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Receiver Operating Characteristics for a Prototype Quantum Two-Mode Squeezing Radar

David Luong, C. W. Sandbo Chang, A. M. Vadiraj, Anthony Damini, C. M. Wilson, Bhashyam Balaji

arXiv:1903.00101v1quant-pheess.SP

TL;DR

The paper addresses the difficulty of applying quantum illumination to radar when joint measurement requires knowing the target distance. It presents a microwave QTMS radar using independent measurements and reports an eightfold integration-time advantage over a classical TMN radar. The prototype’s amplified output is correlated but no longer entangled.

  • Problem

    The original quantum-illumination protocol is impractical for radar because joint, simultaneous measurement is difficult unless target distance is already known.

    Source: §I · p. 1

  • Method

    A JPA generates entangled microwave signals; the prototype independently measures one pulse and transmits the other, then correlates the measurements.

    Source: §I · p. 1

  • Results

    The QTMS radar reduces integration time by a factor of eight relative to the TMN radar.

    Source: §IX · p. 14

  • Takeaways & Limitations

    The experiment demonstrates a significant gain over a similar classical system and shows that microwave quantum radar is not infeasible.

    Source: §IX · pp. 14, 16

    The signals emerging from the refrigerator are not entangled, leaving open whether a better classical system could reproduce the results.

    Source: §IX · p. 16

Abstract

from arXiv · show

We have built and evaluated a prototype quantum radar, which we call a quantum two-mode squeezing radar (QTMS radar), in the laboratory. It operates solely at microwave frequencies; there is no downconversion from optical frequencies. Because the signal generation process relies on quantum mechanical principles, the system is considered to contain a quantum-enhanced radar transmitter. This transmitter generates a pair of entangled microwave signals and transmits one of them through free space, where the signal is measured using a simple and rudimentary receiver. At the heart of the transmitter is a device called a Josephson parametric amplifier (JPA), which generates a pair of entangled signals called two-mode squeezed vacuum (TMSV) at 6.1445 GHz and 7.5376 GHz. These are then sent through a chain of amplifiers. The 7.5376 GHz beam passes through 0.5 m of free space; the 6.1445 GHz signal is measured directly after amplification. The two measurement results are correlated in order to distinguish signal from noise. We compare our QTMS radar to a classical radar setup using conventional components, which we call a two-mode noise radar (TMN radar), and find that there is a significant gain when both systems broadcast signals at -82 dBm. This is shown via a comparison of receiver operator characteristic (ROC) curves. In particular, we find that the quantum radar requires 8 times fewer integrated samples compared to its classical counterpart to achieve the same performance.

I. INTRODUCTION

The introduction presents a microwave prototype inspired by quantum illumination that uses independent measurements, then compares its performance with a classical radar. It also defines the prototype’s scope and practical limitations.

I. INTRODUCTION

  • Because joint measurement is difficult when target distance is unknown, the prototype instead independently measures the two pulses using in-phase and quadrature voltages.The measurements use off-the-shelf equipment.Source: §I · p. 1
  • The QTMS radar generates entangled microwave signals at 6.1445 GHz and 7.5376 GHz, measures one directly, and transmits the other through a horn antenna.The signals are two-mode squeezed vacuum generated by a Josephson parametric amplifier.Source: §I · p. 1
  • The classical TMN radar requires eight times longer integration than QTMS to achieve similar performance, quantified with ROC curves.The comparison uses a classical radar with classically correlated signals.Source: §I · pp. 1–2

A. Claims of This Paper

  • The paper describes an experimental microwave quantum radar protocol, while clarifying that signals exiting the horns are correlated but not entangled.The authors distinguish this from prior optical-wavelength implementations, which they describe as quantum lidars.Source: A. Claims of This Paper · p. 2
  • The apparatus generates entangled microwave signals directly, avoids joint measurement and quantum memories, and uses conventional horn antennas and digitizers.The signals are broadcast, received, and measured after amplification.Source: A. Claims of This Paper · p. 2
  • The authors describe the prototype as non-ideal and adapted from existing apparatus, with further work needed in radar engineering, signal processing, and quantum physics.They do not claim to have invented the JPA.Source: A. Claims of This Paper · p. 2

II. THE BASIC IDEA

Both prototypes transmit one of two correlated noise signals and use matched filtering to distinguish signal from noise. The paper describes how delayed measurements shape the covariance matrix used for filtering.

