Metadata
bibkey: datta2014conditionalsubentropy
authors: Nilanjana Datta, Tony Dorlas, Richard Jozsa, Fabio Benatti
year: 2014
title: "Properties of subentropy"
doi: 10.48550/arXiv.1310.1312
url: https://arxiv.org/abs/1310.1312v1
claim: "Conjecture 2 bounds conditional subentropy of every finite classical-quantum state by minus the natural logarithm of its optimal guessing probability."
strata_touched:
- D5/S3/Quantum/Measurement/ConditionalSubentropyGuessingBound
license: citation-only
triage: anchor
Verified locator
DOI: 10.48550/arXiv.1310.1312
Source: https://arxiv.org/abs/1310.1312v1
The versioned source is the original paper, published as J. Math. Phys. 55, 062203, DOI 10.1063/1.4882935. The simplex-integral expression is equation (4), PDF page 2; conditional subentropy is equation (13), PDF page 8; Conjecture 2 is on PDF page 9.
Source definitions
Equation (2): “Q(ρ) = ⟨H(M(ρ))⟩_M − C_n where C_n := 1/2 + 1/3 + … + 1/n.”
Equation (3): “this uniform measure (normalised to have total volume unity) is given by dx = (n − 1)! dx_1 … dx_{n−1}.”
Equation (4): “Introducing η(y) := −y ln y, we then obtain a third expression for the subentropy from eq.(2): Q(ρ) = n ∫_{Δn} η(λ · x) dx − C_n.”
Equation (13): “we define the conditional subentropy to be Q(A|B)_ρ := Q(ρ_AB) − Q(ρ_B)”.
The source uses natural logarithms. The simplex has coordinates with nonnegative entries and sum at most one; its last barycentric coordinate is one minus their sum. Zero and repeated eigenvalues are included in equation (4).
Conjecture 2
“For a c-q state ρ_XB, the conditional subentropy Q(X|B) satisfies the following bound: Q(X|B) ≤ H_min(X|B) = − ln p_guess(X|B).”
“p_guess(X|B) = max_{{E_x}:POVM} Σ_x p_x Tr(E_x ρ^x_B), where the maximization is over all possible POVMs: {E_x : E_x ≥ 0 ∀ x; Σ_x E_x = I}.”
The matrix family is positive semidefinite and has total trace one. Its classical-quantum state is the block diagonal matrix, and its marginal is the sum of the blocks. The Lean expression uses the supremum of exactly these POVM scores.
Attribution boundary
The source definitions and conjecture are literature statements. The simplex-to-Gaussian correspondence and the resulting proof are repository derivations. Completeness of the literature screen remains ASSUMED-UNVERIFIED; the cited source proves the two-pure-signal-state case in Proposition 6.