Efficient amplification of photonic qubits by optimal quantum cloning

We demonstrate that a phase-independent quantum amplifier of a polarization qubit is a complementary amplifier of the heralded qubit amplifier [N. Gisin, S. Pironio, and N. Sangouard, Phys. Rev. Lett. 105, 070501 (2010)]. It employs the multifunctional cloner in the $1\to 2$ copying regime, capable of providing approximate copies of qubits given by various probability distributions, and is optimized for distributions with axial symmetry. Direct applications of the proposed solution are possible in quantum technologies, doubling the range where quantum information is coherently broadcast. It also outperforms natural nonlinear amplifiers that use stimulated emission in bulk nonlinear materials. We consider the amplifier to be an important tool for amplifying quantum information sent via quantum channels with phase-independent damping.

Figure 1

Shown on the left is the Poincaré sphere of polarization. A qubit is a linear combination of horizontal $|H\rangle$ (the north pole) and vertical $|V\rangle$ (the south pole) polarizations: $|\Psi \rangle =cos(\theta /2)|H\rangle +e^{i\varphi }sin(\theta /2)|V\rangle$. The radius of the sphere is 1. The uniform probability distribution of the polarization qubit is marked in red. Exemplary distributions with axial symmetry are in green and blue. Shown on the right are imperfect copies ${\rho }^{\prime }$ of $\rho =|\Psi \rangle \langle \Psi |$.

Figure 2

Both (a) the cloning-based amplifier and (b) the heralded amplifier can be placed in the communication channel of total transmissivity $\eta ={\eta }_A{\eta }_B$ to fight the losses. We assume that Alice sends a photon to Bob, who wants to receive the quantum message with minimum time spent waiting. Because of losses, only a fraction of photons will reach Bob. If one uses a qubit amplifier, the probability of Bob receiving the signal increases. In the case of the CBA this probability is increased by sending an additional photon generated by a single photon source (SPS). If the HA is used, the probability is increased by informing Bob when he should be ready to receive a photon so that he does not waste time waiting in vain.

Figure 3

Principle of operation of the CBA working at maximum efficiency $P=1$. The CBA has two input modes for the signal qubit ${\rho }_{\mathrm{in}}$ and ancillary qubit ${\rho }_{\mathrm{anc}}$. The output of the CBA is described by state ${\rho }_{1,2}$. The reduced density matrices ${\rho }_1={\mathrm{Tr}}_2{\rho }_{1,2}={\rho }^{\prime }$ and ${\rho }_2={\mathrm{Tr}}_1{\rho }_{1,2}={\rho }^{\prime }$ describe each of the two copies of ${\rho }_{\mathrm{in}}=\rho$ separately. Here we demonstrate how the output state of the amplifier is calculated by modeling losses as beam splitter transformations.

Figure 4

Signal to noise ratio SNR versus gain $G$ for a given input qubit distribution quantified by $\langle {cos}^n\theta \rangle$ for $n=1,2$ fully characterizes the CBA. (a) We assumed distributions with $\langle cos\theta \rangle =0$; if $\langle {cos}^2\theta \rangle =1/3$, the amplifier is universal; if $\langle {cos}^2\theta \rangle =0$ the amplifier is phase-covariant optimized for equatorial qubits. If $\langle {cos}^2\theta \rangle =1$ (red curve) then the signal can be amplified classically with an infinite SNR because the states of $cos\theta =\pm 1$ can be discriminated deterministically. (b) If the state of the qubit is known a priori, e.g., $cos\theta =\pm 1$, $\langle {cos}^2\theta \rangle ={\langle cos\theta \rangle }^2$, the qubit can be amplified with an arbitrarily high gain (red curve). Solid black curves correspond to the experimentally demonstrated amplification discussed in the text.

