The classical capacity of a quantum channel generally requires regularization because input codewords may be entangled across channel uses. We remove this regularization for generalized amplitude-damping channels (GADCs) and determine their exact strong-converse exponent throughout the full parameter range. The exponent is a single-letter transform of the sandwiched Rényi Holevo information and is attained by product input codewords with collective decoding. The key ingredient is a decomposition of the joint output into positive operators whose triangular factors have identical scalar Gram matrices after norm compression. A sharp triangular Schatten-norm compression inequality then establishes maximum output norm multiplicativity for diagonally sandwiched GADCs tensored with arbitrary completely positive maps. This yields additivity of the sandwiched Rényi Holevo information at every order above one. Consequently, the classical capacity equals the known one-copy Holevo information and admits a scalar optimization. Thus, entangled input codewords improve neither the capacity nor the optimal exponential decay of decoding success above capacity.
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v2: determining the exact capacity and its strong converse exponent
The classical capacity of a quantum channel generally requires regularization because input codewords may be entangled across channel uses. We remove this regularization for generalized amplitude-damping channels (GADCs) and determine their exact strong-converse exponent throughout the full parameter range. The exponent is a single-letter transform of the sandwiched Rényi Holevo information and is attained by product input codewords with collective decoding. The key ingredient is a decomposition of the joint output into positive operators whose triangular factors have identical scalar Gram matrices after norm compression. A sharp triangular Schatten-norm compression inequality then establishes maximum output norm multiplicativity for diagonally sandwiched GADCs tensored with arbitrary completely positive maps. This yields additivity of the sandwiched Rényi Holevo information at every order above one. Consequently, the classical capacity equals the known one-copy Holevo information and admits a scalar optimization. Thus, entangled input codewords improve neither the capacity nor the optimal exponential decay of decoding success above capacity.
Determining a noisy quantum channel's classical capacity and entanglement cost generally requires regularization over many channel uses. We remove both regularizations for every qubit-to-qubit channel admitting a pure output. For each such channel, Holevo information, channel entanglement of formation, and parallel entanglement cost are additive with those of any finite-dimensional partner channel. This class includes all qubit-to-qubit channels of Kraus rank at most two. For the amplitude damping channel with damping probability p, the unassisted classical capacity equals the known single-use Holevo information, attained by a binary pure-state ensemble with collective decoding, and the entanglement cost is h2((1+p)/2) ebits per use. The common mechanism is a support criterion for strong superadditivity of entanglement of formation: one marginal has no support on the sector in which both local systems are orthogonal to fixed distinguished vectors. We prove this criterion in arbitrary finite dimensions using a triangular block-matrix entropy inequality and decompositions preserving two expectations. For amplitude damping, we also derive an exact finite-block Holevo deficit, identify the unique optimal average input for p<1, and construct a binary Kraus representation attaining the uniform formation bound.
We determine the unassisted classical capacity of every qubit generalized amplitude-damping channel, for all damping strengths and thermal occupations. It equals the one-shot Holevo capacity and is given by a one-variable maximum. A binary pure-state ensemble attains the one-use optimum, and its independent products attain the optimum at every block length. We also prove that one-shot Holevo capacity, minimum output entropy, and regularized unassisted classical capacity are additive under tensor product with every finite-dimensional completely positive trace-preserving partner channel.
A quantum channel can transmit quantum states, classical messages, or classical messages that remain secret from its environment. We prove exponential strong converses for all three tasks over arbitrary finite-dimensional memoryless quantum channels, at their respective regularized capacities and allowing arbitrary entangled inputs across channel uses and joint decoding. Above quantum or classical capacity, entanglement fidelity or decoding success probability, respectively, decays exponentially with the number of channel uses. Above private capacity, fidelity to an ideal shared secret key decays exponentially; in particular, vanishing secrecy error forces decoding success to vanish. The proofs combine one-shot quantum blowing-up lemmas, which convert codes with low success into reliable codes with a controlled loss in code size, with a low-degree polynomial approximation of the channel's Stinespring image projector. The loss is governed by the projective tensor norm of the approximation across the receiver--environment bipartition. For memoryless channels, the polynomial construction gives exponential accuracy while keeping the rate loss arbitrarily small, turning the corresponding weak converses into strong converses.
The quantum capacity of a noisy channel quantifies the maximum rate at which quantum information can be transmitted reliably. For general channels, its evaluation requires an optimization over arbitrarily many channel uses. Degradable channels form a central exception: their capacity is given by the single-letter coherent information, while antidegradable channels have zero capacity. Nevertheless, even for these fundamental classes, it has remained open whether communication above capacity becomes possible when a fixed non-maximal error is tolerated. Here we resolve this problem by proving an exponential strong converse for every finite-dimensional degradable and antidegradable channel: at any rate above capacity, the fidelity of every coding scheme decays exponentially with the number of channel uses. As an immediate consequence, we establish the first all-code exponential strong converse for the quantum erasure channel throughout its full parameter range, strengthening previous results that applied only to almost all codes. We also show that exponential strong-converse bounds are preserved under receiver post-processing. This yields efficiently computable semidefinite-programming bounds for arbitrary finite-dimensional channels, improved bounds for Pauli channels, and an exact exponential strong converse for a nondegradable multilevel amplitude-damping family.
Abstract In their seminal work, Bennett et al. [1] showed that, with sufficient shared randomness, one noisy channel can simulate another at a rate equal to the ratio of their capacities. We establish that when coding above this channel interconversion capacity, the exact strong converse exponent is characterized by a simple optimization involving the difference of the corresponding Rényi channel capacities with Hölder dual parameters. We further extend this result to the entanglement-assisted interconversion of classical–quantum channels, showing that the strong converse exponent is likewise determined by differences of sandwiched Rényi channel capacities. The converse bound is obtained by relaxing to non-signaling assisted codes and applying Hölder duality together with the data processing inequality for Rényi divergences. Achievability is proven by concatenating refined channel coding and simulation protocols that go beyond first-order capacities, attaining an exponentially small conversion error, remaining robust under small variations in the input distribution, and tolerating a sublinear gap between the conversion rates.
We establish a single-letter and efficiently computable strong converse bound on the classical identification capacity of quantum channels. By equipping the n-fold channel output space with a product state-weighted σ-Euclidean geometry, we allow trace-distance separation constraints for identification codes to be controlled by Euclidean covering estimates. Using Sudakov's inequality, we bound the covering numbers of the n-fold channel outputs via their Gaussian mean widths in the weighted geometry, whose exponential growth in n is governed by the operator norm of a single-letter positive operator. Upon optimizing over all weighing states σ, this yields a strong converse bound on the identification capacity of the channel, which also admits a semidefinite representation. Our method improves the best known converse bounds on the identification capacity of several important examples, such as depolarizing, Pauli, erasure, and amplitude damping channels. We also discuss extensions of this method to more general Euclidean geometries on the output space.