Fidelity, Chernoff Bounds, and State Overlap

Fidelity quantifies state overlap, while the quantum Chernoff exponent gives the optimal symmetric many-copy discrimination rate. Both extend beyond finite matrices, but their normalization, support, algebra, and resource interpretation must be declared before they are compared.

Required background. Start from Relative Entropy for QFT States. Helpful background. Hypothesis Testing and Asymptotic Distinguishability fixes the error model behind asymptotic exponents.

Fidelity and transition probability

This page uses the unsquared fidelity

$$F(\rho,\sigma)=\left\lVert\sqrt{\rho}\sqrt{\sigma}\right\rVert_1, \qquad 0\leq F\leq1.$$

Some literature calls $F^2$ the fidelity. A numerical value is ambiguous unless the convention is stated. For normal states on a general algebra, Uhlmann transition probability is the supremum squared overlap of vector representatives in a common representation; its square root agrees with the convention above in type I. The algebraic construction originates in Uhlmann 1976, pp. 273–279.

Fidelity is symmetric and remains finite even when relative entropy diverges. It measures closeness, not an asymmetric penalty relative to a reference. The structural figure shows this separate overlap branch.

Fidelity is a symmetric one-pair overlap; the Chernoff quantity becomes an optimal symmetric error exponent only for independently repeated tests. Both require a fixed algebra and convention. Schematic.

Chernoff discrimination

For density operators, the quantum Chernoff distance is

$$\xi_{\rm QCB}(\rho,\sigma) =-\log\inf_{0\leq s\leq1} \operatorname{Tr}\rho^s\sigma^{1-s}.$$

For equal priors and $n$ independent copies, the optimal average error obeys $p_{\rm err}^{(n)}\asymp e^{-n\xi_{\rm QCB}}$, as proved by Audenaert et al. 2007, pp. 160501-1–160501-4. This is not a statement about a single continuum field or about correlated spatial regions.

For displaced thermal Gaussian states of finitely many regulated modes, $F$ and the Chernoff overlap can be computed from covariance matrices and displacement vectors. A continuum statement requires convergence of the selected mode algebra and a bound on ultraviolet or high-energy tails. If the states become disjoint in the limiting representation, a finite regulated overlap need not survive.

Trace distance and fidelity are related by the Fuchs–van de Graaf inequalities,

$$1-F(\rho,\sigma) \leq \tfrac12\lVert\rho-\sigma\rVert_1 \leq\sqrt{1-F(\rho,\sigma)^2}.$$

These inequalities inherit the stated unsquared convention.

Conventions and high-energy failure

The lower diagram identifies four ways an overlap comparison can change meaning.

Square versus unsquared fidelity, inequivalent representations, varying mode algebras, and unbounded high-energy inputs can all change an overlap result. A Chernoff exponent also requires the same pair of states on each independent copy. Schematic.

When reporting a QFT overlap, name the algebra, state class, fidelity convention, regulator or split realization, energy control, and whether the statement concerns one copy or an asymptotic exponent.

References

• Audenaert, Koenraad M. R., et al. “Discriminating States: The Quantum Chernoff Bound.” Physical Review Letters 98 (2007): 160501. DOI. Open preprint.
• Uhlmann, Armin. “The ‘Transition Probability’ in the State Space of a *-Algebra.” Reports on Mathematical Physics 9 (1976): 273–279. DOI.

Further reading

• Fuchs, Christopher A., and Jeroen van de Graaf. “Cryptographic Distinguishability Measures for Quantum-Mechanical States.” IEEE Transactions on Information Theory 45 (1999): 1216–1227. DOI.