By C. Macchiavello, G. M. Palma, Anton Zeilinger

The reprinted lawsuits of the ISI TMR community college, held in Villa Gualino, Torino, Italy, July 12-23, 1999. contains the introductory notes, and a bit of 3 introductory papers which clarify the elemental matters in quantum physics that are referred to as quantum info concept.

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4. The 2 x 2 J-unitary and J-contractive matrix functions with entries in the Nevanlinna class will be important when we develop the recurrence relations for the kernels and the orthogonal rational functions. 5. 1. Hardy classes We shall be concerned with complex function theory on the unit circle and the upper half plane. The complex number field is denoted by C We use the following notation for the unit circle, the open unit disk, and the complement 15 16 L Preliminaries of the closed unit disk: T = {z:\z\ = 1}, B = {z:\z\< 1}, E = {z : \z\ > 1}.

In the more general case where some of the ak do coincide, the Gram matrix looks more complicated and involves derivatives of Q^ evaluated at the at. To see this, we give a technical lemma first. 2. Consider the Riesz-Herglotz kernel D(z, w) = (z+w)(z — w)~l for the disk. Then z, w) = 2(k\)z(z - w)-(M\ k > 1, where 3^ denotes the £th derivative with respect to w. We also have [dkwD(z, w)]m = 2(k\)zk(l-wzy(k+1\ k > 1, where the substar transform is with respect to z. Furthermore, if Q^(w) = i I m ^ ( 0 ) + D(t,w)d/jL(t), then for k > 1 QS£\w) = dkw^(w) = JdkwD(t, w)dn{t) = and (substar with respect to t).

Working this out gives 1 - da > 0, where a — (0\u + 012*)"1 (#21* + 022*) and with equality on 9©. This identifies a as an inner function. 5. J-unitary and J-contractive matrices 41 The following theorem describes a simple matrix from the class By. 4. 50). Proof. This is a classical result that can be found, for example, in Potapov [180, pp. 187-188]. • A matrix in N2x2 that is J-contractive in O and J-unitary will be called J-inner, a terminology used by Dym [77]. The set of J-inner matrix functions is denoted as Bj = By n Ty while we set Bj = By HTy.