Quantum State Tomography of Qudits via Hong-Ou-Mandel Interference

We propose a method to perform the quantum state tomography (QST) of an $n$-partite qudit state embedded in single photons based on the Hong-Ou-Mandel interference between the target photon and ancillary probe light. Our method requires only passive beam splitters in the target modes by shifting all of the active devices used in conventional QST methods to the probe preparation system. This feature is useful when the active optical devices for QST have large optical losses. Moreover, even when the probe is prepared by a laser light having a mode mismatch with the target photon, the method offers an accurate estimation of the target state. As a proof-of-principle, we perform the experimental demonstration using a polarization qubit. Regardless of the photon statistics of the probe light, the estimated results of state reconstruction are as accurate as those verified by the conventional QST.

Figure 1

(a) Conventional QST. Projective measurements on the target photons are performed by the active optical devices in the target modes followed by the photon detection. The active optical devices are specialized in the DOF of interest. (b) Concept of HOM QST. The active optical devices are used to switch the probe states and are placed outside the target modes. (c) A detailed view of the interferometer in HOM QST when the target photon is a single qudit. A target photon in state $\rho$ and probe photon in state $|k⟩$ are mixed by a HBS. The depth of the HOM dip (${\mathrm{\Delta }}_k$) reflects the information of the $k$ portion of the target photon as ${\mathrm{\Delta }}_k\propto ⟨k|\rho |k⟩$.

Figure 2

The experimental setup. The setup consists of two stages, namely preparation of the target photon and probe light as well as preparation of the HOM interferometer for HOM QST. In the first stage, the HSPS is generated as the target photon by SPDC. The probe light is prepared in a HSPS or a thermal state via SPDC with or without using heralding signals from $D_4$, or in a coherent state via difference frequency generation. SPF, short-pass filter; DM, dichroic mirror; SHG, second harmonic generation; PPLN/W, periodically poled lithium niobate waveguide; VHG, volume holographic grating.

Figure 3

(a)–(d) Observed HOM dips between the $D$-polarized target photons and (a) $D$-polarized HSPS, (b) $H$-polarized HSPS, (c) $D$-polarized thermal-state, and (d) $D$-polarized coherent-state probes. The measurement times for the data points in HOM QST are 30 s each for (a), (b) HSPS and (c) thermal-state probes and 10 s for the (d) coherent-state probe. (e)–(h) Real parts of the reconstructed density matrices of the $D$-polarized target states using (e) conventional one-qubit QST and HOM QST with (f) HSPS, (g) thermal-state, and (h) coherent-state probes. Each value on the vertical axis corresponds to the real part of the matrix element, e.g., the value for label HV corresponds to $\mathrm{Re}⟨H|\hat{\rho }|V⟩$. The error bars indicate the standard deviations estimated using the Monte Carlo method under the assumption of Poisson statistics for the photon counts. Note, the smaller error bars of Fig. 3 come from the larger count data obtained by the conventional method.

Figure 4

Density matrices reconstructed using (a) conventional QST as well as HOM QST with (b) HSPS, (c) thermal-state, and (d) coherent-state probes. The left and right parts of each figure correspond to the real and imaginary parts of the reconstructed density matrix, respectively.