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Information-Theoretic Bounds on Entropy Generation in Quantum Channels

Tibor Svoboda, Milos Bartos, Igor Novotny

Published Dec 13, 2022• 0 citations
Publisher

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.

DOI
10.5281/zenodo.22698356

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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 systematically characterize quantum entanglement in all sixteen combinations of two-qubit Bell states across three operational regimes: pure superposition, noiseless mixing, and asymmetric mixing of one noisy and one pure Bell state. For pure superpositions, we calculate the von Neumann entropy and show that entanglement depends critically on the relative phase phi between superposed states. For mixed states, we use concurrence C under three physically motivated decoherence channels. The Depolarizing (DP) and Phase Damping (PD) channels are modeled as global, collective operations on the two-qubit Hilbert space, while Amplitude Damping (AD) is treated as independent local decay on each qubit, reflecting the asymmetric, energy-dissipating nature of spontaneous emission. A central and counterintuitive result is that increasing noise can raise concurrence in specific parameter regimes across all three channels. We further map the boundary between mathematical entanglement (C > 0) and operational quantum non-locality using the Horodecki criterion, finding this boundary to be strongly channel-dependent. Under Phase Damping, the Bell-local entanglement regime vanishes entirely for identical and same-bases mixtures, meaning any surviving entanglement guarantees a CHSH violation. Under Depolarizing noise, a large Bell-local region requires mixing probability r > 1/sqrt(2) to recover non-locality at maximum noise. For Amplitude Damping, certain mixtures exhibit a non-monotonic Horodecki parameter M(T): concurrence decreases monotonically with noise, yet the capacity for CHSH violation is lost at intermediate noise levels and restored at high noise, as maximal damping repurifies the noisy branch. These results provide a unified analytical reference for entanglement and non-locality management across Bell-state combinations and decoherence channels relevant to NISQ architectures.
In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi-Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.
Abstract I will investigate the capacities of noisy quantum channels through a combined analytical and numerical approach. First, I introduce novel flagged extension techniques that embed a channel into a higher-dimensional space, enabling single-letter upper bounds on quantum and private capacities. My results refine previous bounds and clarify noise thresholds beyond which quantum transmission vanishes. Second, I present a simulation framework that uses coherent information to estimate channel capacities in practice, focusing on two canonical examples: the amplitude damping channel (which I confirm is degradable and thus single-letter) and the depolarizing channel (whose capacity requires multi-letter superadditivity). By parameterizing input qubit states on the Bloch sphere, I numerically pinpoint the maximum coherent information for each channel and validate the flagged extension bounds. Notably, I capture the abrupt transition to zero capacity at high noise and observe superadditivity for moderate noise levels.
Quantum information decoupling is a fundamental primitive in quantum information theory, underlying various applications in quantum physics. We prove a novel one-shot decoupling theorem formulated in terms of quantum relative entropy distance, with the decoupling error bounded by two sandwiched Rényi conditional entropies. In the asymptotic i.i.d. setting of standard information decoupling via partial trace, we show that this bound is ensemble-tight in quantum relative entropy distance and thereby yields a characterization of the associated decoupling error exponent in the low-cost-rate regime. Leveraging this framework, we derive several operational applications formulated in terms of purified distance: (i) single-letter expressions for the exact error exponents of quantum state merging in the low-entanglement-cost and high-entanglement-distillation-rate regimes, in terms of Petz-Rényi conditional entropies, and (ii) regularized expressions for achievable error exponents of entanglement distillation and quantum channel coding in terms of Petz-Rényi coherent informations. We further prove that these achievable bounds are tight for maximally correlated states and generalized dephasing channels, respectively, for the high distillation-rate/coding-rate regimes.
Entanglement fidelity quantifies how well a quantum channel preserves the correlations between a transmitted system and an inaccessible reference system. We derive closed-form expressions for the entanglement fidelity associated with several standard quantum noise models, including the random Pauli-X, dephasing, depolarizing, Werner-Holevo, generalized Pauli (Weyl), and amplitude-damping channels. For each model, we express the entanglement fidelity in terms of a general input density operator ρρρ, using Schumacher's Kraus-operator approach, which provides a channel-agnostic recipe applicable to any completely positive trace-preserving (CPTP) map with a finite Kraus representation. We then specialize to a communication scenario in which the source emits a two-letter parametric alphabet, thereby making explicit the dependence of entanglement preservation on both channel and source parameters. The resulting expressions enable direct comparisons of channel performance and rankings for representative families of input states, including common qubit states.
We propose a framework to variationally obtain detectable capacity bounds for quantum channels. The proposal of this framework is motivated by the difficulty of estimating the von Neumann entropy of an unknown quantum state. Instead of estimating the von Neumann entropy at the channel output, we propose to estimate the state purity---which can be measured by a single measurement setting---followed by bounding the von Neumann entropy from above and below. This procedure leads to new upper and lower bounds on various communication rates of quantum channels for some fixed input states. Then, by utilizing the variational method to find optimal input states we obtain lower bounds on (i) the quantum capacity of arbitrary channels, (ii) the entanglement-assisted classical capacity of arbitrary channels, and (iii) the classical capacity of covariant channels. Corresponding to these lower bounds, we also obtain upper bounds on (i) NNN-shot coherent information, (ii) the entanglement-assisted classical capacity, and (iii) the NNN-shot Holevo capacity of arbitrary quantum channels. All these bounds can be estimated by a single measurement setting without needing a full process tomography or any further a priori knowledge, e.g., preferred basis of the channel.