Secure communication via quantum illumination

Abstract

In the quantum illumination protocol for secure communication, Alice prepares entangled signal and idler beams via spontaneous parametric downconversion. She sends the signal beam to Bob, while retaining the idler. Bob imposes message modulation on the beam he receives from Alice, amplifies it, and sends it back to her. Alice then decodes Bob’s information by making a joint quantum measurement on the light she has retained and the light she has received from him. The basic performance analysis for this protocol—which demonstrates its immunity to passive eavesdropping, in which Eve can only listen to Alice and Bob’s transmissions—is reviewed, along with the results of its first proof-of-principle experiment. Further analysis is then presented, showing that secure data rates in excess of 1 Gbps may be possible over 20-km-long fiber links with technology that is available or under development. Finally, an initial scheme for thwarting active eavesdropping, in which Eve injects her own light into Bob’s terminal, is proposed and analyzed.

Appendices

Appendix 1

Here we shall derive the bit-error rate for a single channel of the homodyne receiver that Alice uses in the QI-DWDM setup from Sect. 4. We will assume $R_c \ll W_c$. Then, given the value, $k$, of Bob’s message bit, the outputs from Alice’s idler and returned-light homodyne detectors during the $T_c = 1/R_c$ duration bit interval can be regarded as a set of $M_c = W_c/R_c$ statistically independent, identically distributed, zero-mean, 2-D Gaussian random-vector modes $\{\mathbf{x}_m : 1\le m \le M_c\}$ ²⁰ with common covariance matrix

$$\begin{aligned} \Lambda _0^\mathrm{hom} = \frac{1}{4}\left[ \begin{array}{ccc} 2N_{R_S} + 1 &{}&{} C_{R_SR_I} \\ C_{R_SR_I} &{}&{} 2N_{R_I} + 1\end{array}\right] \end{aligned}$$

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when $k=0$, and

$$\begin{aligned} \Lambda _1^\mathrm{hom} = \frac{1}{4}\left[ \begin{array}{ccc} 2N_{R_S} + 1 &{}&{} -C_{R_SR_I} \\ -C_{R_SR_I} &{}&{} 2N_{R_I} + 1\end{array}\right] \end{aligned}$$

(20)

when $k=1$. In these expressions:

$$\begin{aligned} N_{R_S}&= \kappa _d\kappa _A^{\prime } \kappa _2 \kappa _B^{\prime } (G_B\kappa _B \kappa _1 \kappa _A N_S + N_B), \end{aligned}$$

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$$\begin{aligned} N_{R_I}&= \kappa _d\kappa _I N_S, \end{aligned}$$

(22)

and

$$\begin{aligned} C_{R_SR_I} = 2\kappa _d\sqrt{\kappa _A^{\prime }\kappa _2\kappa _B^{\prime }G_B\kappa _B\kappa _1\kappa _A\kappa _IN_S(N_S+1)}. \end{aligned}$$

(23)

When Bob’s message bits are equally likely to be 0 or 1, the minimum error-probability decision rule reduces to the following threshold test:

$$\begin{aligned} \sum _{m=1}^{M_c} x_{m_R}x_{m_I} \begin{array}{c}\text{ decide } \text{0 }\\ \ge \\ < \\ \text{ decide } \text{1 }\end{array} 0. \end{aligned}$$

(24)

The upper bound on Alice’s bit-error probability that we used in Sect. 4 is the Bhattacharyya bound,

$$\begin{aligned} \Pr (e)_\mathrm{Alice}^\mathrm{UB}&\le \frac{1}{2}\left[ \int \!\mathrm{d}\mathbf{x}\,\sqrt{\frac{\exp \!\left[ -\frac{1}{2}\mathbf{x}^T(\Lambda _0^{-1} + \Lambda _1^{-1})\mathbf{x}\right] }{(2\pi )^{2} |\Lambda _0|^{1/2}|\Lambda _1|^{1/2}}}\right] ^{M_c}\end{aligned}$$

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$$\begin{aligned}&= \frac{1}{2}\left( \sqrt{1-\frac{C_{R_SR_I}^2}{(2N_{R_S} + 1)(2N_{R_I}+1)}}\right) ^{M_c}. \end{aligned}$$

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Appendix 2

Here we show how the bit-error-probability bounds for Eve’s heterodyne receiver, plotted in Fig. 10a, were obtained. The upper bound is a Chernoff bound,²¹

$$\begin{aligned} \Pr (e)^\mathrm{UB}_\mathrm{Eve} = \frac{1}{2}\int \!\mathrm{d}\mathbf{x}\,\sqrt{p(\mathbf{x}\mid 0)p(\mathbf{x}\mid 1)}, \end{aligned}$$

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and the lower bound is

$$\begin{aligned} \Pr (e)_\mathrm{Eve}^\mathrm{LB} = \frac{1-\sqrt{1-\left( \int \!\mathrm{d}\mathbf{x}\,\sqrt{p(\mathbf{x}\mid 0)p(\mathbf{x}\mid 1)}\right) ^2}}{2}, \end{aligned}$$

(28)

where $p(\mathbf{x}\mid k)$ for $k = 0,\,1$ is the conditional probability density function for the normalized, complex-valued output $\mathbf{x}$ from bit-interval-matched filtering of Eve’s heterodyne photocurrent.²² For our basis-encoded system, Eve’s conditional probability densities are

$$\begin{aligned} p(\mathbf{x}\mid k) = \sum _{b=1}^{B-1}\frac{\exp \!\left( -{\left| {\mathbf{x} -(-1)^k \sqrt{(1-\kappa )N_\mathrm{in}}\,e^{i\theta ^{(b)}_k}}\right| ^2}/(1\!-\!\kappa )N_B\right) }{B\pi (1-\kappa )N_B},\quad \text{ for } k\!=\!0,1.\nonumber \\ \end{aligned}$$

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