Optimal lower bound for 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 $m$ many states emitted by the quantum stochastic source by breaking the tensor product 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 \lq\lq special block codes\rq\rq\ 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 $m=1$. Lastly, we give a simplified non-adaptive compression technique based on constrained special block codes for general quantum stochastic sources. For quantum stationary sources in particular, we show that the minimum average codeword length per symbol computed over all constrained special block codes is equal to the von-Neumann entropy rate of the source for an asymptotically long block size.