A comparison between quantum and classical noise radar sources
Summary (3 min read)
- 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. .
- 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 .
- The authors analysis shows that any quantum advantage is destroyed by the unavoidable noise added when amplifying either the signal or the idler.
- The authors introduce the models for the quantum and classical systems, within the quantum mechanical description.
- The authors emulate Refs. , , 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 , 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 , .
- 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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Q1. What are the contributions mentioned in the paper "A comparison between quantum and classical noise radar sources" ?
The authors compare the performance of a quantum radar based on two-mode squeezed states with a classical radar system based on correlated thermal noise. In addition, the authors 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.