Hypothesis Testing and Asymptotic Distinguishability

Quantum hypothesis testing turns state comparison into a decision problem with declared errors and measurement resources. Relative entropy emerges as an asymptotic exponent for independent copies; a single QFT region, a detector-limited experiment, or an energy-constrained field theory is not automatically in that asymptotic regime.

Required background. Start from Relative Entropy for QFT States.

One-shot binary tests

For density operators $\rho$ and $\sigma$, a two-outcome effect $0\leq Q\leq1$ accepts $\rho$. The type-I and type-II errors are

$$\alpha(Q)=\operatorname{Tr}[(1-Q)\rho], \qquad \beta(Q)=\operatorname{Tr}(Q\sigma).$$

At tolerance $0<\varepsilon<1$, define

$$D_H^\varepsilon(\rho\Vert\sigma) =-\log\inf_{\alpha(Q)\leq\varepsilon}\beta(Q).$$

On a von Neumann algebra, replace traces by the normal functionals $\omega(1-Q)$ and $\varphi(Q)$ with $Q\in\mathfrak M$. The algebra then specifies all allowed tests. Smooth min- and max-relative entropies optimize related divergences over a metric ball; the metric and smoothing convention must be stated.

The structural figure places the testing branch relative to fidelity, correlation, and recovery measures. Only the testing branch directly encodes the two error probabilities.

Hypothesis-testing divergence is operational for a declared test algebra and error tolerance. Its relative-entropy limit requires an independent-copy resource; overlap, correlation, and recovery branches are related but not interchangeable. Schematic.

Many-copy limit and fluctuations

For faithful finite-dimensional states and fixed $\varepsilon$,

$$\lim_{n\to\infty}\frac1n D_H^\varepsilon(\rho^{\otimes n}\Vert\sigma^{\otimes n}) =D(\rho\Vert\sigma),$$

the quantum Stein lemma proved through Hiai and Petz 1991, pp. 99–114 and the strong converse of Ogawa and Nagaoka 2000, pp. 2428–2433. When the relative-entropy variance

$$V(\rho\Vert\sigma) =\operatorname{Tr}\rho \bigl(\log\rho-\log\sigma-D(\rho\Vert\sigma)\bigr)^2$$

is finite, it controls the leading $\sqrt n$ correction. Infinite-dimensional use additionally needs support, moment, and approximation control.

For a smeared Gaussian excitation, one may define a detector algebra and energy bound, repeat an independently prepared experiment, and verify convergence of the regulated exponents. A single vacuum field does not contain freely addressable tensor copies merely because disjoint regions have been drawn.

Resource failures

The lower figure records what must remain fixed across the asymptotic sequence.

Stein asymptotics do not cover tests whose detector support, smoothing metric, cutoff, or energy window changes with copy number unless that variation is included in a new theorem. Projecting away ultraviolet tails can also alter both errors. Schematic.

Report the preparation model for independent copies, the effect algebra, $\varepsilon$, the energy constraint, and the order of $n\to\infty$ and cutoff removal. These choices determine whether the result is one-shot, second-order, or genuinely asymptotic.

References

• Hiai, Fumio, and Dénes Petz. “The Proper Formula for Relative Entropy and Its Asymptotics in Quantum Probability.” Communications in Mathematical Physics 143 (1991): 99–114. DOI.
• Ogawa, Tomohiro, and Hiroshi Nagaoka. “Strong Converse and Stein’s Lemma in Quantum Hypothesis Testing.” IEEE Transactions on Information Theory 46 (2000): 2428–2433. DOI. Open preprint.