By Cherif Bendjaballah, Osamu Hirota, Serge Reynaud

Quantum conversation concept is receiving a lot recognition because the foundation for functional optical communications and quantum measurements. This booklet is a special survey of the actions of the prime study teams from mathematical physics, details thought, and diverse parts in quantum optics. The contributions take care of either idea and foundations of quantum physics including technical functions.

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**Example text**

1 . 2 . 2 D 3 . 3 . 3 . 4 . 4 5 . 5 . 6 B 7 . 7 . 8 9 m . . . . 3 4 5 6 7 8 9 10 . . . . . 1 . 2 . 2 A 3 . 4 6 7. C 8 10 E m Figure 13: Conﬁguration of all 4 rectangles with central line. We consider the left and right hand corner points in this conﬁguration.

4 . 4 5 . 5 . 6 . 6 7 . 7 . 8 ... ... ...... ... ... ... . .. ... ... ... ... . . ... . . ... . ... . ... ... . ... ... . ..... ... ... ... ... . ... . ... ... ... ... ... ... ... . . ... ... ... ... ... ..... .. 5 . 6 7 . 7 . 8 9 − LL − − − − − − A (0, 7, 2, 9)-rectangle . . . 4 5 6 7 8 9 4 5 6 7 8 9 10 . 6 . 7 . 8 . − − − RRR − − − A (3, 7, 7, 11)-rectangle ..... ...... ... ... ... ... . . ... . ... . ... ... . ... . ...

N + 1 lie in S but i − 1 ∈ S. Then edge i − 1 of P is labelled R and is the leftmost labelled edge in P . (d) The elements of S not in an initial segment as in (b) or a terminal segment as in (c) give rise to edges L or R in P as in (a). Example. Let n = 13. If S = {1, 2, 3, 4, 7, 8, 11}, then P = − − LRRLLRR − − −. See Figure 6 for an explanatory diagram. Applying this bijection S → P to the chamber sets obtained from a reduced word i we obtain a set of 12 n(n − 1) partial quivers associated with i.