Experimental Realization of Optimal Noise Estimation for a General Pauli Channel

We present the experimental realization of the optimal estimation protocol for a Pauli noisy channel. The method is based on the generation of 2-qubit Bell states and the introduction of quantum noise in a controlled way on one of the state subsystems. The efficiency of the optimal estimation, achieved by a Bell measurement, is shown to outperform quantum process tomography.

Figure 1

(a) Experimental setup. Photons $A$ and $B$ are spatially and temporally superimposed on a symmetric beam splitter (BS). The optical path delay $\Delta x$ allows us to vary the arrival time of the photons on the BS. Photons are collected by using an integrated system, composed by a GRIN lens (GL) and a single mode fiber, and then detected by single photon counters. The same setup allows us to perform the ancillary assisted quantum process tomography (AAQPT) after removing the BS. Quantum state tomography (QST) [20] on the output state is performed by using quarter–wave plates (QWPs), half–wave plates (HWPs), and polarizing beam splitters (PBSs). (b) Scheme of the implemented Pauli channel. $t_1$, $t_2$, $t_3$ represent the time intervals of ${\sigma }_x$, ${\sigma }_y$, or ${\sigma }_z$ activation. Both $t_1$, $t_2$, $t_3$ and the repetition time $T$ can be varied by a remote control.

Figure 2

Experimental probabilities of measuring the four Bell states obtained for four different cases of anisotropic noise. The black boxes report the corresponding theoretical values. (a) Identity: noiseless channel. (b) ${\sigma }_y$: only one Pauli matrix, ${\sigma }_y$ is acting on the state $|{\psi }^{-}⟩$. (c) Partially anisotropic DC: ${\sigma }_x$ and ${\sigma }_z$ operate for the same time interval, in fact the probabilities of measuring the states $|{\psi }^{+}⟩$ and $|{\varphi }^{+}⟩$ are equal. (d) Totally anisotropic DC: each Pauli operator operates for a different time interval.

Figure 3

Noise parameter estimation for the DC case. (a) Measured values of $p_{\exp }$ vs $\frac{\delta }{T}$ by implementing the $M^{\prime }$ projective measurements. Continuous red line corresponds to the theoretical behavior. Inset: experimental values of the standard deviations for the optimal protocol implemented by the $M^{\prime }$ projective measurements. They are obtained by propagating the Poissonian uncertainties. The solid line represents the expected theoretical behavior. (b) Experimental probabilities associated to the experimental matrix $\chi$ vs $\frac{\delta }{T}$. Values of $p$ are obtained by maximizing the fidelity $F$ between theoretical and experimental matrix $\chi$. Error bars are calculated by considering the Poissonian uncertainty associated to the coincidence counts, and simulating different matrices of the process, obtaining, in this way, different values of $p$. Inset: experimental values of the standard deviations for the AAQPT. These have been calculated by a simulation based on the Poissonian uncertainty associated to the coincidence counts. The solid line represents the optimal bound. (c),(d) Experimental (left side) and theoretical (right side) matrices $\chi$ for $T=\delta$ and $T\approx 3\delta$.