Molecular communication (MC) enables information transfer via molecules, making it ideal for biomedical applications where traditional methods fall short. In many such scenarios, identifying specific events is more critical than decoding full messages, motivating the use of deterministic identification (DI). This paper investigates DI over discrete-time Poisson channels (DTPCs) with inter-symbol interference (ISI), a realistic setting due to channel memory effects. We consider memory scaling as K = 2κ log n, where κ represents the coding rate and n the code length. We improve the known upper bound on DI capacity under power constraints from $\frac{3}{2} + \kappa $ to $\frac{{1 + \kappa }}{2}$. Additionally, we present the first results on deterministic identification with feedback (DIF) in this context, providing a constructive lower bound. These findings enhance the theoretical understanding of MC and support more efficient, feedback-driven biomedical systems.

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$E_{1}, E_{2}>0$. We find that, in contrast to the case of slowly vanishing errors where the identifiable message length scales as$\Theta(n \log n)$, here linear scaling is restored, now as a function of the reliability exponents. We give upper and lower bounds on the ensuing ratereliability function in terms of (the logarithm of) the packing and covering numbers of the channel output set, which for small error exponents$E_{1}, E_{2}>0$are bounded below and above in terms of the product of the Minkowski dimension and$\log \min \left\{E_{1}, E_{2}\right\}$. 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 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, 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 as$R\,n\log n$with 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:$\frac {1}{4} d \leq R \leq \frac {1}{2} d$. 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.

Motivated by deterministic identification via channels, where the encoder cannot use randomisation, we revisit the problem of identification via quantum channels with the additional restriction that the message encoding must use pure quantum states, rather than general mixed states. Together with the previously considered distinction between simultaneous and general decoders, this suggests a two-dimensional spectrum of different identification capacities, whose behaviour could a priori be very different. We demonstrate two new main results: first, we show that all of the four combinations (pure/mixed encoder, simultaneous/general decoder) have a double-exponentially growing code size, and that indeed the corresponding identification capacities are lower bounded by the classical transmission capacity for a general quantum channel, which is given by the Holevo-Schumacher- Westmoreland theorem. Secondly, we show that the simultaneous identification capacity of a quantum channel equals the simultaneous identification capacity with pure state encodings, thus leaving three linearly ordered identification capacities. By considering some simple examples we finally show that these three are all different: general identification capacity can be larger than pure-state-encoded identification capacity which in turn can be larger than pure-state-encoded simultaneous identification capacity,

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 as$2^{Rn\log n}$with the block length$n$, and that the optimal rate$R$is upper and lower bounded in terms of the covering (aka Minkowski, or Kolmogorov, or entropy) dimension$d$of a certain algebraic transformation of the output set:$\frac{1}{4}d\leq R\leq\frac{1}{2}d$, 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.