In the identification via channels (the ID-channel), under some condition for noisy channels, the sum of two types of error probabilities of identification goes to one at rates of codes above capacity when the length of codes tends to infinity. This is known as a strong converse property of the ID-channel. In this paper we derive an exponential lower bound for the sum of two error probabilities to tend to one. In particular, for the class of discrete memoryless channels, we explicitly compute this lower bound.
In the identification via channels (the ID-channel), under some condition for noisy channels, the sum of two types of error probabilities of identification goes to one at rates of codes above capacity when the length of codes tends to infinity. This is known as a strong converse property of the ID-channel. In this paper we derive an exponential lower bound for the sum of two error probabilities to tend to one. In particular, for the class of discrete memoryless channels, we explicitly compute this lower bound.
Studies two problems in the theory of identification via channels. The first problem concerns the identification via channels with noisy feedback. Whereas for Shannon's transmission problem the capacity of a discrete memoryless channel does not change with feedback, it is known that the identification capacity is affected by feedback. The authors study its dependence on the feedback channel. They prove both, a direct and a converse coding theorem. Although a gap exists between the upper and lower bounds provided by these two theorems, the known result for channels without feedback and the known result for channels with complete feedback, are both special cases of these two new theorems, because in these cases the bounds coincide. The second problem is the identification via wiretap channels. A secrecy identification capacity is defined for the wiretap channel. A "dichotomy theorem" is proved which says that the second-order secrecy identification capacity is the same as Shannon's capacity for the main channel as long as the secrecy transmission capacity of the wiretap channel is not zero, and zero otherwise. Equivalently, one can say that the identification capacity is not lowered by the presence of a wiretapper as long as 1 bit can be transmitted (or identified) correctly with arbitrarily small error probability. This is in strong contrast to the case of transmission.>
A minimax converse for the identification via channels is derived. By this converse, a general formula for the identification capacity, which coincides with the transmission capacity, is proved without the assumption of the strong converse property. Furthermore, the optimal second-order coding rate of the identification via channels is characterized when the type I error probability is non-vanishing and the type II error probability is vanishing. Our converse is built upon the so-called partial channel resolvability approach; however, the minimax argument enables us to circumvent a flaw reported in the literature.
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 p, that vanish as p→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.
In the theory of identification via noisy channels randomization in the encoding has a dramatic effect on the optimal code size, namely, it grows double-exponentially in the blocklength, whereas in the theory of transmission it has the familiar exponential growth. We consider now instead of the discrete memoryless channel (DMC) more robust channels such as the familiar compound (CC) and arbitrarily varying channels (AVC). They can be viewed as models for jamming situations. We make the pessimistic assumption that the jammer knows the input sequence before he acts. This forces communicators to use the maximal error concept and also makes randomization in the encoding superfluous. Now, for a DMC W by a simple observation, made by Ahlswede and Dueck (1989), in the absence of randomization the identification capacity, say C/sub NRI/(W), equals the logarithm of the number of different row-vectors in W. We generalize this to compound channels. A formidable problem arises if the DMC W is replaced by the AVC W. In fact, for 0-1-matrices only in W we are-exactly as for transmission-led to the equivalent zero-error-capacity of Shannon. But for general W the identification capacity C/sub NRI/(W) is quite different from the transmission capacity C(W). An observation is that the separation codes of Ahlswede (1989) are also relevant here. We present a lower bound on C/sub NRI/(W). It implies for instance for W={(/sub 0 1//sup 1 0/), (/sub /spl delta/ (1-/spl delta/)(1 0)/)}, /spl delta//spl isin/(0, 1/2 ) that C/sub NRI/(W)=1, which is obviously tight. It exceeds C(W), which is known to exceed 1-h(/spl delta/), where h is the binary entropy function. We observe that a separation code, with worst case average list size L~ (which we call an NRA code) can be partitioned into L~2/sup ne/ transmission codes. This gives a nonsingle-letter characterization of the capacity of AVC with maximal probability of error in terms of the capacity of codes with list decoding. We also prove that randomization in the decoding does not increase C/sub I/(W) and C/sub NRI/(W). Finally, we draw attention to related work on source coding.
Classical identification via a noisy channel is a communication task in which the receiver is not required to reconstruct the full transmitted message, but only to decide whether it coincides with a message of interest. This relaxation allows the number of identifiable messages to grow doubly exponentially with the blocklength. For quantum channels, the resulting (doubly exponential) identification capacity CID can strictly exceed the ordinary (exponential) transmission capacity C.
In this paper, we prove that the entanglement-assisted transmission capacity CE is a strong converse bound for this task: CID≤CE. For sufficiently low-noise channels, this bound can also be achieved via the Hayden-Winter (quantum) identification + fingerprinting codes. This yields an exact characterization CID=CE of identification capacity for such channels. However, for general channels, we prove that this upper bound can be strict. We exhibit an explicit family of transpose-depolarizing channels for which CID<CE. As a consequence, we also obtain the first example of strict superadditivity of the identification capacity CID.
Deterministic identification (DI) for the discrete-time Poisson channel, subject to an average and a peak power constraint, is considered. It is established that the code size scales as 2(nlogn)R, where n and R are the block length and coding rate, respectively. The authors have recently shown a similar property for Gaussian channels [1]. Lower and upper bounds on the DI capacity of the Poisson channel are developed in this scale. Those imply that the DI capacity is infinite in the exponential scale, regardless of the dark current, i.e., the channel noise parameter.