State-dependent linear-optical qubit amplifier

We propose a linear-optical setup for heralded qubit amplification with tunable output qubit fidelity. We study its success probability as a function of output qubit fidelity, showing that at the expense of lower fidelity, the setup can considerably increase probability of successful operation. These results are subsequently applied in a proposal for state-dependent qubit amplification. Similar to state-dependent quantum cloning, the a priori information about the input state allows us to optimize the qubit amplification procedure to obtain a better fidelity versus success probability trade-off.

Figure 1

Conceptual scheme of a heralding qubit amplifier. The input state is transformed according to Eq. (1). D, detector; EPR, ancillary photons; G, amplifier: FF, feed forward.

Figure 2

Scheme for a state-dependent linear-optical qubit amplifier as described in the text. EPR, source of entangled ancillary photon pairs; PBS, polarizing beam splitter; PPBS, partially polarizing beam splitter (defined in the text); WP, wave plate; PDF, polarization-dependent filter; D, standard polarization analysis detection block (for reference see [33]).

Figure 3

Success probability $P_{\mathrm{succ}}$ given by Eq. (16) as a function of output-state fidelity $F_{\mathrm{QFF}}$ given by Eq. (17) in the case of infinite gain is depicted for four different input states as described in the text.

Figure 4

Success probability $P_{\mathrm{succ}}$ as a function of both output-state fidelity and amplification gain $G_{\mathrm{FF}}$ is depicted for four different input states: (a) $\theta =\pi /2$, (b) $\theta =2\pi /5$, (c) $\theta =\pi /3$, and (d) $\theta =\pi /4$. THR stands for the threshold of unreachable area.

Figure 5

Probability density function $g=g(\theta ,\kappa )$ given by Eq. (18) over the Poincaré sphere for various values of the $\kappa$ parameter used in subsequent numerical simulations: (a) $\kappa =0$, (b) $\kappa =1$, (c) $\kappa =3$, (d) $\kappa =10$. Labels $|H\rangle$, $|D\rangle$, and $|R\rangle =(|H\rangle +i|V\rangle )/2$ denote the position of the horizontal, diagonal, and right-hand circular-polarization states, respectively.

Figure 6

Maximum achievable success probability $\langle P\rangle$ as a function of average fidelity $\langle F\rangle$ for various values of average overall gain [(a) $\langle G\rangle =3$ dB, (b) $\langle G\rangle =10$ dB, (c) $\langle G\rangle =20$ dB, (d) $\langle G\rangle \to \infty$] and state knowledge described by parameter $\kappa$ of the probability density function $g=g(\theta ,\kappa )$ given by Eq. (18).

Figure 7

Merit function $M$ given by Eq. (19) depicted for various parameters $\kappa$ and average gains: (a) $\langle G\rangle =3$ dB, (b) $\langle G\rangle =10$ dB, (c) $\langle G\rangle =20$ dB, (d) $\langle G\rangle \to \infty$.