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arXiv:2502.02389

Rate-reliability tradeoff for deterministic identification

Pau Colomer, Christian Deppe, Holger Boche, Andreas Winter

Published Feb 4, 2025• 0 citations
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Abstract

We investigate deterministic identification over arbitrary memoryless channels under the constraint that the error probabilities of first and second kind are exponentially small in the block length $\mathbf{n}$, controlled by reliability exponents $\mathbf{E_1,E_2 \geq 0}$. In contrast to the regime of slowly vanishing errors, where the identifiable message length scales linearithmically as $\mathbf{Θ(n\log n)}$, here we find that for positive exponents linear scaling is restored, now with a rate that is a function of the reliability exponents. We give upper and lower bounds on the ensuing rate-reliability function in terms of (the logarithm of) the packing and covering numbers of the channel output set, which for small error exponents $\mathbf{E_1,E_2>0}$ can be expanded in leading order as the product of the Minkowski dimension of a certain parametrisation the channel output set and $\mathbf{\log\min\{E_1,E_2\}}$. These allow us to recover the previously observed slightly superlinear identification rates, and offer a different perspective for understanding them in more traditional information theory terms. We also show that even if only one of the two errors is required to be exponentially small, the linearithmic scaling is lost. We further illustrate our results with a discussion of the case of dimension zero, and extend them to classical-quantum channels and quantum channels with tensor product input restriction.

DOI
10.48550/arxiv.2502.02389

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We investigate deterministic identification over arbitrary memoryless channels under the constraint that the error probabilities of first and second kind are exponentially small in the block length n\mathbf{n}n, controlled by reliability exponents E1,E2≥0\mathbf{E_1,E_2 \geq 0}E1​,E2​≥0. In contrast to the regime of slowly vanishing errors, where the identifiable message length scales linearithmically as Θ(nlog⁡n)\mathbf{Θ(n\log n)}Θ(nlogn), here we find that for positive exponents linear scaling is restored, now with a rate that is a function of the reliability exponents. We give upper and lower bounds on the ensuing rate-reliability function in terms of (the logarithm of) the packing and covering numbers of the channel output set, which for small error exponents E1,E2>0\mathbf{E_1,E_2>0}E1​,E2​>0 can be expanded in leading order as the product of the Minkowski dimension of a certain parametrisation the channel output set and log⁡min⁡{E1,E2}\mathbf{\log\min\{E_1,E_2\}}logmin{E1​,E2​}. These allow us to recover the previously observed slightly superlinear identification rates, and offer a different perspective for understanding them in more traditional information theory terms. We also show that even if only one of the two errors is required to be exponentially small, the linearithmic scaling is lost. We further illustrate our results with a discussion of the case of dimension zero, and extend them to classical-quantum channels and quantum channels with tensor product input restriction.
We investigate deterministic identification over arbitrary memoryless channels under the constraint that the error probabilities of first and second kind are exponentially small in the block length n, controlled by reliability exponents E1,E2≥ 0. In contrast to the regime of slowly vanishing errors, where the identifiable message length scales linearithmically as Θ(n log n), here we find that for positive exponents linear scaling is restored, now with a rate that is a function of the reliability exponents. We give upper and lower bounds on the ensuing rate-reliability function in terms of (the logarithm of) the packing and covering numbers of the channel output set, which for small error exponents E1,E2> 0 can be expanded in leading order as the product of the Minkowski dimension of a certain parametrisation the channel output set and log min{E1,E2}. These allow us to recover the previously observed slightly superlinear identification rates, and offer a different perspective for understanding them in more traditional information theory terms. We also show that even if only one of the two errors is required to be exponentially small, the linearithmic scaling is lost. We further illustrate our results with a discussion of the case of dimension zero, and extend them to classical-quantum channels and quantum channels with tensor product input restriction.
Following initial work by JaJa, and Ahlswede and Cai, and inspired by a recent renewed surge in interest in deterministic identification (DI) via noisy channels, we consider the problem in its generality for memoryless channels with finite output, but arbitrary input alphabets. Such a channel is essentially given by (the closure of) the subset of its output distributions in the probability simplex. Our main findings are that the maximum number of messages thus identifiable scales super-exponentially as2Rnlog⁡n2^{Rn\log n}2Rnlognwith the block lengthnnn, and that the optimal rateRRRis upper and lower bounded in terms of the covering (aka Minkowski, or Kolmogorov, or entropy) dimensiondddof a certain algebraic transformation of the output set:14d≤R≤12d\frac{1}{4}d\leq R\leq\frac{1}{2}d41​d≤R≤21​d, Along the way, we present a Hypothesis Testing Lemma that shows it is sufficient to ensure pairwise reliable distinguishability of the output distributions to construct a DI code. Although we do not know the exact capacity formula, we can conclude that the DI capacity exhibits super-activation: there exist channels whose capacity is zero, but whose product has positive capacity. These results are then generalised to classical-quantum channels with finite-dimensional output quantum system (but arbitrary input alphabet), and in particular to quantum channels on finite-dimensional quantum systems under the constraint that the identification code can only use tensor product inputs.
Following initial work by JaJa, Ahlswede and Cai, and inspired by a recent renewed surge in interest in deterministic identification (DI) via noisy channels, we consider the problem in its generality for memoryless channels with finite output, but arbitrary input alphabets. Such a channel is essentially given by its output distributions as a subset in the probability simplex. Our main findings are that the maximum length of messages thus identifiable scales superlinearly asR nlog⁡nR\,n\log nRnlognwith the block length n, and that the optimal rate R is bounded in terms of the covering (aka Minkowski, or Kolmogorov, or entropy) dimension d of a certain algebraic transformation of the output set:14d≤R≤12d\frac {1}{4} d \leq R \leq \frac {1}{2} d41​d≤R≤21​d. Remarkably, both the lower and upper Minkowski dimensions play a role in this result. Along the way, we present a Hypothesis Testing Lemma showing that it is sufficient to ensure pairwise reliable distinguishability of the output distributions to construct a DI code. Although we do not know the exact capacity formula, we can conclude that the DI capacity exhibits superactivation: there exist channels whose capacities individually are zero, but whose product has positive capacity. We also generalise these results to classical-quantum channels with finite-dimensional output quantum system, in particular to quantum channels on finite-dimensional quantum systems under the constraint that the identification code can only use tensor product inputs.
A fundamental quantity of interest in Shannon theory, classical or quantum, is the error exponent of a given channel W and rate R: the constantE(W,R)E(W,R)E(W,R)which governs the exponential decay of decoding error when using ever larger optimal codes of fixed rate R to communicate over ever more (memoryless) instances of a given channel W. Nearly matching lower and upper bounds are well-known for classical channels. Here I show a lower bound on the error exponent of communication over arbitrary classical-quantum (CQ) channels which matches Dalai’s sphere-packing upper bound for rates above a critical value, exactly analogous to the case of classical channels. This proves a conjecture made by Holevo in his investigation of the problem. Unlike the classical case, however, the argument does not proceed via a refined analysis of a suitable decoder, but instead by leveraging a bound by Hayashi on the error exponent of the cryptographic task of privacy amplification. This bound is then related to the coding problem via tight entropic uncertainty relations and Gallager’s method of constructing capacity-achieving parity-check codes for arbitrary channels. Along the way, I find a lower bound on the error exponent of the task of compression of classical information relative to quantum side information that matches the sphere-packing upper bound of Cheng et al. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi, at least for linear randomness extractors.
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 ppp, that vanish as p→1p\to 1p→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.
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.