II. THE BASIC IDEA

  • Both prototypes generate two correlated noise signals, measure one as a reference, transmit the other, and declare detection when the matched-filter output exceeds a threshold.The quantum and classical setups differ in how the correlated noise signals are generated; subsequent measurement and analysis are treated identically.Source: §II · pp. 2–3
  • The signals need only be correlated, not identical, and stronger correlations increase the matched-filter output.The authors relate the basic concept to noise radar.Source: §II · p. 3

A. Covariance Matrices in a Single-Channel System

  • The covariance matrix is formed from measured in-phase and quadrature voltages, with signal 2 serving as the matched-filter reference rather than as an array-processing channel.The two signals are generally measured at different times, with delay related to the transmitted signal’s free-space path length.Source: A. Covariance Matrices in a… · p. 3
  • Assuming stationary generated noise, the delayed covariance matrix’s R12(τ) block encodes the cross-signal correlation used for matched filtering.The experiment operates at fixed range, so τ is fixed for its results.Source: A. Covariance Matrices in a… · p. 3

III. TWO-MODE NOISE RADAR AS A BASELINE FOR COMPARISON

The TMN radar is a classical baseline designed to approximate the QTMS radar, with the two setups differing only in how correlated signals are generated. Its free-space test uses facing horns 0.5 m apart, with acknowledged limitations in the antenna arrangement and propagation environment.

III. TWO-MODE NOISE RADAR AS A BASELINE FOR COMPARISON

  • The TMN and QTMS setups are identical apart from signal generation, making TMN a valid baseline for comparison.The TMN block diagram retains the basic structure of the QTMS setup while specifying its components in more detail.Source: §III · p. 3
  • The classical source mixes a 6.84105 GHz carrier with band-limited Gaussian noise centered at 0.69655 GHz to produce correlated sidebands at 6.1445 GHz and 7.5376 GHz.Both radars use the same amplifier chain.Source: §III · p. 3
  • The 6.1445 GHz sideband is recorded directly, while the 7.5376 GHz sideband is transmitted through a horn and measured at a receiving horn.The digitized receiver data is correlated with the 6.1445 GHz measurement record.Source: §III · p. 3
  • The transmit and receive horns face each other across 0.5 m, making the current arrangement a one-way ranging setup rather than a target-detection apparatus.The experiment does not isolate the horns or eliminate multipath.Source: §III · pp. 3–4, Figure 3 · p. 4
  • The authors report −63 dBm total power at the transmit horn, including amplifier noise, and estimate −82 dBm for the Gaussian-generator signal after discounting that noise.They describe the resulting −19 dB signal-to-noise ratio as crude and do not use SNR in the paper.Source: §III · p. 4
  • At present, both frequencies are transmitted, but the 6.1445 GHz signal is attenuated because it falls outside the X-band antenna passband.The authors propose C-band antennas and filtering out the 6.1445 GHz component as future changes.Source: §III · p. 4

A. Covariance Matrix for the TMN Radar

The TMN covariance model starts from mixing a carrier with Gaussian noise to form two correlated sidebands, then represents additive noise in their measured quadratures. Its covariance structure includes anticorrelated quadratures, distinguishing it from conventional noise radar.

A. Covariance Matrix for the TMN Radar

  • Mixing a carrier with bandpass Gaussian noise produces sidebands at ω1 = ω0 + ω and ω2 = ω0 − ω.The two sidebands are the signals received at the digitizers, with possible amplitude reduction from transmission or direct measurement.Source: A. Covariance Matrix for the TMN Radar · pp. 4–5
  • The model adds independent zero-mean Gaussian noise processes to the measured signal components to represent system and environmental AWGN.The factors A1 and A2 account for amplifier gains and losses from system components and antenna distance.Source: A. Covariance Matrix for the TMN Radar · p. 5
  • The matrix expresses the Pearson correlation coefficient ρ using the signal variances and a space-saving definition s = σ1σ2.The authors use the matrix form to make an explicit expression for ρ.Source: A. Covariance Matrix for the TMN Radar · p. 5
  • The derived TMN covariance matrix has a negative E[Q1Q2] entry, indicating that the two quadratures are anticorrelated.The authors contrast this with conventional noise radars, whose identical transmitted and recorded signals would have positively correlated quadratures.Source: A. Covariance Matrix for the TMN Radar · p. 5

IV. QUANTUM NOISE AND ENTANGLEMENT

Quantum noise limits the correlation attainable by classically generated signals, while entanglement allows quantum noise in paired signals to be correlated. The paper uses this mechanism to motivate higher signal correlation in QTMS radar, while noting that its explanation is not a rigorous treatment of entanglement.

