Process reconstruction: From unphysical to physical maps via maximum likelihood

We show that the method of maximum likelihood (MML) provides us with an efficient scheme for the reconstruction of quantum channels from incomplete measurement data. By construction this scheme always results in estimations of channels that are completely positive. Using this property we use the MML for a derivation of physical approximations of unphysical operations. In particular, we analyze the optimal approximation of the universal NOT gate as well as the physical approximation of a quantum nonlinear polarization rotation.

Figure 1

(Color online) A schematic representation of a reconstruction of a single-qubit channel. Input (test) states of the single-qubit channels are represented by the Bloch sphere (the state space of a single qubit). At the output of the single-qubit channel (modeled as an ellipsoid—i.e., the Bloch sphere that is “deformed” by the action of the channel) a complete measurement of test states is performed. The complete measurement is performed via the projective measurement of $\sigma$ operators. Based on correlations between input and output states of the test qubits the action of the quantum channel (a CP map) is determined (estimated).

Figure 2

(Color online) Unital CP maps are embedded in the set of all positive unital maps (cube). The CP maps form a tetrahedron with four unitary transformations in its corners (extremal points), $I$, $x$, $y$, and $z$, corresponding to the Pauli $\sigma$ matrices. The unphysical UNOT operation $({\lambda }_1={\lambda }_2={\lambda }_3=-1)$ and its optimal completely positive approximation quantum universal NOT gate $({\lambda }_1={\lambda }_2={\lambda }_3=-1∕3)$ are shown.

Figure 3

(Color online) The distance $d({\mathcal{E}}_{\mathrm{NOT}},{\mathcal{E}}_{\mathit{est}})$ as a function of the number of measured outcomes, $N$, in logarithmic scale. We used six input states (eigenvectors of ${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$) and measured ${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$. The distance converges to the theoretical value $1∕3$, which corresponds to the optimal universal NOT.

Figure 4

(Color online) We present analytical as well as numerical results of an approximation of a nonlinear map ${\mathcal{E}}_{\theta }$ for different values of the parameter $\theta$ (measured in radians). The numerical (“experimental”) results shown in the graph in terms of a set of discrete points with error bars are obtained via the MML. The theoretical approximation ${\tilde{\mathcal{E}}}_{\theta }$ of the nonlinear NPR map is characterized by the parameter $\lambda$, which is plotted (solid line) in the figure as a function of the parameter $\theta$. In the inset (a) is the Bloch sphere transformation for $\theta =3$ obtained by MML and in (b) the same transformation obtained analytically.