Direct Reconstruction of the Quantum Density Matrix by Strong Measurements

New techniques based on weak measurements have recently been introduced to the field of quantum state reconstruction. Some of them allow the direct measurement of each matrix element of an unknown density operator and need only $O(d)$ different operations, compared to $d^2$ linearly independent projectors in the case of standard quantum state tomography, for the reconstruction of an arbitrary mixed state. However, due to the weakness of these couplings, these protocols are approximated and prone to large statistical errors. We propose a method which is similar to the weak measurement protocols but works regardless of the coupling strength: our protocol is not approximated and thus improves the accuracy and precision of the results with respect to weak measurement schemes. We experimentally apply it to the polarization state of single photons and compare the results to those of preexisting methods for different values of the coupling strength. Our results show that our method outperforms previous proposals in terms of accuracy and statistical errors.

Figure 1

The reconstruction of the density matrix ${\varrho }_{\mathcal{S}}$ needs two bidimensional pointers. Measurement $A$ ($B$) consists of a unitary evolution, $U_{A,j}$ ($U_B$), and a projective measurement on the pointer $A$ ($B$). Finally, a projective measurement on the state $|a_k⟩$ is performed.

Figure 2

Experimental scheme.

Figure 3

Trace distance between reconstructed states and reference states for different input purity: $T({\varrho }^W,{\varrho }^Q)$ (circles), $T({\varrho }^{\mathrm{I}},{\varrho }^Q)$ (triangles), $T({\varrho }^{\mathrm{II}},{\varrho }^Q)$ (squares).

Figure 4

Trace distance between reconstructed states and reference state, for the input states (top) $|D⟩$ and (bottom) maximally mixed. The solid line represents the theoretical trace distance between the expected value of ${\varrho }^W$ and the experimental ${\varrho }^Q$.

Figure 5

Mean square statistical error on the reconstructed states, for the input states (top) $|D⟩$ and (bottom) maximally mixed. The lines represent the theoretical expectations for $\delta {\varrho }^W$ (points), $\delta {\varrho }^{\mathrm{I}}$ (dashes and points), and $\delta {\varrho }^{\mathrm{II}}$ (dashes).