Classical Identification Capacity

• Classical Identification Capacity is the asymptotic rate at which a channel can support reliable yes/no tests for message identity, differing fundamentally from full message recovery.
• It distinguishes randomized encoding that enables doubly exponential hypothesis growth from deterministic encoding where capacity scales with factors like Minkowski dimension and channel geometry.
• The theory bridges classical and quantum channels, applying to models such as Poisson and colored Gaussian channels, and highlights critical trade-offs in error criteria and encoding strategies.

Searching arXiv for recent and foundational papers on identification capacity, including classical channels, deterministic identification, Poisson/Gaussian models, and quantum-channel converse results. Classical identification capacity is the asymptotic rate at which a channel can support reliable yes/no tests of message identity rather than full message recovery. In the identification paradigm, the sender selects a message $i$, but the receiver’s task is not to decode $i$ exactly; instead, for each hypothesis $j$, the receiver wants to decide whether $j$ was sent. This change in objective produces a different asymptotic theory from ordinary transmission. With randomized encoding, the maximal number of identifiable messages can grow doubly exponentially in blocklength, so the relevant rate is $\frac{1}{n}\log\log N$. With deterministic encoding, the characteristic growth for continuous-input settings is instead $2^{R n \log n}$, leading to the rate scale $\frac{1}{n\log n}\log N$ (Labidi et al., 2023, Colomer et al., 2024).

1. Core definitions and asymptotic regimes

For a memoryless channel $W$, a randomized identification code assigns to each message $i$ an encoding distribution $P_i$ on $i$0-letter inputs and a test region $i$1. The two error criteria are the type-I error, requiring $i$2, and the type-II error, requiring $i$3 for all $i$4. In this regime, one defines capacity through

$i$5

or equivalently through the achievability of code sizes $i$6 for every $i$7 and all sufficiently large $i$8 (Labidi et al., 2023, Singh, 3 Jun 2026).

Deterministic identification restricts each message to a single codeword rather than an encoding distribution. In that setting, the doubly exponential scale typically disappears. For channels with finite output and arbitrary input alphabet, the relevant asymptotic quantity is

$i$9

reflecting the super-exponential but sub-doubly-exponential growth $j$0 (Colomer et al., 2024). For colored Gaussian channels with deterministic encoding under peak power, the corresponding definition is formulated through code sizes

$j$1

and the supremum of achievable $j$2 (Salariseddigh, 6 Apr 2026).

A central conceptual distinction follows immediately. In Shannon transmission, reliable communication concerns exact decoding and scales as $j$3. In identification, the size of the hypothesis set can be much larger, but the appropriate rate notion changes with the encoding model. This suggests that “more messages” and “larger capacity” are not interchangeable statements.

2. Randomized identification and equality with transmission capacity

For discrete memoryless channels, the classical benchmark is that randomized identification capacity equals Shannon capacity. The Poisson-channel analysis makes this analogy explicit: Ahlswede–Dueck and Han–Verdú showed for any DMC that randomized ID capacity equals Shannon capacity, and the discrete-time Poisson channel (DTPC) obeys the same principle despite its continuous input alphabet and Poisson tails (Labidi et al., 2023).

The DTPC considered there has input alphabet $j$4, output alphabet $j$5, dark current $j$6, and transition law

$j$7

subject to the peak constraint $j$8 and the average constraint $j$9. Its main theorem states

$j$0

where

$j$1

If $j$2 is a capacity-achieving input law, then the optimal identification rate is

$j$3

Thus, although randomized identification codes can support approximately $j$4 hypotheses, the identifying rate measured by $j$5 is limited by exactly the same constant $j$6 as the transmission rate (Labidi et al., 2023).

