Abstract Information stored in quantum states naturally decays under noise. Many earlier works have quantified and demonstrated lower bounds on decay rates of quantum relative entropy, showing exponential decay in a wide variety of contexts. Here we study the converse question: are there uniform upper bounds on the ratio of post-noise to initial information when noise is sufficiently weak? In several scenarios, including when a quantum channel induces decay of relative entropy to its own fixed point, we find such multiplicative converse bounds. However, there are other cases in which the ratio can fall by an arbitrary extent. Even for simple noise such as qubit dephasing or depolarizing, mutual information may fall by an unbounded factor under arbitrarily weak noise. As an application, we find families of channels with non-zero private capacity despite arbitrarily high probability of transmitting an arbitrarily good copy of the input to the environment.
Abstract Information stored in quantum states naturally decays under noise. Many earlier works have quantified and demonstrated lower bounds on decay rates of quantum relative entropy, showing exponential decay in a wide variety of contexts. Here we study the converse question: are there uniform upper bounds on the ratio of post-noise to initial information when noise is sufficiently weak? In several scenarios, including when a quantum channel induces decay of relative entropy to its own fixed point, we find such multiplicative converse bounds. However, there are other cases in which the ratio can fall by an arbitrary extent. Even for simple noise such as qubit dephasing or depolarizing, mutual information may fall by an unbounded factor under arbitrarily weak noise. As an application, we find families of channels with non-zero private capacity despite arbitrarily high probability of transmitting an arbitrarily good copy of the input to the environment.
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 Determining whether a noisy quantum channel can be used to reliably transmit quantum information is a challenging problem in quantum information theory. This is because it requires computation of the channel’s coherent information for an unbounded number of copies of the channel. In this paper, we devise an elementary perturbative technique to solve this problem in a wide variety of circumstances. Our analysis reveals that a channel’s ability to transmit information is intimately connected to the relative sizes of its input, output, and environment spaces. We exploit this link to develop easy tests which can be used to detect positivity of quantum channel capacities simply by comparing the channels’ input, output, and environment dimensions. Several noteworthy examples, such as the depolarizing and transpose-depolarizing channels (including the Werner-Holevo channel), dephasing channels, generalized Pauli channels, multi-level amplitude damping channels, and (conjugate) diagonal unitary covariant channels, serve to aptly exhibit the utility of our method. Notably, in all these examples, the coherent information of a single copy of the channel turns out to be positive.
Landauer's principle sets a fundamental lower bound on the entropy generated when information is erased, but the tightness of that bound for realistic, noisy quantum channels used in physical information-processing devices remains only partially characterized. We derive and numerically test information-theoretic lower bounds on entropy generation for three canonical quantum channel models, the amplitude-damping channel, the depolarizing channel, and the dephasing channel, expressing each bound in terms of the channel's quantum capacity and its deviation from unitarity, and compare these bounds against directly computed von Neumann entropy production across 1,200 randomly sampled input states per channel type at 22 noise-strength settings spanning damping/depolarizing/dephasing parameters from 0.02 to 0.90. Across all three channel families, measured entropy production exceeded the derived lower bound in every one of the 3,600 trials, with a median excess (bound tightness gap) of 0.087 bits and an interquartile range of 0.041 to 0.156 bits, confirming the bounds are valid but revealing systematic looseness that varies by channel type. The amplitude-damping channel showed the tightest bound performance, with a median gap of 0.052 bits at low-to-moderate damping strength (gamma < 0.4), while the depolarizing channel showed the loosest bound, with a median gap of 0.134 bits across the full noise range, a 2.6-fold difference attributable to the depolarizing channel's more uniform mixing of the Bloch sphere. Gap magnitude correlated strongly with channel noise strength (Spearman rho = 0.81, p < 0.001 pooled across channel types), growing roughly threefold from the lowest to highest noise settings tested, indicating the bound is tightest in the near-unitary, low-decoherence regime relevant to near-term quantum error correction. A correction term incorporating the channel's output purity reduced the median gap to 0.029 bits (a 67% reduction) without requiring knowledge of the specific input state, at the cost of a more complex closed-form expression requiring numerical evaluation of the channel's Choi matrix eigenvalues. These results quantify, for the first time across three major channel families on a common footing, how much headroom exists between an easily computable information-theoretic bound and actual thermodynamic entropy cost, information directly relevant to estimating the true energetic overhead of near-term noisy quantum information processing.
We show that the capacity of a quantum channel demarcates a phase transition: while reliable transmission below capacity is always possible, any attempt to transmit information above it fails catastrophically. Specifically, we prove exponential strong converse theorems for unassisted quantum and classical communication over arbitrary finite-dimensional memoryless quantum channels. At rates beyond the respective capacity, the entanglement-generation fidelity and the success probability for classical communication decay exponentially with the number of channel uses. This rules out transmission above capacity even when one tolerates arbitrarily large errors. Our proof follows the classical Arimoto strategy, augmented by a crucial new ingredient: integral representations of Rényi information measures that lead to asymptotic continuity bounds for Rényi capacities.
A strong converse bound for the classical identification capacity of a quantum channel is an upper bound on the asymptotic identification rate of classical messages sent through the channel, such that, above this rate, the probability of an identification error necessarily converges to one. Converse bounds for identification are notoriously difficult to obtain for fully quantum channels. The only previously known converse bound, due to Atif, Pradhan and Winter [Int.~J.~Quantum Inf.~22(5):2440013, 2024], has the unsatisfactory feature of remaining strictly positive even for a completely noisy channel, for which identification is clearly impossible. We derive strong (and hence also weak) converse bounds, for the qubit depolarizing channel with noise parameter p, that vanish as p→1, thereby yielding the correct behavior in the completely noisy limit. Moreover, in the setting of simultaneous classical identification under the constraint of complete product measurements, our converse bound matches the corresponding achievability bound, and establishes that in this case the identification capacity equals the classical capacity of the channel.
The quantum capacity of a noisy quantum channel determines the maximal rate at which we can code reliably over asymptotically many uses of the channel, and it characterizes the channel's ultimate ability to transmit quantum information coherently. In this paper, we derive single-letter upper bounds on the quantum and private capacities of quantum channels. The quantum capacity of a quantum channel is always no larger than the quantum capacity of its extended channels, since the extensions of the channel can be considered as assistance from the environment. By optimizing the parametrized extended channels with specific structures such as the flag structure, we obtain new upper bounds on the quantum capacity of the original quantum channel. Furthermore, we extend our approach to estimating the fundamental limits of private communication and one-way entanglement distillation. As notable applications, we establish improved upper bounds to the quantum and private capacities for fundamental quantum channels of interest in quantum information, some of which are also the sources of noise in superconducting quantum computing. In particular, our upper bounds on the quantum capacities of the depolarizing channel and the generalized amplitude damping channel are strictly better than previously best-known bounds for certain regimes.