Overarching framework between Gaussian quantum discord and Gaussian quantum illumination

We cast the problem of illuminating an object in a noisy environment into a communication protocol. A probe is sent into the environment, and the presence or absence of the object constitutes a signal encoded on the probe. The probe is then measured to decode the signal. We calculate the Holevo information and bounds to the accessible information between the encoded and received signal with two different Gaussian probes—an Einstein-Podolsky-Rosen (EPR) state and a coherent state. We also evaluate the Gaussian discord consumed during the encoding process with the EPR probe. We find that the Holevo quantum advantage, defined as the difference between the Holevo information obtained from the EPR and coherent state probes, is approximately equal to the discord consumed. These quantities become exact in the typical illumination regime of low object reflectivity and low probe energy. Hence we show that discord is the resource responsible for the quantum advantage in Gaussian quantum illumination.

Figure 1

Diagram of illumination setup. (a) With probability $p_0$, there is an object located in a noisy environment. The object is partially reflective (modeled as a beam splitter with reflectivity $\epsilon$). A probe is sent towards the object. The probe is mixed with the noisy environment and reflected to the detector. (b) With probability $p_1$, an object is not present, in which case there is nothing to reflect the probe to the detector. Hence, only noise is detected. (c) An equivalent description of illumination whereby first noise is injected. Then, encoding is performed on the probe, whereby with a probability $p_0$, an identity operation is performed on the probe (after noise injection) and the environment noise, and (d) with probability $p_1$, a swap operation is performed on the probe and environment. In quantum illumination, we also have an idler initially entangled with the probe which is used to perform a joint measurement. Single-mode illumination is when there is no idler. ${\rho }_{\mathrm{env}}$ is the noisy environment and ${\tilde{\rho }}_{\mathrm{env}}$ is the environment with the mean photon number scaled by $1/(1-\epsilon )$.

Figure 2

Information vs object reflectivity $\epsilon$ when probe has mean photon number (a) $0.5$ and (b) 0.01. The environment noise has mean photon number 4. Each plot has two insets showing zoomed portions. Insets (ii) show the upper and lower bounds for $A_q$, with the true value lying somewhere in the shaded region. Insets (iii) show that ${\chi }_s, {\chi }_c, A_s$, and $A_c$ differ slightly, despite appearing as a single line in the main plot.

Figure 3

The Holevo information obtained when an EPR state is used for illumination, but a local Gaussian measurement is first performed on mode $B$. The measurement consists of beam splitter with transmissivity $t$, followed by conjugate homodyne measurements on both outputs. The maximum gives ${\chi }_c$, which occurs up to numerical precision at $t=0.5$, which corresponds to a heterodyne measurement. Parameters are $\epsilon =0.1, \overline{n}=0.5$, and ${\overline{n}}_{\mathrm{env}}=4$.

Figure 4

The quantities ${\chi }_q-{\chi }_s, {\chi }_q-{\chi }_c, A_q-A_s$, and $A_q-A_c$ compared to the consumed Gaussian discord ${\delta }_{\mathrm{con}}^{\mathrm{G}}\left(A\right|B)$. The average photon number of the probe is (a) 0.5 and (b) 0.01. The mean photon number of the environment noise is 4. Each plot has two insets showing zoomed portions. Insets (ii) show ${\delta }_{\mathrm{con}}^{\mathrm{G}}\left(A\right|B), {\chi }_q-{\chi }_s$, and ${\chi }_q-{\chi }_c$. Insets (iii) shows upper and lower bounds of $A_q-A_s$ and $A_q-A_c$.

Figure 5

The accessible information and Holevo information quantities (top) and consumed Gaussian discord and quantum advantage (bottom) vs the mean energy of the probe. The environment-noise mean photon number is 4 and object reflectivity $\epsilon =0.1$.

Figure 6

One million small perturbations were made on a coherent state. This is a histogram of the Holevo information when the top 500 000 states are used in single-mode illumination. The vertical line shows the Holevo information of the coherent state. Parameters are $n_{\mathrm{env}}=4, \epsilon =0.5, \overline{n}=0.5$.

Figure 7

Accessible information when the probe is a squeezed coherent state with mean photon number $\overline{n}=0.5$, squeezing $r$, and displacement $\alpha =\sqrt{\overline{n}-{sinh}^2\left(r\right)}$. The object has reflectivity $\epsilon =0.1$ and the mean photon number of the noise ${\overline{n}}_{\mathrm{env}}$ is 4. The optimal squeezing is $r=0.00279$, which gives information $0.0015\%$ higher than with no squeezing.

Figure 8

Lower bound of $A_s$ as a function of probe average photon number $\overline{n}={\left|\alpha \right|}^2$ when the noise mean photon number ${\overline{n}}_{\mathrm{env}}=4$, and the object has reflectivity $\epsilon$ of $1/2, 1/10$, and $1/100$ (top). The same plot with each line scaled by $1/\epsilon$ so that linearity can be compared (bottom).