Relative tomography of an unknown quantum state

We propose a tomography scheme capable of reconstructing the quantum state of an unknown mode of light. The complex mode structure of the investigated field is expressed by a single number—the mismatch between the probe and signal. This parameter is estimated as a part of the quantum tomography protocol. This opens up ways for reconstructions utilizing interference between independent sources.

Figure 1

Quantum-state estimation in the presence of mismatch. Panel (a): the true Wigner function for the signal small-amplitude Schrödinger cat state (12) $(\chi =1)$. Panel (b): “stretched” point-by-point reconstruction for the true mismatch $\mu =0.5$. Panel (c): true elements of the density matrix in the Fock-state basis for the signal small-amplitude Schrödinger cat state of panel (a). Panel (d): differences ${\Delta }_{mn}=\mid {\rho }_{mn}-{\rho }_{mn}^{\mathit{\text{true}}}\mid$ between the true ${\rho }^{\mathit{\text{true}}}$ and the reconstructed $\rho$ elements of the signal density matrix. The estimated value $\mu =0.49$ has been adopted for ML inversion.

Figure 2

Error analysis of the mismatch estimation and the corresponding ML reconstruction. Panel (a): the histogram of estimated values of the mismatch parameter for 1000 runs for the constant true mismatch of $\mu =0.5$. Panel (b): intervals of estimated mismatches for the stochastically behaving true values of the mismatch described by the true value $\mu =0.5$ and variance $\delta \mu$. Each averaging was done over 100 runs of the reconstruction procedure. Panel (c): Hilbert-Schmidt distance (HSD) (13) between ${\rho }^{\mathit{\text{true}}}$ and ML estimation $\rho$ plotted as a function of the estimated mismatch $\mu$ for the constant true value of the mismatch $\mu =0.5$. Compare the abscissas of panels (a) and (c). Panel (d): the mean HSD (13) and variance for the true mismatch which changes stochastically as in panel (b).