Andreas Winter

Pau Colomer, Christian Deppe, Holger Boche et al.

Deterministic identification (DI) has emerged as a promising paradigm for large-scale and goal-oriented communication systems. Despite significant progress, a fundamental open problem has remained unresolved: a persistent gap between the best known lower and upper bounds on the DI capacity, as well as on the corresponding rate-reliability tradeoff bounds. In this paper, we finally close this gap for Gaussian channels $\mathcal{G}$ by constructing an optimised code that achieves the known upper bound. This allows us to establish that the linearithmic capacity for deterministic identification is $\dot{C}_{\text{DI}}(\mathcal{G})=\frac{1}{2}$. Furthermore, we analyse the rate-reliability tradeoff and show that the proposed scheme matches the known upper bounds to first order, thereby closing the existing gap in reliability performance for all admissible error decay regimes. Finally, we demonstrate the existence of an optimum universal code, which does not require knowledge of the channel parameters and yet achieves capacity.

Xingyi He, S. Sandeep Pradhan, Andreas Winter

This paper establishes the exact strong converse exponent of the soft covering problem in the classical setting. This exponent characterizes the slowest achievable convergence speed of the total variation to one when a code of rate below mutual information is applied to a discrete memoryless channel for synthesizing a product output distribution. The proposed exponent is expressed through a new two-parameter information quantity, differing from the more commonly studied Rényi divergence or Rényi mutual information. In addition, we demonstrate the non-tightness of random coding for rates both below and above mutual information. Discussions on the latter start with noiseless channels, where we develop a deterministic code construction that outperforms random codes in error exponents. We further observe that the conventional formulation, which assumes a uniform distribution over messages, inherently introduces a discrepancy in error exponents depending on whether the components of the target distribution are rational or irrational numbers. To eliminate this discrepancy, we propose a new formulation in which messages are allowed to be distributed non-uniformly, and the rate is given by the logarithm of the smallest nonzero message probability (corresponding to Rényi entropy $H_{-\infty}$ of order $-\infty$). The exact error exponent is characterized in this formulation for noiseless channels. Furthermore, for noisy channels, we provide a high-rate improvement in achievability and derive a converse bound on the error exponent.

Pau Colomer, Christian Deppe, Holger Boche et al.

For memoryless channels with continuous input alphabets, deterministic identification (DI) typically exhibits a linearithmic ($n\log n$) message growth. However, the exact DI capacity has long remained open due to a persistent gap between the best known achievability and converse bounds. This gap was recently closed for AWGN channels via a novel code construction optimising the "galaxy" codes. Here, we extend this approach to the Bernoulli channel and subsequently to any channel $W$ whose image contains a continuous curve of output probability distributions, and hence admits a reduction to the Bernoulli channel restricted to a subinterval of inputs. As a consequence, we prove that the converse bound is tight and establish $\dot{C}_{\text{DI}}(W) = \frac 12$ for this broad class of channels, thereby closing the long-standing capacity gap. A similar gap was also observed for the DI rate-reliability tradeoff. We analyse the tradeoff between rate and error of the proposed code and derive improved lower bounds on the reliability function, approaching the converse at leading order in the regime of small error exponents.

Marco Fanizza, Niklas Galke, Josep Lumbreras et al.

Matrix product operators allow efficient descriptions (or realizations) of states on a 1D lattice. We consider the task of learning a realization of minimal dimension from copies of an unknown state, such that the resulting operator is close to the density matrix in trace norm. For finitely correlated translation-invariant states on an infinite chain, a realization of minimal dimension can be exactly reconstructed via linear algebra operations from the marginals of a size depending on the representation dimension. We establish a bound on the trace norm error for an algorithm that estimates a candidate realization from estimates of these marginals and outputs a matrix product operator, estimating the state of a chain of arbitrary length $t$. This bound allows us to establish an $O(t^2)$ upper bound on the sample complexity of the learning task, with an explicit dependence on the site dimension, realization dimension and spectral properties of a certain map constructed from the state. A refined error bound can be proven for $C^*$-finitely correlated states, which have an operational interpretation in terms of sequential quantum channels applied to the memory system. We can also obtain an analogous error bound for a class of matrix product density operators on a finite chain reconstructible by local marginals. In this case, a linear number of marginals must be estimated, obtaining a sample complexity of $\tilde{O}(t^3)$. The learning algorithm also works for states that are sufficiently close to a finitely correlated state, with the potential of providing competitive algorithms for other interesting families of states.