# A comparison between quantum and classical noise radar sources

Abstract: We compare the performance of a quantum radar based on two-mode squeezed states with a classical radar system based on correlated thermal noise. With a constraint of equal number of photons $N_S$ transmitted to probe the environment, we find that the quantum setup exhibits an advantage with respect to its classical counterpart of $\sqrt{2}$ in the cross-mode correlations. Amplification of the signal and the idler is considered at different stages of the protocol, showing that no quantum advantage is achievable when a large-enough gain is applied, even when quantum-limited amplifiers are available. We also characterize the minimal type-II error probability decay, given a constraint on the type-I error probability, and find that the optimal decay rate of the type-II error probability in the quantum setup is $\ln(1+1/N_S)$ larger than the optimal classical setup, in the $N_S\ll1$ regime. In addition, we consider the Receiver Operating Characteristic (ROC) curves for the scenario when the idler and the received signal are measured separately, showing that no quantum advantage is present in this case. Our work characterizes the trade-off between quantum correlations and noise in quantum radar systems.

## Summary (3 min read)

### Introduction

- Amplification of the signal and the idler is considered at different stages of the protocol, showing that no quantum advantage is achievable when a large-enough gain is applied, even when quantum-limited amplifiers are available.
- The authors work characterizes the trade-off between quantum correlations and noise in quantum radar systems.
- A first proposal for implementing a microwave QI protocol was advanced in Ref. [11].
- Noise radar is an old concept that operates by probing the environment with a noisy signal and crosscorrelating the returns with a retained copy of the transmitted signal [15].
- The authors analysis shows that any quantum advantage is destroyed by the unavoidable noise added when amplifying either the signal or the idler.

### II. THEORY

- The authors introduce the models for the quantum and classical systems, within the quantum mechanical description.
- The authors emulate Refs. [14], [21], where the classical and quantum noise radar were first studied.

### A. Quantum preliminaries

- The commutation relation [q̂, p̂] = îI implies that the quadratures can not be measured simultaneously with arbitrary precision, due to the Heisenberg uncertainty relation.
- These mode designations are used interchangeably for both the quantum and classical system.
- The classicallycorrelated noise (CCN) system uses two sources of thermal noise, â0 and â1, at temperatures T0 and T1, respectively.
- In the following, the authors will refer as N0 (N1) the average number of photons for the mode â0 (â1).
- A TMSV state is closely related to a classicallycorrelated thermal noise, as also here both signal and idler photons are Bose-Einstein distributed.

### B. Covariance and correlation matrices

- As the states considered here are Gaussian, their statistics is entirely determined by the first and second order quadrature moments.
- (4) This is the case for both the classical and the entangled thermal noise states.
- (5) These coefficients, also referred to as Pearson’s correlation coefficients, characterize the linear dependence between the quadratures X̂i and X̂j .

### C. Quantum relative entropy

- The quantum relative entropy defines an information measure between two quantum states.
- This quantity is related to the performance in the asymmetric binary hypothesis testing via the quantum Stein’s lemma.
- The authors rely on quantum relative entropy computations and its variance in order to quantify the performance in the asymptotic setting, i.e., when M 1.
- This is in contrast to the original treatment based on the Chernoff bound [2], which provides an estimation of the average error probability when the prior probabilities of target absence or presence are equal.
- In a typical radar scenario, the prior probabilities are not the same, and may be even unknown.

### A. Probing the environment

- The signal mode is transmitted to probe the environment where an object may be present or absent.
- Here, η can be interpreted as the ratio between transmitted power and received power, including the effects of atmospheric attenuation, the antenna gain and the target radar cross section.
- The authors use a beamsplitter model to take into account the environmental losses.
- For the current calculation, the reflection coefficient is assumed to be non-fluctuating.
- The authors also assume 〈âI âB〉 = 0, i.e., the returned signal preserves some correlations with the idler mode only if the object is present.

### B. Cross-correlation coefficient

- The covariance matrix of the system composed of the received signal and the idler is easily computable using Eq. (4) and Eq. (8).
- For both the classical and the quantum noise radars considered here, applying Eq. (5) gives us the correlation matrix R = ( I κD(θ) κD(θ)T I ) , (9) where 0 ≤ κ ≤ 1 is the amplitude of the cross-correlation coefficient, and D is a matrix with determinant |D| = ±1.
- For a fair comparison between the quantum and classical systems, the authors introduce a constraint on the transmitted power of the signal modes, i.e., they set NS = ξN0 + (1− ξ)N1. (11) This constraint can be interpreted as giving both systems equal LPI properties.
- Eq. (12), for given noise transmitting power, defines the correlations of a class of classical noise radars, labeled by the beamsplitter parameter ξ.

### C. Quantum advantage

- This setting as been used as benchmarking in the recent microwave quantum illumination experiments [12], [13].
- Show the functional behaviour of the correlation coefficients for the entangled and classically-correlated thermal noise source, depending on the transmitting power NS .
- The quantum advantage decays slowly with increasing NS , keeping an advantage also for finite NS , until virtually disappearing for NS ' 10.
- Here, the phase-shift of the signal mode due to the propagating path has been set to zero.
- This strategy shows the same asymptotic performance as in the case of known θ.

### D. The effect of amplification

- In the following, the authors consider three Gaussian amplifying schemes.
- 1) Amplification before transmitting the signal mode:.
- The classical and quantum covariance matrix transforms identically.
- In addition, in the weak signal regime where the quantum advantage is relevant, the effective SNR is low even before amplification.
- As can be intuitively understood, amplification at the receiver end can not increase correlations with the idler reference.

### E. Receiver Operating Characteristic performance

- The authors analyze the quantum and classical noise radar in the asymmetric setting, i.e., when the prior probabilities are not equal, under a different perspective.
- The authors derive the ROC curves in the case when the idler and the signal are separately measured with heterodyne detection.
- A substantial advantage can be also found for moderate values of NS (see Fig. 5).
- The variance of the quantum relative entropy provides the convergence rate of the error probability exponent to its asymptotic value, for the type-I error probability constrained to be lower than ε.
- The authors do not provide the explicit formula here.

### IV. CONCLUSION AND OUTLOOK

- The authors have compared the performance of a quantum noise radar based on two-mode squeezed states and a class of noise radars based on thermal states.
- The authors have found that, given a constraint on the transmitting power, a quantum advantage in the ROC curve and their asymptotic performances is possible.
- A quantum advantage appears when the idler and signal are allowed to be measured jointly.
- The authors results show that amplification is not a good strategy to overcome the technical difficulties that one must face in a practical quantum radar implementation.
- Interfacing a large bandwidth entangled microwave signal with the environment, and developing low-noise power detectors is crucial for performing a QI experiment with a quantum advantage.

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