IV. QUANTUM NOISE AND ENTANGLEMENT

  • Quantum noise is unavoidable in measured I and Q voltages, and its relative importance increases as signal amplitude decreases.The figure contrasts sinusoids whose amplitudes differ by a factor of 10, showing quantum noise becoming significant at low signal powers.Source: §IV · pp. 5–6, Figure 4 · p. 6
  • In classically generated sidebands, quantum noise is independent between the signals, limiting their correlation, particularly at low power.Only the classical noise component is correlated between the sidebands.Source: §IV · p. 6
  • Entanglement allows quantum noise in signal pairs to be correlated, and the authors describe this as boosting correlation between transmitted and recorded signals.They characterize the QTMS radar as producing a higher correlation coefficient ρ than the TMN radar.Source: §IV · p. 6
  • The paper uses the correlation explanation as an operational engineering description of entanglement, not as its definition.A rigorous treatment of entanglement is beyond the paper's scope.Source: §IV · p. 6

A. The Entangled Signal: Two-mode Squeezed Vacuum

The prototype uses two-mode squeezed vacuum (TMSV), a pair of entangled frequency modes described probabilistically through quadrature means and covariance. Its squeezing redistributes noise among quadrature combinations, and its covariance has a mathematical correspondence to the TMN model without making the two signal generators physically equivalent.

A. The Entangled Signal: Two-mode Squeezed Vacuum

  • TMSV comprises entangled beams at two frequencies whose measured quadratures follow a multivariate Gaussian distribution specified by their mean and covariance matrix.The individual beams are indistinguishable from thermal noise; their correlations appear only when the beams are correlated.Source: A. The Entangled Signal: Two-mode… · pp. 6–7
  • TMSV has no associated waveform and is described as a stochastic process because its quadrature voltages are random and have zero mean.The paper says waveform is a classical concept not used to describe quantum-mechanical behavior.Source: A. The Entangled Signal: Two-mode… · p. 7
  • Squeezing reduces quantum noise in certain linear combinations of quadratures at the cost of increasing noise in other combinations.For φ = 0, (I1 − I2)/2 and (Q1 + Q2)/2 have variance e^-2r, while (I1 + I2)/2 has variance e^2r.Source: A. The Entangled Signal: Two-mode… · p. 7
  • The TMN and TMSV covariance matrices have essentially the same structure, with the TMN parameters set to σ1^2 = σ2^2 = cosh 2r and ρ = tanh 2r.The mathematical correspondence does not mean the signals are physically the same; only an entangled signal generator can produce the TMSV covariances.Source: A. The Entangled Signal: Two-mode… · p. 7
  • The paper reports tests verifying entanglement at the source before amplification or transmission, while noting that verification requires precise calibration and accounting for added noise.The authors state that this level of precision is not currently necessary or approached in practical radar systems.Source: A. The Entangled Signal: Two-mode… · p. 7

V. EXPERIMENTAL SETUP

The QTMS setup uses a JPA-generated entangled signal, then amplifies and splits it between a horn antenna and digitizer. Its post-refrigerator arrangement matches the TMN radar, while the signal-generation step differs.

V. EXPERIMENTAL SETUP

  • After the dilution refrigerator, the QTMS setup is the same as the TMN radar setup.The signal-generation step is the difference between the two systems.Source: §V · pp. 7–8
  • The JPA generates the TMSV signal; after amplification, the signal is split between a horn antenna and a digitizer.The JPA is the source of the entangled signal and the setup's most unusual component for a radar engineer.Source: Figure 5 · p. 8, §V · pp. 8–9
  • The JPA sits in the dilution refrigerator's coldest stage, cooled to approximately 7 mK to produce an electromagnetic vacuum above 4 GHz.The JPA is a superconducting device.Source: §V · p. 8, Figure 7 · p. 8
  • The same amplifier chain is used for QTMS and TMN, and two circulators isolate the JPA from extraneous signals.The HEMT begins the amplifier chain; apart from it, the chain is at room temperature.Source: §V · p. 9
  • Amplification makes the outgoing signal easier to detect but destroys entanglement; higher correlations persist in the experimental results.The SNTJ and bias tee are used for calibration to confirm that the JPA signal is entangled, and play no other role.Source: §V · p. 9

VI. ENTANGLEMENT GENERATION USING A JOSEPHSON PARAMETRIC AMPLIFIER

The JPA generates TMSV by applying parametric amplification to the quantum vacuum, using a magnetically tunable superconducting resonator. The experiment's JPA requires cryogenic operation and produces signals at two stated frequencies.

