Quantum Illumination

Quantum illumination is entanglement-assisted discrimination between two lossy, noisy channels: one describing an absent target and one describing a weakly reflecting target. A source prepares correlated signal–idler mode pairs. The signal interrogates a target region, the idler is retained, and a receiver jointly processes the collected return and retained idler to decide which channel acted.

Its best-known result is striking but narrow. For many independent low-brightness mode pairs, a weak specular reflector of known phase, and a bright thermal background, an ideal two-mode-squeezed transmitter with an optimum joint receiver has an error-probability exponent four times that of an optimum-reception coherent-state transmitter at the same transmitted signal energy. This factor of four is conventionally called a 6 dB error-exponent advantage. It can survive even when loss and noise make the received return–idler state separable.

That theorem does not establish a six-decibel improvement in received power, range, cross section, or arbitrary radar performance. It assumes a declared hypothesis pair, known mode matching, a particular background model, an ideal idler, and an optimum receiver. Unknown phase, speckle, range search, Doppler, clutter, receiver loss, idler storage, and wall-clock duty cycle can change or remove the advantage. Quantum illumination is therefore best understood as a precise result in quantum channel discrimination, not as a synonym for every sensor using correlated radiation.

Required background. Quantum Measurement as Estimation introduces priors and decision losses; Density Operators for Quantum Information supplies mixed states and trace distance; Two-Mode Entanglement develops the signal–idler state; POVMs defines general measurement outcomes.

Target absence and presence as quantum channels

Two hypotheses

For one matched temporal-frequency-spatial mode, let $\hat a_S$ denote the transmitted signal, $\hat a_I$ the retained idler, and $\hat a_R$ the collected return. A standard specular-target model is

$$\begin{aligned} H_0 &: \quad \hat a_R = \hat b_0, \\[3pt] H_1 &: \quad \hat a_R = \sqrt{\kappa}\,e^{i\phi}\hat a_S + \sqrt{1-\kappa}\,\hat b_1. \end{aligned}$$

Here $H_0$ means target absent, $H_1$ means target present, $\kappa\ll1$ is the round-trip transmissivity into the collected mode, and $\phi$ is the return phase. The bath modes are thermal and independent of the source. In the usual no-passive-signature convention, they are chosen so that the received background brightness is $N_B$ under either hypothesis:

$$\langle \hat b_0^\dagger\hat b_0\rangle=N_B, \qquad \langle \hat b_1^\dagger\hat b_1\rangle = \frac{N_B}{1-\kappa}.$$

Consequently,

$$\begin{aligned} \langle \hat a_R^\dagger\hat a_R\rangle_{H_0} &=N_B, \\ \langle \hat a_R^\dagger\hat a_R\rangle_{H_1} &=N_B+\kappa N_S, \end{aligned}$$

where $N_S=\langle\hat a_S^\dagger\hat a_S\rangle$ is signal photons per mode. This convention prevents an artificial detector from deciding between $H_0$ and $H_1$ merely because inserting the target changes the background level. A different physical background model is legitimate, but it defines a different discrimination problem and must be used for both quantum and classical systems.

Priors, costs, and errors

Let $\Pi_0$ and $\Pi_1$ be the receiver’s two decision operators, with $\Pi_0+\Pi_1=\mathbb I$. The false-alarm and miss probabilities are

$$P_{\mathrm{FA}} = \operatorname{Tr}(\Pi_1\rho_0), \qquad P_{\mathrm{M}} = \operatorname{Tr}(\Pi_0\rho_1).$$

For prior probabilities $\pi_0$ and $\pi_1$ and equal costs for the two error types, the Bayes error is

$$P_e = \pi_0P_{\mathrm{FA}} + \pi_1P_{\mathrm{M}}.$$

Equal priors are mathematically convenient, but many sensing systems operate at low target prevalence and impose a stringent false-alarm constraint. The appropriate output is then a receiver-operating characteristic or a Neyman–Pearson miss probability at fixed $P_{\mathrm{FA}}$, not only an equal-prior error number. Threshold tuning after seeing test data invalidates either comparison.

Many matched modes

A broadband pulse or observation interval contains many approximately orthogonal mode pairs. In the ideal independent-and-identically-distributed model,

$$\rho_j^{(M)} = \rho_j^{\otimes M}, \qquad j\in\{0,1\}.$$

For duration $T$ and usable bandwidth $W$, the order of magnitude is $M\sim TW$, up to real-mode versus complex-mode conventions and polarization factors. The transmitted photon budget is

$$N_{\mathrm{tx}}=MN_S, \qquad E_{\mathrm{tx}}=\hbar\omega MN_S$$

for a narrowband carrier. Reporting $N_S$ without $M$, or $M$ without source brightness, does not specify the exposure.

The Gaussian Quantum-Illumination Source

Two-mode squeezed vacuum

The canonical Gaussian transmitter uses $M$ two-mode squeezed-vacuum pairs. For each pair,

$$\lvert\psi\rangle_{SI} = \sum_{n=0}^{\infty} \sqrt{\frac{N_S^n}{(N_S+1)^{n+1}}} \lvert n\rangle_S\lvert n\rangle_I.$$

Each marginal is thermal,

$$\langle \hat a_S^\dagger\hat a_S\rangle = \langle \hat a_I^\dagger\hat a_I\rangle =N_S,$$

but the pair has phase-sensitive cross correlation

$$\langle \hat a_S\hat a_I\rangle = \sqrt{N_S(N_S+1)}$$

in a convenient phase convention. A classical signal–idler state with a positive Glauber–Sudarshan representation obeys the Cauchy–Schwarz bound

$$\bigl\lvert \langle \hat a_S\hat a_I\rangle \bigr\rvert^2 \leq N_SN_I.$$

For equal signal and idler brightness $N_I=N_S$, a two-mode squeezed vacuum has correlation squared $N_S(N_S+1)$ rather than at most $N_S^2$. The relative gap is especially large when $N_S\ll1$. Quantum illumination encodes target presence in the weak remnant of this initially nonclassical correlation.

