Quantum Illumination Advantage for Classification Among an Arbitrary Library of Targets

Abstract

Quantum illumination (QI) is the task of querying a scene using a transmitter probe whose quantum state is entangled with a reference beam retained in ideal storage, followed by optimally detecting the target-returned light together with the stored reference, to make decisions on characteristics of targets at stand-off range, at precision that exceeds what is achievable with a classical transmitter of the same brightness and otherwise identical conditions. Using tools from perturbation theory, we show that in the limit of low transmitter brightness, high loss, and high thermal background, there is a factor of four improvement in the Chernoff exponent of the error probability in discriminating any number of apriori-known reflective targets when using a Gaussian-state entangled QI probe, over using classical coherent-state illumination (CI). While this advantage was known for detecting the presence or absence of a target, it had not been proven for the generalized task of discriminating between arbitrary target libraries. In proving our result, we derive simple general analytic expressions for the lowest-order asymptotic expansions of the quantum Chernoff exponents for QI and CI in terms of the signal brightness, loss, thermal noise, and the modal expansion coefficients of the target-reflected light's radiant exitance profiles when separated by a spatial mode sorter after entering the entrance pupil of the receiver's aperture.

Figures (2)

• Figure 1: Diagram illustrating the stand-off target discrimination task for a set of known targets $S$. The light emitted from the source is a signal of duration $T$ and bandwidth $W$, supporting $M<TW$ quasi-monochromatic frequency modes, which are initiated in a product of $M$ identical states of mean photon number $N_S$, leading to a product of $M$ identical states exciting $M$ orthonormal target-dependent spatiotemporal-polarization modes $\{\vec{\psi}_1^{(i)}(x,y)f_m(t)\}_{m=1}^M$ defined on the image plane. The targets are placed in a thermal background characterized by an approximately fixed Planck-law-governed mean photon number per mode $N_B$ over the spectral profile of the signal, and target $i$ is known to induce a round-trip transmissivity $\kappa^{(i)}$. The target dependence of the image plane modes is reflected in the spatial component $\psi^{(i)}_1$, which decomposes into a fixed orthonomal spatial mode set $\{\zeta_k(x,y)\}_{k=1}^n$ via the coefficient vector $\overrightarrow{C}^{(i)}.$ The identical states comprising the target return product state, when represented in the $n$-mode $\zeta$- basis, are denoted $\hat{\rho}^{(i)},$ with a one-to-one correspondence with $\overrightarrow{C}^{(i)}.$ The receiver performs a collective measurement on the product $\left(\hat{\rho}^{(i)}\right)^{\otimes M},$ the outcome on which a guess $\tilde{i}$ of the true hypothesis $H_i\in S$ is based.
• Figure 2: Comparison of the form of the target return light for CI and QI. For each frequency mode $m,$ the corresponding signal mode at the source is excited in (\ref{['fig:CI_diagram']}) a coherent state with real displacement $\sqrt{N_S},$ resulting in the $n$ receiver modes $\{\vec{\zeta}_k(x,y)f_m(t)\}_{k=1}^n$ being excited in the state $\hat{\rho}^{(i)}_C,$ and (\ref{['fig:QI_diagram']}) a signal-idler mode pair emitted from a SPDC source, excited in a two-mode-squeezed-vacuum of signal mean photon number $N_S$, resulting in the $n+1$ orthonormal receiver modes $\{\vec{\zeta}_k(x,y)f_m(t)\}_{k=1}^n\cup\{\vec{\zeta}_I(x,y)f_m(t)\}$ being excited in the state $\hat{\rho}^{(i)}_Q.$