VI. ENTANGLEMENT GENERATION USING A JOSEPHSON PARAMETRIC AMPLIFIER

  • The JPA is a resonant cavity with a SQUID whose resonance frequencies can be changed by applying a magnetic field.Modulating the field at the sum of two frequencies enables parametric amplification.Source: §VI · p. 9
  • Parametric amplification of the quantum vacuum produces the entangled TMSV signal; in this prototype, the JPA performs spontaneous downconversion without an input signal.A magnetic field modulated at the sum of two resonance frequencies enables parametric amplification.Source: §VI · p. 9
  • The JPA is a 33 mm quarter-wavelength coplanar-waveguide resonator, terminated by a SQUID and coupled to the input/output line through a small capacitance.It is fabricated from thin-film aluminum and mounted on a printed circuit board approximately 3 cm long.Source: §VI · p. 9
  • The JPA cannot operate at room temperature, and excessive thermal noise at its input prevents it from generating an entangled signal.The paper notes that even room-temperature superconducting materials would be unlikely to remove the thermal-noise constraint.Source: §VI · pp. 9–10
  • The experiment's JPA output frequencies are 6.1445 GHz and 7.5376 GHz, tunable to some extent by changing the magnetic-field strength.The frequencies are two resonance frequencies after the SQUID is biased with a static magnetic field.Source: §VI · p. 10

VII. SIGNAL PROCESSING, MATCHED FILTERING, AND DETECTOR FUNCTIONS

The processing uses covariance between the two frequency modes for matched filtering and considers several detector functions, which may behave differently when experimental covariance matrices depart from the ideal form. The functions are not claimed to be exhaustive or optimal.

VII. SIGNAL PROCESSING, MATCHED FILTERING, AND DETECTOR FUNCTIONS

  • Matched filtering uses the off-diagonal covariance block R12(τ); detector functions can recover ρ in the ideal case but may perform differently on experimental data.The measured sample covariance matrices do not fit the ideal form exactly.Source: §VII · p. 10
  • The data are phase-corrected to φ = 0 by fitting the phase to one second of sample covariance data and rotating the measurements.The digitizers' limited phase coherence causes the phase to drift randomly over time.Source: §VII · p. 10
  • For target detection, the off-diagonal elements indicate target presence when nonzero and absence when zero.The statement follows the paper's phase preprocessing, which sets φ = 0.Source: §VII · p. 10
  • Detector 5 uses only signal powers, discarding cross-mode covariance and matched filtering; the authors expect no quantum–classical difference with it.Detectors 1 and 2 depend on phase, while Detectors 3–5 do not.Source: §VII · p. 10
  • The authors use the same mathematical formulas and computer code for classical and quantum measurements, without the calibration required to test entanglement.The detector list is not exhaustive, and none of its functions is claimed to be optimal.Source: §VII · p. 10

VIII. RESULTS

The results section presents Detector 1 ROC curves for QTMS and TMN at roughly −82 dBm, across multiple integration counts. It also reports the signal power settings used for the comparison.

VIII. RESULTS

  • 5,000 to 100,000 integrated samples are compared in Detector 1 ROC curves for QTMS and TMN, both operating at roughly −82 dBm.The counts correspond to integration times from 5 ms to 0.1 s.Source: §VIII · p. 10, Figure 10 · p. 11
  • 1,000 integrated samples are used for the Detector 1 ROC curves comparing QTMS and TMN at roughly −82 dBm.The figure presents a ROC curve for each radar.Source: Figure 11 · p. 11
  • −145.43 dBm is the stated JPA output power that the authors found seemed to maximize the performance difference between the quantum and classical setups.After amplification, the signal power fed into the transmit horn is roughly −82 dBm for both radars.Source: §VIII · pp. 10–11

A. QTMS vs. TMN Radar

The QTMS radar matches the TMN radar’s ROC performance with eight times fewer integrated samples. Detector 1 performs best, while Detector 5 shows no quantum advantage because it discards amplitude correlations.

