Priority Choice Experimental Two-Qubit Tomography: Measuring One by One All Elements of Density Matrices

In standard optical tomographic methods, the off-diagonal elements of a density matrix ρ are measured indirectly. Thus, the reconstruction of ρ, even if it is based on linear inversion, typically magnifies small errors in the experimental data. Recently, an optimal tomography solution measuring all the elements of ρ one-by-one without error magnification has been theoretically proposed. We implemented this method for two-qubit polarization states. For comparison, we also experimentally implemented other well-known tomographic protocols, either based solely on local measurements (of, e.g., the Pauli operators and James-Kwiat-Munro-White projectors) or with mutually unbiased bases requiring both local and global measurements. We reconstructed seventeen separable, partially and maximally entangled two-qubit polarization states. Our experiments show that our method has the highest stability against errors in comparison to other quantum tomographies. In particular, we demonstrate that each optimally-reconstructed state is embedded in an uncertainty circle of the smallest radius, both in terms of trace distance and disturbance. We explain how to experimentally estimate uncertainty radii for all the implemented tomographies and show that, for each reconstructed state, the relevant uncertainty circles intersect indicating the approximate location of the corresponding physical density matrix.

Here we show explicitly all the density matrices discussed in the Letter, which are reconstructed with the optimal tomographic protocol and those based on: (i) mutually unbiased bases, (ii) the James-Kwiat-Munro-White projectors, (iii) the tensor products of the Pauli operators, and (iv) the standard separable basis corresponding to all the eigenvectors of the Pauli operators. We also present the coefficient matrices, observation vectors corresponding to coincidence counts, the estimated variances for the observations, and the error radii for each reconstructed matrix. Finally, we compare the reconstructed matrices graphically, where we show the relative trace distances between the reconstructed states and they error radii.
Thus, a two-qubit density matrix ρ is represented as a real vector x = (x 1 , ..., x 16 ) with its elements given as follows The coefficient matrices depend on the choice of the equations used for reconstructing a given density matrix. Below we list the transposed (for typographic reasons) coefficient matrices for the four analyzed tomographic protocols:

III. OBSERVATION VECTORS
The observation vectors correspond to photon coincidence counts. In reality we measure disturbed quantities b ≡ b + δ b instead of b. The observation vectors are column vectors. For convenience we arrange them in arrays, where each column corresponds to one of the 17 reconstructed states.
Note that the values of b listed below are not normalized and cannot be interpreted as probabilities. The elements of each vector b were registered over 5 seconds. This means that if an element of b is a sum or a difference of n projectors the measurement for each of the n projectors took 5/n seconds. In this way the measurements for observation vectors of the same length take the same amount of time. To obtain the frequencies we can divide these values by the total number of photon coincidences counted or by a sum of coincidences counted for a set of projectors forming a basis.
The set of such projectors is not unique. In our calculations we use the unnormalized coincidences and normalize the reconstructed density matrices.

C. Relative trace distances between the reconstructed states
In order to compare the quality of the matrices reconstructed with different tomographic protocols, we have also calculated the relative trace distances for the respective states in each protocols. Here we omitted the JKMW protocol, because it provides the largest error radius. Having three relative distances (for the remaining three protocols) it is possible to visualize the relative distances between the matrices and their error radii on a plane.