Entanglement-based linear-optical qubit amplifier

We propose a linear-optical scheme for an efficient amplification of a photonic qubit based on interaction of the signal mode with a pair of entangled ancillae. In contrast to a previous proposal for qubit amplifier by Gisin et al. [Phys Rev. Lett. 105, 070501 (2010)], the success probability of our device does not decrease asymptotically to zero with increasing gain. Moreover, we show how the device can be used to restore entanglement deteriorated by transmission over a lossy channel and calculate the secure key rate for device-independent quantum key distribution.

Figure 1

Scheme for entanglement-based linear-optical qubit amplifier as described in the text. D${}_1$ and D${}_2$ are standard polarization analysis detection blocks (for reference see [21]).

Figure 2

Success probability is depicted as a function of gain for three different input states parametrized by ${\left|\alpha \right|}^2$. For comparison, the success probability of the Gisin et al. scheme [13] is presented (in this case ${\left|\alpha \right|}^2=0.5$). Note that the success probability of our amplifier converges asymptotically to a nonzero value for any state with ${\left|\alpha \right|}^2\ne 1$. Success probability is also plotted as a function of nominal gain $G_{\mathrm{nom}}$ for the case of ${\left|\alpha \right|}^2=0.95$. Note that according to its definition (3), the nominal gain is upper bounded by the value of 20 in this particular case (blue X symbol).

Figure 3

Negativity of entanglement depicted as a function of amplification gain for several different channel transmissivities $T$. A maximally entangled state formed of superposition of vacuum and qubit state is subjected to a channel with transmissivity $T$ resulting in entanglement loss. Suitably set amplification gain can increase the amount of entanglement. The wide gray curve joins the maxima of negativity for all values of transmissivity $T$ and subsequent optimal gains.

Figure 4

The optimal gain $G_{\mathrm{opt}}$ for various entanglement measures as function of channel transmissivity $T$. Setting the optimal gain allows to obtain the largest possible value of the selected entanglement measure for a given loss parametrized by $T$.

Figure 5

The entanglement measures before ($C,N,S$) and after ($C_{\mathrm{opt}},N_{\mathrm{opt}},S_{\mathrm{opt}}$) optimal amplification as functions of channel transmissivity $T$.

Figure 6

Negativity and success probability tradeoff obtained using coherent attenuation before transferring the state through a lossy channel. This tradeoff is depicted for three different values of channel transmissivity $T$. Even though this strategy allows us to increase the negativity arbitrarily close to $\frac{1}{2}$, the product of negativity and success probability is maximized when no coherent attenuation is used.

Figure 7

Key rate per laser pulse for device-independent quantum key distribution versus Bob's channel loss and dark counts per second in heralding detectors. Assuming 100 ps timing resolution in the heralding detectors leads to ${10}^{-10}$ and ${10}^{-8}$ dark count probability per pulse for 1 and 100 dark count/s, respectively. Our entangled photon amplifier allows more key to be extracted than the Gisin et al. scheme, and even shows better scaling with loss. It additionally delivers approximately 12 times the key rate of the Pitkanen et al. scheme.