Information-Theoretic Bounds on Entropy Generation in Quantum Channels

Abstract

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.