Notation

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For two hermitian operators we write:

\[ A \leq B \]

if \(\langle\psi|B - A|\psi\rangle \geq 0\) for all \(|\psi\rangle\). In the special case that one operator is just a multiple of the Identity, e.g. \(A=aI\), we write:

\[ a \leq B \]

We define the Kronecker delta to be

\[ \delta_{ij} = \begin{cases} 0 & \text{if } i \neq j, \\ 1 & \text{if } i = j. \end{cases} \]

The Fourier transform is denoted like that:

\[ \FT \sum_{k=0}^{N-1} x_k |k\rangle = \frac{1}{\sqrt{N}} \sum_{k,j=0}^{N-1} x_k e^{\frac{2\pi\ii}{N}kj} |j\rangle . \]