SIC-POVM - Properties - Superoperator

Superoperator

In using the SIC-POVM elements, an interesting superoperator can be constructed, the likes of which map . This operator is most useful in considering the relation of SIC-POVMs with spherical t-designs. Consider the map

 \begin{align} \mathcal{G}: \mathcal{B}(\mathcal{H}) &\rightarrow \mathcal{B}(\mathcal{H})\\ A &\mapsto \displaystyle \sum_\alpha |\psi_\alpha \rangle \langle \psi_\alpha | A |\psi_\alpha \rangle \langle \psi_\alpha | \end{align}

This operator acts on a SIC-POVM element in a way very similar to identity, in that

 \begin{align} \mathcal{G}(\Pi_\beta) &= \displaystyle \sum_\alpha \Pi_\alpha \left| \langle \psi_\alpha | \psi_\beta \rangle \right|^2 \\ &= \displaystyle \Pi_\beta + \frac{1}{d+1} \sum_{\alpha \neq \beta} \Pi_\alpha \\ &= \displaystyle \frac{d}{d+1} \Pi_\beta + \frac{1}{d+1} \Pi_\beta + \frac{1}{d+1} \sum_{\alpha \neq \beta} \Pi_\alpha \\ &= \displaystyle \frac{d}{d+1} \Pi_\beta + \frac{d}{d+1}\sum_\alpha \frac{1}{d}\Pi_\alpha \\ &= \displaystyle \frac{d}{d+1} \left( \Pi_\beta + I \right) \end{align}

But since elements of a SIC-POVM can completely and uniquely determine any quantum state, this linear operator can be applied to the decomposition of any state, resulting in the ability to write the following:

where

From here, the left inverse can be calculated to be, and so with the knowledge that

,

an expression for a state can be created in terms of a quasi-probability distribution, as follows:

 \begin{align} \rho = I | \rho ) &= \displaystyle \sum_\alpha \left \frac{ (\Pi_\alpha|\rho)}{d} \\
&= \displaystyle \sum_\alpha \left \frac{ \mathrm{Tr}(\Pi_\alpha\rho)}{d} \\
&= \displaystyle \sum_\alpha p_\alpha \left \quad \text{ where } p_\alpha = \mathrm{Tr}(\Pi_\alpha\rho)/d\\
&= \displaystyle -I + (d+1) \sum_\alpha p_\alpha |\psi_\alpha \rangle \langle \psi_\alpha | \\
&= \displaystyle \sum_\alpha \left |\psi_\alpha \rangle \langle \psi_\alpha |
\end{align}

where is the Dirac notation for the density operator viewed in the Hilbert space . This shows that the appropriate quasi-probability distribution (termed as such because it may yield negative results) representation of the state is given by

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