Sequential Analysis of a finite number of Coherent states

We investigate an advantage for information processing of ordering a set of states over making a global quantum processing with a fixed number of copies of coherent states. Suppose Alice has $N$ copies of one of two quantum states ${\sigma }_0$ or ${\sigma }_1$ and she gives these states to Bob. Using the optimal sequential test, the SPRT, we ask if processing the states in batches of size $l$ is advantageous to optimally distinguish the two hypotheses. We find that for the symmetric case $\{\mathrm{\mid }\gamma ⟩,\mathrm{\mid }-\gamma ⟩\}$ there is no advantage of taking any batch size $l$. We give an expression for the optimal batch size $l_{\text{opt}}$ in the assymetric case. We give bounds $l_{\text{min}}$ and $l_{\text{max}}$ for when $P_S\approx 1$.