Optimal Lower Bound of the Average Indeterminate Length Lossless Quantum Block Encoding

Consider a general quantum stochastic source that emits at discrete time steps quantum pure states which are chosen from a finite alphabet according to some probability distribution which may depend on the whole history. Also, fix two positive integers $m$ and l. We encode any tensor product of ml many states emitted by the given quantum source by breaking it into $m$ many blocks where each block has length l, and considering sequences of $m$ many isometries so that each isometry encodes one of these blocks into the Fock space, followed by the concatenation of their images. We only consider certain sequences of such isometries that we call “special block codes” in order to ensure that the string of encoded states is uniquely decodable. We compute the minimum average codeword length of these encodings which depends on the quantum source and the integers m, l, among all possible special block codes. Our result extends the result of [Bellomo, Bosyk, Holik and Zozor, Scientific Reports 7.1 (2017): 14765] where the minimum was computed for one block, i.e. for m = 1.