Results on channels, entropy inequalities and the generalized amplitude-damping channel

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Kraus representation — Theorem 1.2. Every completely positive trace-preserving map between finite-dimensional matrix algebras is X -> sum V_l X V_l^* with sum V_l^* V_l = I, from the spectral decomposition of its Choi operator.
• Diagonal bound and concavity of entropy — Proposition 2.1. The von Neumann entropy of a state is at most the Shannon entropy of its diagonal in any orthonormal basis, and it is concave.
• Subadditivity in a product basis — Corollary 2.2. The entropy of a state on a tensor product is at most the sum of the Shannon entropies of the diagonals of its marginals in chosen bases.
• Holevo information and minimum output entropy — Lemma 3.1. Pure ensembles suffice for the Holevo information, pure inputs attain the minimum output entropy, both are unitarily invariant, and both behave superadditively under tensor products.
• The regularized Holevo information is a limit — Proposition 4.2. chi(N^{(x)n}) is superadditive and bounded by n log_2 of the output dimension, so chi(N^{(x)n})/n converges to its supremum.
• Pure inputs of the damping channel — Lemma 5.2. A pure qubit input with excited population p leaves A_{gamma,nu} with determinant gamma nu (1 - nu) + gamma (1 - gamma)(p - nu)^2.
• Classical capacity of generalized amplitude damping — Theorem 5.3. chi(A^{(x)n}) = n chi(A) for every generalized amplitude-damping channel, chi(A) is an explicit one-variable maximum, and A is additive with every finite-dimensional partner; with the coding theorem, C(A) = chi(A).
• Entropy differences as log-determinant integrals — Lemma 1.1. For positive semidefinite U, V of equal trace, S(U) - S(V) is the integral over r > 0 of [log det(I + rU) - log det(I + rV)] r^-2.
• Joint convexity of a Schur-complement log-determinant — Lemma 2.1. log det(D + Z E^-1 Z^*) - log det D is jointly convex in positive definite D and E, for fixed Z.
• The scalar deficit — Proposition 4.1. The function H(x,y,z) - g(yz), extended homogeneously, is concave and superadditive on the nonnegative octant.
• Convexity of s -> g(s^2) — Lemma 5.2. The entropy of a qubit state as a function of the square root of its determinant is nondecreasing and convex.
• Zero-block entropy inequality — Theorem 1.1. For M = [[A, B], [C, 0]] with block masses x, y, z summing to one, S(MM^*) is at least the weighted block entropies plus g(yz).
• Thermal block entropy inequality — Theorem 2.1. The entropy of the block matrix of a thermal damping branch decomposition is at least the weighted branch entropies plus g(u), by a mixture of two zero-block matrices with a common scalar term.
• Convexity of the one-qubit output entropy — Lemma 1.1. The output entropy e(p) of a pure input of population p is convex in p.
• One damping use next to a partner — Lemma 2.1. For a pure input of A (x) N split by the qubit population p, the output entropy is at least e(p) plus the weighted output entropies of the two branches under N.
• Site-by-site output entropy bound — Corollary 3.2. Every input of n uses of A, entangled or mixed, has output entropy at least the sum of e(p_j) over the site populations.
• Holevo information of many uses — Theorem 4.1. chi(A^{(x)n}) = n chi(A) = (n / ln 2) max_p f(p), attained by products of two equiprobable pure states with opposite phases.
• Holevo additivity with a partner — Theorem 1.1. chi(A (x) N) = chi(A) + chi(N) for every finite-dimensional channel N.
• Minimum output entropy — Theorem 2.1. S_min(A) = g(gamma nu (1 - nu)) and S_min(A (x) N) = S_min(A) + S_min(N).
• Regularized additivity — Theorem 3.1. The regularized Holevo information of A (x) N is chi(A) plus that of N; with the coding theorem, C(A (x) N) = C(A) + C(N).
• The unital damping channel — Corollary 4.3. For nu = 1/2 the Holevo information is 1 - h((1 + sqrt(1 - gamma))/2)/ln 2 bits, attained by the two equal superpositions.

Reading order

Five lessons, to be read in order. Every statement about Holevo information and minimum output entropy is proved in full; the classical coding theorem is the named hypothesis (CT) of the first lesson, used only to state the results for the classical capacity.