IEEE Aerospace and Electronic Systems Magazine - November 2020 - 22

Feature Article:

DOI. No. 10.1109/MAES.2020.3004019

Noisy Receivers for Quantum Illumination
Athena Karsa, Stefano Pirandola, University of York

INTRODUCTION
Quantum illumination (QI) [1]-[5] is an entanglementbased protocol able to detect the presence of a low-reflectivity object embedded in bright thermal noise, even in the
case where the signal employed is itself very weak. Using
an optimum quantum receiver, it offers a 6 dB advantage
in error probability exponent over the best possible classical strategy using the same transmitted energy. Such an
advantage persists despite the fact that all entanglement is
lost during the process [6]-lending itself to being of particular use in the microwave regime where the ambient
background is inherently high [7].
To date, the specifics of such an optimum receiver for
QI remains unknown without access to a quantum computer. There have, however, been several proposals for
practical receiver designs, the best of which are the suboptimal optical parametric amplifier and phase-conjugating
(PC) receivers [8], achieving up to 3 dB in performance
advantage. The 3 dB performance deficit is owing to the
fact that these receivers operate based on Gaussian local
operations, which are known to be not optimal for general
mixed-state discrimination [9], [10]. Employing nonlinear
operations, Zhuang et al. [11]-[13] used sum-frequency
generation (SFG) alongside a feed-forward (FF) mechanism to show that the full QI advantage could theoretically
be attained, however the physical implementation of the
FF-SFG receiver is yet out of reach.
Even though low signal energy is one of the key ingredients for QI's advantage, this inherent property of microwave photons make their detection difficult such that
single photon counting forms a great obstacle for any
experiment in the microwave domain. This is despite the
fact that this task is generally quite straightforward in
other regimes with efficient optical photon counters being
widely available [16]. The actual measurement procedure
Authors' current addresses: Athena Karsa, Stefano Pirandola, Department of Computer Science, University of York,
York, YO10 5GH , U.K. (e-mail: ak1674@york.ac.uk).
Manuscript received April 20, 2020, revised June 9,
2020; accepted June 12, 2020, and ready for publication
June 18, 2020.
Review handled by Marco Frasca.
0885-8985/20/$26.00 ß 2020 IEEE
22

forms a crucial and fundamental design aspect of any
QI receiver, particularly in the microwave domain, with
interesting progress being demonstrated by recent experiments[14], [15].
Further to question regarding receiver design, idler
storage poses another issue particularly with respect to target-ranging problems. In QI, an entangled photon pair is
created with one forming the signal and the other, the
idler, stored for later joint measurement. In scenarios
where the range, and thus return time of the signal, is
unknown, or even a measure to be determined, idler storage forms a crucial aspect of the protocol necessary for its
success.
A potential solution is to perform measurement on the
idler photon, mitigating issues associated with its storage,
and combine the result with that of the returning signal. In
microwave QI, these measurement results take the form of
quadrature voltages which may be used to reconstruct the
annihilation operators of the modes; in turn, these may be
postprocessed to simulate potential receivers for QI, such
as the digital PC receiver [14]. Despite the fact that the
collected data can be used in this way, real-time implementation of such a strategy cannot beat the optimal performance of coherent states, as already discussed in [14]
and further investigated here.
In this article, we consider the QI protocol using a
generic source modeled as a two-mode Gaussian state with
arbitrary quadrature correlations. Keeping in the domain of
Gaussian linear operations, we study the PC receiver in
terms of its effective signal-to-noise ratio (SNR) for our
generic source. We consider various cases of added noise
from, for example, the application of a heterodyne measurement on one or both of the source's modes, comparing the
performance of various receivers and determine their absolute performance capabilities relative to the optimal classical
method using coherent states with homodyne detection.

BASICS OF THE QI PROTOCOL
Consider the production of M independent signal-idler
ðkÞ ðkÞ
mode pairs, f^
aS ; a^I g; 1 k M, with mean number
of photons per mode given by NS and NI for the signal
and idler modes, respectively. The signal (S) mode is sent
out to some target region while the idler (I) mode is

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