A. QTMS vs. TMN Radar

  • At pFA = 0.001, pD is approximately 0.95 for QTMS and 0.05 for TMN at 100,000 samples, after near-equal performance at 5,000.The plotted curves extend only to pFA = 10^-4 because the experiment did not collect enough data to go lower.Source: A. QTMS vs. TMN Radar · p. 11
  • The eightfold sample advantage also appeared at 1,000 samples, though that comparison was not shown.The authors say the factor seems to hold for other sample counts as well.Source: A. QTMS vs. TMN Radar · p. 11
  • The authors describe the comparison as apples-to-apples, using the same frequencies, horns, detection setup, and signal processing despite the TMN setup’s practical weaknesses.They identify setup simplicity, multipath, near-field operation, and very low power as factors behind the TMN radar’s poorer performance than most practical radars.Source: A. QTMS vs. TMN Radar · p. 11
  • 8 times fewer integrated samples let the QTMS radar match the TMN radar’s ROC performance with Detector 1.The comparison uses 50,000 and 100,000 QTMS samples versus 400,000 and 800,000 TMN samples.Source: A. QTMS vs. TMN Radar · p. 11, Figure 12 · p. 12

B. Comparison of Detector Functions

  • Detector 1 was best among the five detectors; with more samples, performance differences narrowed because rotation preprocessing concentrates covariance in I1I2 and Q1Q2.The authors attribute the narrowing to residual covariance in I1Q2 and Q1I2 being suppressed as sample integration increases.Source: B. Comparison of Detector Functions · p. 13
  • Detector 5 showed virtually no QTMS–TMN ROC difference because it uses power correlations and discards amplitude correlations.The authors conclude that amplitude correlations are crucial in their detection scheme.Source: B. Comparison of Detector Functions · p. 13, Figure 14 · p. 13

C. Comparison with Analytic Expressions for a Conventional Noise Radar

Analytic noise-radar formulas generally fit the experimental ROC curves, though the QTMS fit deviates somewhat. Detector 1 time series and histograms also show clearer separation of on/off data for QTMS than TMN.

C. Comparison with Analytic Expressions for a Conventional Noise Radar

  • The analytic curves generally fit the experimental ROC data, nearly perfectly for TMN and with some deviation for QTMS.The authors expect the QTMS deviation because their radar is not a conventional noise radar, though it is similar.Source: C. Comparison with Analytic… · p. 13, Figure 15 · p. 14
  • 2.4 at 50,000 samples and 2.5 at 75,000 samples were the fitted ratios ρ_quantum/ρ_classical; the authors interpret the ratios, not the individual ρ values, as meaningful.Numerical instability prevented direct use of N = 50,000 or 75,000, so the authors chose a nominal N and varied ρ to fit the curves.Source: C. Comparison with Analytic… · p. 13
  • The rudimentary analysis estimates QTMS correlations at about 2.4 to 2.5 times the TMN correlations.The authors express the improvement as a correlation-coefficient comparison for a comparable noise radar.Source: C. Comparison with Analytic… · p. 13

D. Time Series and Histograms of Detector 1 Outputs

  • Detector 1 time series separate on/off data more clearly for QTMS than TMN, and the histograms show clear QTMS separation versus substantial TMN overlap.The histogram comparison also supports the reported 2.4–2.5 correlation ratio; one large fluctuation affects the QTMS 100,000-sample time series.Source: D. Time Series and Histograms of… · p. 14

IX. CONCLUSION

The prototype generates TMSV signals entirely in the microwave regime and detects transmissions at powers as low as −82 dBm. Against a parallel classical system, it demonstrates an eightfold reduction in integration time, while leaving important experimental and deployment limitations unresolved.

IX. CONCLUSION

  • The QTMS prototype generates TMSV signals in the microwave regime without optical downconversion and detects transmitted signals at powers as low as −82 dBm.The signals become unentangled after the amplifier chain, despite having been entangled at the JPA source.Source: §IX · pp. 14, 16
  • The QTMS radar reduces integration time by a factor of eight relative to a parallel classical radar using carrier-and-Gaussian-noise signals.The authors describe the comparison as apples-to-apples and attribute the quantum-system improvement to correlations in the entangled signals.Source: §IX · p. 14
  • The authors avoid comparison with ideal quantum-illumination error-exponent claims, instead evaluating against a physical laboratory system using ROC curves.They note their experiment is not traditional quantum illumination and that no uniquely optimum radar exists across criteria.Source: §IX · p. 14
  • The experiment used one-way ranging rather than reflecting signals from targets, leaving target detection at different ranges for future work.The authors also say the prototype is not immediately field-deployable because it requires a cryogenic refrigerator, cool-down time, and substantial SWaP.Source: §IX · pp. 14, 16
  • The authors do not claim an optimal quantum radar and identify room to improve signal generation and detector functions.They also state that high-resolution SAR imaging, array processing for direction-finding, and space-time adaptive processing have not yet been demonstrated.Source: §IX · p. 16
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