forgot some modulus there
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@ -261,11 +261,11 @@ and $s=0$ are obtained with probability $\frac{1}{2}$ and after choosing a $j
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\begin{equation}
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\begin{equation}
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\begin{aligned}
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\begin{aligned}
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P(s=0) &= \hbox{Tr}\left(\frac{I + g_a}{2}\ket{\psi}\bra{\psi}\right) \\
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P(s=0) &= \left|\hbox{Tr}\left(\frac{I + g_a}{2}\ket{\psi}\bra{\psi}\right) \right|\\
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&= \hbox{Tr}\left(\frac{I + g_a}{2}S^{(j)} \ket{\psi}\bra{\psi}\right)\\
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&= \left|\hbox{Tr}\left(\frac{I + g_a}{2}S^{(j)} \ket{\psi}\bra{\psi}\right)\right|\\
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&= \hbox{Tr}\left(S^{(j)}\frac{I - g_a}{2}\ket{\psi}\bra{\psi}\right)\\
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&= \left|\hbox{Tr}\left(S^{(j)}\frac{I - g_a}{2}\ket{\psi}\bra{\psi}\right)\right|\\
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&= \hbox{Tr}\left(\frac{I - g_a}{2}\ket{\psi}\bra{\psi}S^{(j)}\right)\\
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&= \left|\hbox{Tr}\left(\frac{I - g_a}{2}\ket{\psi}\bra{\psi}S^{(j)}\right)\right|\\
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&= \hbox{Tr}\left(\frac{I - g_a}{2}\ket{\psi}\bra{\psi}\right)\\
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&= \left|\hbox{Tr}\left(\frac{I - g_a}{2}\ket{\psi}\bra{\psi}\right)\right|\\
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&= P(s=1)
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&= P(s=1)
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\end{aligned}
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\end{aligned}
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\notag
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\notag
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