Figure 1

Picturing the vector quartet of Eq. (2.5) as pointing from the center to four nonadjacent corners of a cube or as the vectors normal on the faces of a tetrahedron.Reuse & Permissions

Figure 2

Schematic setup of an ellipsometer that implements the standard six-state POVM; see text.Reuse & Permissions

Figure 3

(a) Principle of a four-path interferometer for minimal ellipsometry and (b) optical network for the implementation; see text.Reuse & Permissions

Figure 4

Quantum computation network for minimal qubit tomography. The bottom qubit is the one of interest; it enters in the state $\rho$ whose properties are to be determined. The two top qubits are auxiliary qubits that enter in state 0 of the computational basis and are eventually measured. The probabilities for the four different measurement results—00, 01, 10, and 11—are the $p_j$’s of Eq. (2.10) with the ${\vec{a}}_j$’s of Eq. (2.6). The network uses eight gates of the generalized Hadamard type (3.1), three of them controlled by one of the auxiliary qubits, and a controlled phase gate.Reuse & Permissions

Figure 5

The likelihood function as a probability density for the Pauli vector $\vec{s}$. The plot shows the “likelihood clouds” for simulated measurements for which the most likely Pauli vector has length $s=0.84$ and is pointing toward one of the corners of the cube of Fig. 1. A total of $100\text{qubits}$ have been detected for the bottom right corner, 200 for the bottom left corner, 400 for the top rear corner, and 800 for the top front corner. The successive shrinking of the cloud is clearly visible.Reuse & Permissions

Figure 6

Mean value of the distance from the ML estimator $\vec{S}$ to the true Pauli vector $\vec{s}$ for 40 simulated experiments with up to 6000 detected qubits per run. The top plot is for $\vec{s}=0$, the bottom plot for $\vec{s}=-{\vec{a}}_4$. The solid lines indicate one standard deviation to each side of the mean value, as obtained from the large-$N$ approximations for ${\left(\Delta \vec{s}\right)}^2$ in Eqs. (4.21, 4.24), respectively.Reuse & Permissions

Figure 7

Mean estimation errors of the POVM tetrahedrons that, on the Bloch sphere, differ from the exact parallel (squares) and antiparallel (circles) orientations by the known angle given on the abscissa.Reuse & Permissions

Figure 8

For each value of ${\kappa }^2$ between ${\kappa }^2=0$ (perfect antialignment) and ${\kappa }^2=1∕18$ (perfect alignment), the possible values of the coefficient of the $1∕N$ term in Eq. (6.8) are in the area bounded by the two curves. The smallest coefficient obtains for the case of perfect alignment, when Eq. (6.9) applies.Reuse & Permissions

Figure 9

Mean error (curves $a$, decreasing) and mean fidelity (curves $b$, increasing) of the minimal qubit tomography as a function of $N$, the size of the measured ensemble. Solid lines: the proposed network of Figs. 3, 4 with the adaptive parallel strategy described in the text. Dashed lines: the quantum limit of Eq. (6.4). Notice that the proposed optical network attains the quantum limit asymptotically for large $N$.Reuse & Permissions

Figure 10

Relative difference $\Delta F$, in percent, between the errors of the random and parallel adaptive strategies (black dots) and least-squares linear fit (straight line). The lines connecting the dots guide the eye but have no further significance. The irregularities stem from the limited number of input states (200 000) used for the averaging.Reuse & Permissions