Approximate Unitary and Truncated Optimum Transformations

Since unitarity is the key property of the transformations it is clear that an optimum truncated transformation is the better suited the better this condition is fulfilled, i.e., the smaller the operator norm UU — 11 is. In the following, this principle will be exploited to determine the optimum parametrization of truncated unitary matrices. If we consider a truncated transformation U 

Because of the unitarity conditions all other coefficients would automatically vanish as well, and U would be the identity transformation. The coefficient a-y has therefore necessarily to be chosen to be different from zero, and since defines only a simple scaling of W, we may choose = 1. 

If the expansion for U is driven to higher orders, the xmitarity condition reads 

This expression will in general be minimal if the first term in parentheses vanishes. As a quadratic expression for ag, it has two solutions. In order to achieve the smallest deviation of U from unitarity possible, we select the smaller of these two solutions, which reads 

Having previously fixed ag according to Eq. (11.70), the transformation U closest to unitarity including the next higher terms of the series expansion of 

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