Fig. 3 | Nature Communications

Fig. 3

From: Experimental optical phase measurement approaching the exact Heisenberg limit

Fig. 3

Density matrices of the experimental state and \({\rho }_{{\mathrm{opt}}}\). a Real part of the state matrix \(\rho _{{\mathrm{exp}}}\) reconstructed with polarisation state tomography. The fidelity of the state with the optimal state \(|\psi _{{\mathrm{opt}}}\rangle\), Eq. (2), is \(\langle \psi _{{\mathrm{opt}}}|\rho _{{\mathrm{exp}}}{\mathrm{|}}\psi _{{\mathrm{opt}}}\rangle = 0.980 \pm 0.003\), and the purity is \({\mathrm{Tr}}\left[ {\rho _{{\mathrm{exp}}}^2} \right] = 0.965 \pm 0.006\). The density matrix was calculated from ~50,000 twofold coincidence events. Uncertainties in fidelity and purity represent 95% confidence intervals calculated with Monte-Carlo simulation22. Imaginary components (not shown) are ≤0.013. b Real part of the ideal optimal state \(\rho _{{\mathrm{opt}}}\). Note that \({\mathrm{Im}}(\rho _{{\mathrm{opt}}}) = 0\)

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