Definitions in quantum computation and quantum information

mode: we can simply understand this as different eigenvectors in Dirac basis: \(|0\rangle\) is a mode, and \(|1\rangle\) is also a mode.

classical-quantum states^[1]: \(\chi^{(A)}=\sum_{i} p_{i}|i\rangle\left\langle\left. i\right|^{A} \otimes \tau_{i}^{B}\right.\) where the states \(\left\{|i\rangle^{A}\right\}\) form an orthonormal basis for subsystem \(A\), and \(\left\{\tau_{i}^{B}\right\}\) denote a set of arbitrary states for subsystem \(B\), while \(\left\{p_{i}\right\}\) stands for a probability distribution.

Helstrom bound:The quantum multiparameter Cramér–Rao bound, also known as Helstrom bound^[2].

Bloch representation for d-dimensional density matrix^[2:1]: Bloch representation is another well-used tool in quantum information theory. For a \(d\)-dimensional density matrix, it can be expressed by

where \(\vec{r}=\left(r_{1}, r_{2} \ldots, r_{m}, \ldots\right)^{\mathrm{T}}\) is the Bloch vector \(\left(\left.\vec{r}\right|^{2} \leqslant 1\right)\) and \(\vec{\kappa}\) is a \(\left(d^{2}-1\right)\)-dimensional vector of \(\mathfrak{s u}(d)\) generator satisfying \(\operatorname{Tr}\left(\kappa_{i}\right)=0\).