Source
Fuchs (1995), PhD thesis, LIT-018, §3.3, read in full in NOTE-021. The result was published separately as Fuchs & Caves (OSID), the thesis’s ref. [149].
What was actually shown
Definition. A quantum distinguishability is a classical measure of distinguishability extremized over every POVM, with the number of outcomes free (Eqs. 3.3–3.7). For the statistical overlap F(p0,p1) = Σ_b √p0(b)√p1(b), the Bhattacharyya coefficient (Eq. 2.40), the quantum version is its minimum over measurements.
Theorem (§3.3.1). min_E Σ_b √(tr ρ0E_b)√(tr ρ1E_b) = tr√(ρ1^{1/2}ρ0ρ1^{1/2}) (Eq. 3.28). The proof bounds each outcome’s term by a Schwarz inequality with a free unitary inserted, sums the measurement away with Σ_b E_b = 1, and fixes the unitary by polar decomposition so the equality condition becomes an eigen-equation (Eqs. 3.54–3.74).
The optimal measurement. Projectors onto the eigenbasis of M = ρ1^{-1/2}√(ρ1^{1/2}ρ0ρ1^{1/2})ρ1^{-1/2} (Eq. 3.67). Its eigenvalues are m_b = (tr ρ0E_b / tr ρ1E_b)^{1/2} (Eq. 3.92), so M acts as an operator likelihood ratio. M is the Kubo–Ando geometric mean ρ1⁻¹ # ρ0 (Eq. 3.106).
Checks. For pure states F = |⟨ψ0|ψ1⟩| (Eq. 3.36). The naive bound tr(ρ0^{1/2}ρ1^{1/2}) is unattainable when the states do not commute (Eqs. 3.45–3.53). F is symmetric, unitarily invariant, between 0 and 1, and multiplicative on products (Eq. 4.58). The Bures distance is d_B² = 2 − 2F (Eq. 3.29).
What could have come out otherwise. The minimum over POVMs could have been strictly below every projective value, or could have had no closed form, as happens for the Rényi overlaps the thesis leaves unsolved (§3.4).
What this does not say
- Anything about trace distance. NOTE-021 corrects the seeded skim: the thesis has no trace-distance results, and the inequalities 1 − F ≤ D ≤ √(1 − F²) come later (Fuchs & van de Graaf, 1999).
- That the fidelity is squared. The convention here is unsquared. Jozsa’s “fidelity” is its square (pp. 49, 115). Formulas that mix the two conventions differ by a square.
- That every quantum distinguishability is solved. The quantum Chernoff bound (Eq. 3.26) and the Rényi overlaps (§3.4) are defined and not solved. The quantum Kullback information has only bounds.
- That the projective version was new. The thesis concedes that the projective-measurement version could already be assembled from Wootters, Alberti and Uhlmann (p. 49). What it claims is the derivation over all POVMs, the statistical reading and the optimal measurement.
- Infinite dimension. The proof is written for finite dimension.
Connections
- Its multiplicativity and monotonicity under partial trace are the properties on which the no-broadcasting theorem rests (THEORY-695, same thesis).
- Quantum thermodynamics’ smooth entropies use this unsquared fidelity (LIT-010, NOTE-011; THEORY-416).