequalities, but only after the generator domain and state differentiability are established.

References

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

Further reading

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11 (1976): 809–833. DOI.
• Lindblad, Göran. “Completely Positive Maps and Entropy Inequalities.” Communications in Mathematical Physics 40 (1975): 147–151. DOI.