Figure 5

Implementation of the cloning-based quantum amplifier (CBA). A random number generator (RNG) provides a number $r\in [0,1]$ and switches the device between its two regimes of work: optimal quantum cloning (if $r\le 1-\epsilon$) and the mixing of input qubit $\rho$ with the central state $\sigma$ (if $r>1-\epsilon$). The latter regime requires removing the PDBS and filters F from beam divider assembly (BDA). Also shown are the polarization-dependent beam splitter (PDBS), the beam displacer (BD), the beam splitter (BS), the half-wave plate (HWP), the quarter-wave plate (QWP), and the neutral density filter (F). After the signal $\rho$ and ancillary photons are prepared (by means of HWPs and a QWP) they interact on a PDBS. Next, if the photons exit at separate ports, a specific polarization component (set by HWPs) of both photons is attenuated in the respective BDAs (another pair of HWPs restores the original polarization). Just before exiting the CBA the intensity in both output ports is balanced by a pair of neutral density filters F. Finally, a polarization analysis is performed.

Figure 6

Implementation of the polarization-dependent beam splitter: BD, beam displacer; BS, symmetric beam splitter; HWP (QWP${}_{1\left(2\right)}$), half- (quarter-) wave plate; PS${}_{H\left(V\right)}$, phase shifter. By manipulating the phase shifts $\nu$ and $\chi$ of $H$ and $V$ polarization components, respectively, the interferometer acts as a PDBS with variable reflection and transmittance for each of the $H$ and $V$ polarizations. To remove the phase difference $\nu -\chi$ between the phases of the $V$- and $H$-polarization components of both output modes, we use QWP${}_1$ and QWP${}_2$ (or Pockels cells) set to shift phases of V-polarized photons by $\chi -\nu$. It can be verified that the setup output may be expressed in terms of annihilation operators as $a_{2,s}=\sqrt{{\eta }_s}{a}_{\mathrm{anc},s}-\sqrt{1-{\eta }_s}{a}_{\mathrm{in},s}$ and $a_{1,s}=\sqrt{{\eta }_s}{a}_{\mathrm{in},s}+\sqrt{1-{\eta }_s}{a}_{\mathrm{anc},s}$, where ${\eta }_s={cos}^2\chi ,{cos}^2\nu$ for $s=H,V$, respectively. If there is no phase shift, the PDBS acts as if it were removed.

Figure 7

Signal-to-noise ratio and amplification gain $G$ measured for the CBA with the mixing parameter $\epsilon =1/2$ as a function of the input qubit distribution on the Poincaré sphere, parametrized by $\langle {cos}^2\theta \rangle$ with $\langle cos\theta \rangle =0$, i.e., mirror phase-covariant regime (see Fig. 9 for the $\langle {cos}^2\theta \rangle ={\langle cos\theta \rangle }^2$ regime). Dashed lines indicate the value of fidelity and gain for the universal CBA. The details are given in the main text.

Figure 8

Signal-to-noise ratio as a function of gain $G$ measured by the device from Fig. 5 for input qubit distributions $\langle {cos}^2\theta \rangle =1/3$ (left) and $\langle {cos}^2\theta \rangle =0$ (right) for $\langle cos\theta \rangle =0$. Solid curves show theoretical predictions for the ideal setup components, assuming the device operates on a scale from a purely quantum $Q_1=(-3.01,3.01)$ and $Q_2=(-1.76,3.83)$ ($\epsilon =0$) to a classical $C_{1\left(2\right)}=(3.01,0)$ ($\epsilon =1$) regime. The marked areas of $G>0$ dB and $\text{SNR}>0$ dB highlight the part of the regime where the device works as the universal and phase-covariant CBA.

Figure 9

Signal-to-noise ratio and amplification gain $G$ measured for the CBA with the mixing parameter $\epsilon =1/2$ as a function of the input qubit distribution on the Poincaré sphere, parametrized by $\langle {cos}^2\theta \rangle$ with $\langle {cos}^2\theta \rangle ={\langle cos\theta \rangle }^2$, i.e., the phase-covariant regime. Red dashed lines indicate the values of (a) SNR and (b) gain for the universal CBA. The solid blue line in (a) indicates the SNR above which the CBA provides improvement in the product state capacity [see Eq. (1)] for $\eta \ll 1$. The black curves correspond to the theoretically predicted values.