The proof structure mirrors the standard identification paradigm. Achievability uses an optimal input distribution $j$7, subcodes of size approximately $j$8, a decoding rule based on empirical information density relative to the induced output law $j$9, and concentration tools such as Chernoff or Hoeffding bounds. The converse is a strong converse obtained through the information-spectrum method: for every $\frac{1}{n}\log\log N$0,

$\frac{1}{n}\log\log N$1

with Chebyshev’s inequality and moment bounds for Poisson distributions controlling the relevant sums (Labidi et al., 2023).

The same mechanism extends to a state-dependent DTPC with i.i.d. state $\frac{1}{n}\log\log N$2, unknown to both parties. Defining the averaged channel

$\frac{1}{n}\log\log N$3

one obtains

$\frac{1}{n}\log\log N$4

so the identification capacity is again the Shannon capacity of the averaged channel (Labidi et al., 2023).

3. Deterministic identification and dimensional characterizations

Deterministic identification has a markedly different asymptotic structure. For memoryless channels with finite output but arbitrary input alphabet, the channel is represented by its output distributions $\frac{1}{n}\log\log N$5 as a subset $\frac{1}{n}\log\log N$6. The maximum deterministic code size satisfies

$\frac{1}{n}\log\log N$7

and the corresponding $\frac{1}{n}\log\log N$8-rate is controlled by the Minkowski dimension of $\frac{1}{n}\log\log N$9 (Colomer et al., 2024).

More precisely, if $2^{R n \log n}$0 and $2^{R n \log n}$1 denote the lower and upper Minkowski dimensions, then

$2^{R n \log n}$2

If the ordinary Minkowski dimension

$2^{R n \log n}$3

exists, this becomes

$2^{R n \log n}$4

The converse uses disjoint total-variation balls of radius $2^{R n \log n}$5 around the output distributions corresponding to codewords, while the achievability argument combines packings in $2^{R n \log n}$6 with classical codes of large Hamming distance (Colomer et al., 2024).

A key technical ingredient is the “Hypothesis-Testing Lemma.” For codewords $2^{R n \log n}$7 whose induced output distributions satisfy $2^{R n \log n}$8, one chooses as the test region for $2^{R n \log n}$9 its conditional-entropy typical set

$\frac{1}{n\log n}\log N$0

Classical typicality then yields small first-kind error, while trace-distance separation and equipartition estimates control second-kind error (Colomer et al., 2024).

The dimensional perspective clarifies why deterministic identification can be super-exponential without becoming doubly exponential. The growth is tied to the metric complexity of the single-letter output set rather than to mutual information. The same detailed exposition states that there is no “superactivation” of deterministic identification capacity for classical channels, because Minkowski dimension under Cartesian product adds and channels with zero single-letter dimension retain zero dimension after product formation (Colomer et al., 2024). A plausible implication is that deterministic identification is governed more by output-set geometry than by the combinatorial amplification characteristic of randomized identification.

4. Continuous-input classical channels: Poisson and colored Gaussian models

Continuous-input models sharpen the contrast between randomized and deterministic identification. The DTPC with molecule-counting receivers is motivated by event-driven molecular communications, where the conventional Shannon capacity may not be the appropriate metric and identification is proposed as an alternative performance measure. Yet, once randomized encoding is permitted, the DTPC again satisfies $\frac{1}{n\log n}\log N$1 under peak and average power constraints (Labidi et al., 2023).

The colored Gaussian channel with inter-symbol interference presents the deterministic counterpart. The model has $\frac{1}{n\log n}\log N$2 taps, $\frac{1}{n\log n}\log N$3, Toeplitz convolution matrix $\frac{1}{n\log n}\log N$4, and additive colored Gaussian noise $\frac{1}{n\log n}\log N$5 on dimension $\frac{1}{n\log n}\log N$6. The singular-value spectrum of $\frac{1}{n\log n}\log N$7 is polynomially bounded as

$\frac{1}{n\log n}\log N$8

with $\frac{1}{n\log n}\log N$9, and the input obeys the per-symbol peak-power constraint $W$0 (Salariseddigh, 6 Apr 2026).

