Divergence radii and the strong converse exponent of classical-quantum channel coding with constant compositions

Abstract

There are different inequivalent ways to define the Rényi capacity of a channel for a fixed input distribution P . In a 1995 paper [16] Csiszár has shown that for classical discrete memoryless channels there is a distinguished such quantity that has an operational interpretation as a generalized cutoff rate for constant composition channel coding. We show that the analogous notion of Rényi capacity, defined in terms of the sandwiched quantum Rényi divergences, has the same operational interpretation in the strong converse problem of classical-quantum channel coding. Denoting the constant composition strong converse exponent for a memoryless classical-quantum channel W with composition P and rate R as sc(W, R, P ), our main result is that

The Augustin Capacity and Center

“…For constant composition codes on the memoryless classical-quantum channels, the Augustin information for orders less than one arises in the expression for the sphere packing exponent and the Augustin information for orders greater than one arises in the expression for the strong converse exponent, as recently pointed out by Dalai [3] and by Mosonyi and Ogawa [4], respectively. For the constant composition codes on the discrete stationary product channels, these observations were made implicitly by Csiszár and Körner in [5, p. 172] and by Csiszár in [2].…”

Refined Strong Converse for the Constant Composition Codes

“…Thus Arimoto's bound is tight, in terms of the exponential decay rate of the probability of correct decoding with block length, for all rates above the channel capacity. An analogous result is derived for constant composition codes on DSPCs in [22], for the Gaussian channel in [23], for classical-quantum channels in [18], for classical data compression with quantum side information in [20], and for constant composition codes on classical-quantum channels in [24]. Although Arimoto's bound, given in (1), is tight in terms of the exponential decay rate of the correct decoding probability with block length, the prefactor multiplying the exponentially decaying term can be improved.…”

“…Proof of Theorem 1. The existence of a unique order α * satisfying (28) was proved and its value was determined in §IV, see (23), (24), and (25).…”

On the Existence of the Augustin Mean

“…Hence, (19) follows from (21) and the fact that f α ∈ B τα ≤1 (q p ). Furthermore, f α τα = 1 as a result of ( 19), (20), and (21).…”