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A comparison between quantum and classical noise radar sources Robert Jonsson Roberto Di Candia Martin Ankel 1Department of Microtechnology Department of Communications 1Department of Microtechnology and Nanoscience and Networking and Nanoscience Chalmers University of Technology Aalto University Chalmers University of Technology 2Radar Solutions, Saab AB Helsinki, Finland 2Radar Solutions, Saab AB Goteborg,¨ Sweden Email: roberto.dicandia@aalto.fi Goteborg,¨ Sweden Email: [email protected] Anders Strom¨ Goran¨ Johansson Radar Solutions, Saab AB Department of Microtechnology Goteborg,¨ Sweden and Nanoscience Chalmers University of Technology Goteborg,¨ Sweden Abstract—We compare the performance of a quantum radar in the last decades [10], paving the way of implementing these based on two-mode squeezed states with a classical radar system ideas for building a first prototype of quantum radar. based on correlated thermal noise. With a constraint of equal The possible benefits of a quantum radar system are generally number of photons N transmitted to probe the environment, we S understood to be situational. In an adversarial scenario, it find that the quantum setup exhibitsp an advantage with respect to its classical counterpart of 2 in the cross-mode correlations. is beneficial for a radar system to be able to operate while Amplification of the signal and the idler is considered at different minimizing the power output, in order to reduce the probability stages of the protocol, showing that no quantum advantage for the transmitted signals to be detected. This property is is achievable when a large-enough gain is applied, even when commonly referred to as Low Probability of Intercept (LPI), quantum-limited amplifiers are available. We also characterize the minimal type-II error probability decay, given a constraint and it is a common measure to limit the ability of the enemy on the type-I error probability, and find that the optimal decay to localize and discover the radar. The low signal levels rate of the type-II error probability in the quantum setup required for QI are in principle excellent for acquiring good is ln(1 + 1=NS ) larger than the optimal classical setup, in LPI properties. However, there are several challenges to face the NS 1 regime. In addition, we consider the Receiver in order to achieve this goal. A first proposal for implementing Operating Characteristic (ROC) curves for the scenario when the idler and the received signal are measured separately, showing a microwave QI protocol was advanced in Ref. [11]. The that no quantum advantage is present in this case. Our work protocol relies on an efficient microwave-optical interface for characterizes the trade-off between quantum correlations and the idler storage and the measurement stage. This technology is noise in quantum radar systems. promising for this and other applications, however it is still in its infancy. Furthermore, the signal generation requires cryogenic I. INTRODUCTION technology, which must be interfaced with a room-temperature environment. Recently, a number of QI-related experiments The quantum Illumination (QI) protocol [1]–[7] uses en- have been carried out in the microwave regime [12], [13], arXiv:2008.12711v1 [quant-ph] 28 Aug 2020 tanglement as a resource to improve the detection of a low- showing that some correlations of an entangled signal-idler reflectivity object embedded in a bright environment. The system are preserved after the signal is sent out of the dilution protocol was first developed for a single photon source [1], refrigerator. While these results are a good benchmark for future and it was then extended to general bosonic quantum states QI experiments, they strictly rely on the amplification of the and thermal bosonic channels [2]. Here, a 6 dB advantage signal and idler. This has been shown to rule out any quantum in the effective signal-to-noise ratio (SNR) is achievable advantage with respect to an optimal classical reference [14]. when using two-mode squeezed states instead of coherent In this work, we discuss the role of quantum correlations states. This gain has been recently shown to be optimal, and amplification in the QI protocol, providing a comparative and reachable exclusively in the regime of low transmitting analysis of quantum and classical noise radars in different power per mode [8], [9]. The QI protocol has possible scenarios. Noise radar is an old concept that operates by applications in the spectrum below the Terahertz band, as probing the environment with a noisy signal and cross- here the environmental noise is naturally bright. In particular, correlating the returns with a retained copy of the transmitted microwave quantum technology has been very well developed signal [15]. A quantum noise radar operates similarly to its conventional counterpart, but differs in the use of a two-mode (C) T0 a^ entangled state as noise source [12], [13]. An advantage of S the quantum noise radar over the classical counterpart can be a^0 declared if stronger correlations can be achieved, when both systems illuminate the environment with equal power. In the (ξ; ') microwave regime, the two-mode squeezed state used for noise correlations can be generated with superconducting circuits with a^1 a Josephson Parametric Amplifier (JPA) at T ' 20 mK [16], (C) T1 a^ [17]. On the one hand, using quantum correlated signals I generated by a JPA enhances the signal-to-noise ratio in the low transmitting power per mode regime. On the other hand, Fig. 1. Preparation of classically-correlated thermal noise. A beamsplitter Josephson parametric circuits are able to generate correlated with reflection coefficient ξ and phase turning angle ' generates the signal and entangled signals with large bandwidth [18]–[20]. This and idler modes from the modes a^0 and a^1. These modes are in a thermal state with T0 and T1 effective noise temperatures, respectively. The output allows, in principle, a system to operate in the low power-per- modes are correlated provided that T0 6= T1. mode regime, where quantum radars show fully their advantage. Here, we analyze the performance of a JPA-based noise radar in different scenarios which include different sources of noise. In general, the quantum state of a thermal noise mode at Our analysis shows that any quantum advantage is destroyed temperature T can be represented by the density operator 1 by the unavoidable noise added when amplifying either the X N n ρth = jni hnj ; (1) signal or the idler. We also show when the idler and signal are (N + 1)n+1 measured separately, the entanglement initially present in the n=0 signal-idler system is not properly exploited, and no quantum where the average number1 of photons is defined by the thermal advantage can be retained. The latter happens even without equilibrium Bose-Einstein statistics at temperature T , i.e., N = −1 amplifying the signal or the idler. Our work complements the [exp (2πf=T ) − 1] . In the following, we will refer as N0 analysis done in Ref. [14] with the explicit calculations of the (N1) the average number of photons for the mode a^0 (a^1). cross-correlation coefficients and the optimal asymptotic ROC The thermal modes a^0 and a^1 pass through a beamsplitter, as (C) performance in the microwave regime. shown in Fig. 1. This generates a signal mode a^S and an (C) idler mode a^I , related to the inputs as [22] ! p p II. THEORY (C) i' a^S p ξ 1 p− ξe a^0 (C) = −i' : (2) a^ − 1 − ξe ξ a^1 In this section, we introduce the models for the quantum and I classical systems, within the quantum mechanical description. Here, ξ 2 (0; 1) is the reflection coefficient and ' is the phase In this step, we emulate Refs. [14], [21], where the classical turning angle of the beamsplitter, in the following set to be and quantum noise radar were first studied. In all expressions, zero. One can think of this process as a noise signal, generated by a thermal source at temperature T , sent as input of a we assume the natural units (~ = 1, kB = 1). 0 power divider placed in an environment at temperature T1. The output modes a^(C) and a^(C) are in a thermal state with A. Quantum preliminaries S I ξN0 + (1 − ξ)N1 and ξN1 + (1 − ξ)N0 average number of T 6= T N 6= N A single, narrowband mode of the electric field, at microwave photons, respectively. If 1 0, or, equivalently, 1 0, frequency f, is defined with an operator (in suitable units) then the outputs are classically-correlated, regardless of the ξ as E^ =q ^cos 2πft +p ^sin 2πft; where q^ and p^ are the in- value of . 2) Entangled thermal noise: A Two-Mode Squeezed Vac- phase and quadrature operators, respectively. The quadratures y uum (TMSV) state j iTMSV is represented in the Fock basis are related to the bosonicp annihilation (a^) and creationp (a^ ) y y as operators as q^ = (^a +^a)= 2 and p^ = i(^a −a^)= 2, where 1 s n y ^ ^ X NS [^a; a^ ] = I. The commutation relation [^q; p^] = iI implies j iTMSV = n+1 jniS jniI ; (3) that the quadratures can not be measured simultaneously with n=0 (NS + 1) arbitrary precision, due to the Heisenberg uncertainty relation. where NS is the average number of photons in both the signal In the following, we represent the quadratures of the two- and idler mode. A TMSV state is closely related to a classically- ^ T modes of the electric field by the vector X = (^qS; p^S; q^I ; p^I ) , correlated thermal noise, as also here both signal and idler where the indices S and I refer to the signal and idler mode, photons are Bose-Einstein distributed.

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