Rate-reliability tradeoff for deterministic identification

TL;DR

This paper explores the tradeoff between rate and reliability in deterministic identification over channels, showing how exponential error constraints influence the achievable message rate and providing bounds related to the channel's output set complexity.

Contribution

It introduces bounds on the rate-reliability function for deterministic identification, linking it to the geometry of the output set and extending results to quantum channels.

Findings

Linear scaling of message rate with block length for positive error exponents.

Identification rates relate to Minkowski dimension and error exponents.

Loss of linearithmic scaling when only one error probability is exponentially small.

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 $\mathbf{\Theta(n\log n)}$, 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.