Experimental Detection of Quantum Channel Capacities

We present an efficient experimental procedure that certifies nonvanishing quantum capacities for qubit noisy channels. Our method is based on the use of a fixed bipartite entangled state, where the system qubit is sent to the channel input. A particular set of local measurements is performed at the channel output and the ancilla qubit mode, obtaining lower bounds to the quantum capacities for any unknown channel with no need of quantum process tomography. The entangled qubits have a Bell state configuration and are encoded in photon polarization. The lower bounds are found by estimating the Shannon and von Neumann entropies at the output using an optimized basis, whose statistics is obtained by measuring only the three observables ${\sigma }_x\otimes {\sigma }_x$, ${\sigma }_y\otimes {\sigma }_y$, and ${\sigma }_z\otimes {\sigma }_z$.

Figure 1

Experimental setup. A Sagnac interferometric source of polarization-entangled photons sends the $S$ qubit through the noisy channel, while the $A$ qubit remains untouched. Both qubits are measured in a joint photon counter system. Here PPKTP is a periodically poled potassium titanyl phosphate nonlinear crystal, PBS a polarizing beam splitter, $M$ a mirror, $L$ a converging lens, HWP a half-wave plate, QWP a quarter-wave plate, and DM a dichroic mirror.

Figure 2

Noisy channels. (a) Experimental scheme for a polarization ADC. Given a single input qubit $|\psi ⟩=\alpha |H⟩+\beta |V⟩$, it uses two HWPs inside a SI to apply a rotation on $|V⟩$, while $|H⟩$ remains unrotated, achieving $K_0$ and $K_1$. The dashed line inside the MZI represents the rotated $\sqrt{\gamma }$ portion of $|V⟩$. (b) Top: Experimental scheme for a PDC. It applies $\mathbb{I}$ and ${\sigma }_z$ over $|\psi ⟩$ by using an unrotated HWP and a variable voltage LC. Bottom: Experimental scheme for the PC and DC. They apply $\mathbb{I}$, ${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$ over $|\psi ⟩$ by using a 45°-rotated HWP and the same LC.

Figure 3

Minimal bounds $Q_{\mathrm{DET}}$ to the quantum channel capacity. For the entire set of data, the red points represent the experimental values, and the continuous blue line corresponds to the ideal simulation of a pure input state $|{\mathrm{\Phi }}^{+}⟩$ with fidelity $F=1$. Green shaded areas correspond to a region of $Q_{\mathrm{DET}}$ for an input Werner state ${\rho }_W$ within one standard deviation of fidelity $F_{\exp }=0.979\pm 0.011$. (a) ADC: $Q_{\mathrm{DET}}$ versus the damping parameter $\gamma$. (b) PDC: $Q_{\mathrm{DET}}$ versus $p$ for the statistical mixture of $\mathbb{I}$ and ${\sigma }_z$, with probability vector $\overrightarrow{p}=\{P_{\mathbb{I}},P_x,P_y,P_z\}=[1-(p/2),0,0,(p/2)]$. (c) DC: $Q_{\mathrm{DET}}$ versus $p$ for the balanced mixture of $\mathbb{I}$, ${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$, with $\overrightarrow{p}=[1-p,(p/3),(p/3),(p/3)]$. (d) PC: $Q_{\mathrm{DET}}$ versus $p$ for the mixture of $\mathbb{I}$, ${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$, with $\overrightarrow{p}=[1-(p/6),(p/12),(p/18),(p/36)]$. The error bars on the mapping probability are significant only in case (a), where an uncertainty of 0.5° in the rotation angle of ${\mathrm{HWP}}_V(\gamma )$ was propagated, while this error was negligible in cases (b)–(d). In case (a), the error bars on $Q_{\mathrm{DET}}$ were calculated from Poissonian statistics on one set of points, obtaining negligible values. In cases of (b)–(d), error bars were obtained from an average of at least eight sets of points. (See [27] for more details.)