Statistical estimation of the quality of quantum-tomography protocols

We present a complete methodology for testing the performances of quantum tomography protocols. The theory is validated by several numerical examples and by the comparison with experimental results achieved with various protocols for whole families of polarization states of qubits and ququarts including pure, mixed, entangled, and separable.

Figure 1

Tetrahedron and distributions fidelity loss over the Poincaré-Bloch sphere for the protocol based on tetrahedron geometry. Color bar shows level of average fidelity loss $L$.

Figure 2

Shapes of solids and distributions of average fidelity loss over the Poincaré-Bloch sphere for the protocol based on cube and octahedron geometry. Color bar shows level of average fidelity loss $L$.

Figure 3

Shapes of solids and distributions of average fidelity loss over the Poincaré-Bloch sphere for the protocol based on dodecahedron and icosahedron. Color bar shows level of average fidelity loss $L$.

Figure 4

Shapes of solids and distributions of average fidelity loss over the Poincaré-Bloch sphere for the protocol based on fullerene and its dual polyhedron. Color bar shows level of average fidelity loss $L$.

Figure 5

The universal statistical fidelity loss distribution for 200 numerical experiments (histogram) compared with the theoretical expectation (solid curve). Sample size of each experiment is 1 million.

Figure 6

Density distribution of the scaled fidelity $z$ for three qubit state with different value of mixture GHZ and white noise states. The measurement protocol is based on dodecahedron. Sample size $n$ is equal to 1 million.

Figure 7

Density distribution of the scaled fidelity $z$ for Bell and GHZ states. The measurement protocol is based on tetrahedron. Sample size $n$ is equal to 1 million.

Figure 8

Experimental setup for different tomographic reconstructions of qubits with variable degree of mixture. Here H, V are Glan-Thompson prisms, $h_0$ is a thick quartz plate, $h_1$ is a thick tilt plate, $h_d$ is a driving phase plate, $h_{(}2,3)$ are tomographic phase plates, and D is a detector.

Figure 9

Calculated condition numbers $K$ and maximum losses $\max L$ on the Poincaré-Bloch sphere vs the optical thickness $\delta$ of a phase plate for the B9 protocol. The dashed and dash-dotted lines are for the R4 and J4 protocols, respectively .

Figure 10

Difference between the (a) and (b) optimal and (c) and (d) nonoptimal approach for protocol B9. (a) and (c) and (b) and (d) correspond to ${20}^{\circ }$ and $1^{\circ }$ steps of the plate orientation, respectively.

Figure 11

Reconstruction of pure qubit states by protocol B9 at different thicknesses of the plate. Vertical bars show 1% and 99% quantiles for fidelity distributions. Dotted lines connecting lower bar ends point out critical significant levels.

Figure 12

Theoretical distribution of the real and imaginary parts of the density matrix for states with varying degrees of purity. The spectral widths of the spectrum are (a) 1.6 nm, (b) 7 nm, and (c) 13 nm. Distribution (d), for 22 nm, corresponds almost to a completely mixed state.

Figure 13

Reconstruction of qubits states with various degree of mixture. Vertical bars show 1% and 99% quantiles for fidelity distributions. Dotted lines connecting lower bar ends point out critical significant levels. (a) Mixed 3%, (b) mixed 30%, (c) mixed 66%, and (d) mixed 100%

Figure 14

Experimental setup for different tomographic reconstructions of photon pairs with variable polarization entanglement and degree of mixture. Ar laser: argon laser with wavelength $351\text{nm}$, M: mirror, V: vertical oriented Glan-Thompson prism, BBO: nonlinear barium borate crystals, UVF: ultraviolet filter, ${\lambda }_p/2$, ${\lambda }_p/4$: half-wave plates, quarter-wave plates, L: lens with focus $20\text{cm}$, BS: beamsplitter, IF: interfilter, D: avalanche photodetectors, CC: coincidence circuit.

Figure 15

Dependence of ${\log }_{10}\left(K\right)$ on plates thicknesses for protocol B144.

Figure 16

Dependence of the average fidelity on number of registered events forming the sample for Bell state reconstructed by protocols J16, K16, and B144.

Figure 17

Density distribution of the scaled fidelity $z$ (lower abscissa) at $n=3\times {10}^3$ for Bell state reconstructed by protocols J16, K16, and B144. Upper abscissa presents regular fidelity.

Figure 18

Reconstruction of ququart states with various degree of entanglement. Vertical bars show 1% and 99% quantiles for fidelity distributions. Dotted lines connecting lower bar ends point out critical significant levels.

Figure 19

Reconstruction of ququart states with various degree of mixture. Vertical bars show 1% and 99% quantiles for fidelity distributions. Dotted lines connecting lower bar ends point out critical significant levels. (a) $p=0.3$, (b) $p=0.7$, and (c) $p=1$.

Figure 20

Analysis of precision losses for protocol proposed in [35, 53].