Under assumptions $W$1–$W$2 and $W$3, the identification capacity $W$4 admits super-exponential codebook sizes

$W$5

and satisfies

$W$6

The achievability proof constructs a packed codebook in $W$7, passes to the “convoluted codebook” $W$8, uses the minimum-distance estimate

$W$9

and decodes with the Mahalanobis-distance test

$i$0

Type-I and type-II errors vanish by Chebyshev bounds on $i$1-type deviations (Salariseddigh, 6 Apr 2026).

In the memoryless white-noise case $i$2, one recovers

$i$3

consistent with earlier deterministic-identification results for the AWGN channel under peak power (Salariseddigh, 6 Apr 2026). As $i$4 increases or $i$5 increases, the achievable region shrinks and vanishes at $i$6. This suggests that channel memory and spectral ill-conditioning act as geometric penalties on deterministic identification packings.

5. Classical identification over quantum channels

The phrase “classical identification capacity” also refers to the transmission of classical messages in identification mode through quantum channels. For a finite-dimensional quantum channel $i$7, an $i$8 classical identification code consists of code states $i$9 and effects $P_i$0 such that

$P_i$1

(Singh, 3 Jun 2026).

For stationary memoryless classical–quantum channels $P_i$2, the established identity is

$P_i$3

and for fixed $P_i$4 there is a strong converse of the form

$P_i$5

The same equality between identification and transmission capacity extends to compound memoryless cq-channels: $P_i$6 and, for arbitrarily varying cq-channels, one has the dichotomy

$P_i$7

These formulas show that the equality between randomized identification capacity and transmission capacity survives substantial channel uncertainty (Boche et al., 2018).

Secure identification introduces a second observer. For compound wiretap cqq-channels and arbitrarily varying wiretap cqq-channels, the secret identification capacity obeys a dichotomy: it is $P_i$8 when the secrecy capacity is $P_i$9, and otherwise equals the transmission capacity of the main channel. In the compound case,

$i$00

This identifies a threshold phenomenon absent from ordinary randomized identification without secrecy constraints (Boche et al., 2018).

6. Strong converses and converse geometry

Strong converse bounds for classical identification over quantum channels have become a distinct line of work. For the qubit depolarizing channel

$i$01

a strong-converse upper bound on the unrestricted identification capacity is

$i$02

where

$i$03

As $i$04, this bound tends to $i$05, matching the completely noisy limit (Ye et al., 31 Mar 2026).

Under simultaneous identification with complete product measurements, the same channel admits an exact capacity formula: $i$06 and a strong converse holds at that rate. The proof reduces product-basis measurements on $i$07 to an $i$08-fold binary symmetric channel with crossover probability $i$09, then uses classical soft-covering to control the number of distinguishable output distributions (Ye et al., 31 Mar 2026).

A more general converse framework is given by the Gaussian mean-width method. Fixing a full-rank state $i$10, one defines the weighted inner product

$i$11

the weighted adjoint $i$12, and the single-letter positive operator

$i$13

Combining the domination $i$14, Gaussian mean-width estimates in the product geometry, and Sudakov’s inequality yields the single-letter strong converse

$i$15

This bound admits a semidefinite-program representation and improves previously known converse bounds for several channels, including depolarizing, Pauli, erasure, and amplitude damping channels (Singh, 3 Jun 2026).

One recurring misconception is that identification “exceeds” transmission capacity because it permits vastly more messages. The classical theory distinguishes sharply between message count and rate definition. Randomized identification indeed supports doubly-exponential hypothesis sets, but its asymptotic rate remains $i$16 for DMCs, the DTPC, and memoryless cq-channels. Deterministic identification, by contrast, replaces the $i$17 scale by $i$18 and is governed by geometric quantities such as Minkowski dimension or packing under Mahalanobis distance (Labidi et al., 2023, Colomer et al., 2024, Salariseddigh, 6 Apr 2026).