Optimal lower bound of the average indeterminate length lossless block encoding

Consider a general quantum 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 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

maps 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 concatenation is also an

isometry, and hence the string of alphabet states is uniquely decodable. We compute the minimum average 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,

($m=1$).