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BACKGROUND

The Peres-Mermin magic square consists of different combinations of tensor products of Pauli operators being applied on an arbitrary bipartite state. Suppose we're considering the entry in the third row and first column, namley $\hat{\sigma}_x \otimes \hat{\sigma}_z$. Assuming the arbitrary state under consideration is called $\hat{\rho}$, what operations gives us the value ($\pm 1$) in that entry?

Is it \begin{equation} \mbox{Tr}\{ \hat{\sigma}_x \otimes \hat{\sigma}_z \cdot \hat{\rho}\}? \end{equation}

That can't be right since $\hat{\sigma}_x \otimes \hat{\sigma}_z$ isn't a von Neumann projector, but rather a unitary transformation, in which case the entry should be

\begin{equation} \mbox{Tr}\{ [ \hat{\sigma}_x \otimes \hat{\sigma}_z ]^{\dagger} \cdot \hat{\rho} \cdot \hat{\sigma}_x \otimes \hat{\sigma}_z\}. \end{equation}

That can't be right either since there is no projector per se, hence the trace won't make sense as a measurement.

QUESTION

What is the function $f$($\hat{\sigma}_x \otimes \hat{\sigma}_z$, $\hat{\rho}$) that produces one of the $\pm 1$ outcomes?

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