Covariance matrices under the hypotheses

Use $X=(a+a^\dagger)/\sqrt2$ and $P=(a-a^\dagger)/(i\sqrt2)$, so $[X,P]=i$ and vacuum variance is $1/2$. Order the covariance as $(X_R,P_R,X_I,P_I)$ and define

$$\mathbf Z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.$$

For known phase $\phi=0$, the return–idler covariance matrices are

$$V_{RI}^{(0)} = \begin{pmatrix} (N_B+\tfrac12)\mathbb I_2 & 0\\ 0 & (N_S+\tfrac12)\mathbb I_2 \end{pmatrix}$$

and

$$V_{RI}^{(1)} = \begin{pmatrix} (N_B+\kappa N_S+\tfrac12)\mathbb I_2 & \sqrt{\kappa N_S(N_S+1)}\,\mathbf Z \\ \sqrt{\kappa N_S(N_S+1)}\,\mathbf Z & (N_S+\tfrac12)\mathbb I_2 \end{pmatrix}.$$

When $N_B\gg\kappa N_S$, the return marginal changes only slightly. The main signature available to a joint receiver is the off-diagonal block. Under $H_0$ it vanishes; under $H_1$ it is proportional to $\sqrt{\kappa N_S(N_S+1)}$.

Advantage after entanglement is gone

In this convention, the thermal-loss channel is entanglement breaking when $N_B\geq\kappa$. Its added quadrature noise is $Y=[N_B+(1-\kappa)/2]I_2$; the phase-insensitive attenuator is entanglement breaking when $Y\geq(1+\kappa)I_2/2$, which gives this threshold. The returned mode and stored idler are then separable even under $H_1$. This does not make the two conditional states identical. The initial entangled source created a phase-sensitive correlation stronger than any equal-energy classical source could create; attenuation leaves a weak but measurable remnant of that correlation.

The operational resource statement is therefore:

Initial signal–idler entanglement enables a stronger transmitter correlation, while target detection uses distinguishability between the two final return–idler states.

It is misleading to say that entanglement itself travelled out and back intact. It is also unnecessary to identify one surviving correlation measure, such as discord, as the unique cause of the advantage. The performance claim is settled by a matched channel-discrimination bound.

Quantum Hypothesis Testing

Helstrom limit

For arbitrary priors, the minimum Bayes error over all two-outcome POVMs is

$$P_e^\star = \frac12 \left[ 1- \left\lVert \pi_1\rho_1^{(M)} - \pi_0\rho_0^{(M)} \right\rVert_1 \right].$$

For equal priors this becomes

$$P_e^\star = \frac12 \left[ 1- \frac12 \left\lVert \rho_1^{(M)}-\rho_0^{(M)} \right\rVert_1 \right].$$

The corresponding Helstrom measurement projects onto the positive and negative eigenspaces of $\pi_1\rho_1^{(M)}-\pi_0\rho_0^{(M)}$. This defines the optimum, but directly diagonalizing a many-mode mixed-state difference does not by itself give a practical optical receiver.

Quantum Chernoff exponent

For equal priors and independent copies, define

$$Q := \inf_{0\leq s\leq1} \operatorname{Tr} \left( \rho_0^s\rho_1^{1-s} \right), \qquad \xi_Q:=-\ln Q.$$

The quantum Chernoff bound is

$$P_e^\star(M) \leq \frac12Q^M = \frac12e^{-M\xi_Q}.$$

More importantly, the exponent is asymptotically exact:

$$\lim_{M\to\infty} -\frac1M\ln P_e^\star(M) = \xi_Q.$$

An exponent is not an exact finite-sample error probability. Prefactors, non-identical modes, calibration uncertainty, and threshold estimation matter before the asymptotic regime is reached.

The Error-Exponent Hierarchy

The standard asymptotic regime

The familiar comparison assumes

$$N_S\ll1, \qquad \kappa\ll1, \qquad N_B\gg1, \qquad M\gg1,$$

with known return phase and ideal mode matching. Let

$$\gamma := \frac{\kappa N_S}{N_B}.$$

For a coherent-state transmitter with optimum reception, the leading error exponent is

$$\xi_{\mathrm{coh}} \simeq \frac{\gamma}{4}, \qquad P_{e,\mathrm{coh}} =\exp[-M\xi_{\mathrm{coh}}+o(M)].$$

This is not merely a comparison with direct detection. A coherent state is the reference classical transmitter for the specified Gaussian channel, and it is paired here with optimum quantum reception. The original positive-$P$ analysis also bounds arbitrary classical signal–idler transmitters, but that general bound is looser than the exact coherent-state result. An experiment claiming the best classical benchmark must therefore state its allowed transmitter, idler, receiver, amplifier, and reference resources rather than infer that claim from a comparison with one convenient classical apparatus.

Structured pairwise quantum-illumination receivers such as ideal OPA and phase-conjugate receivers attain

$$\xi_{\mathrm{pair}} \simeq \frac{\gamma}{2}.$$

The optimum joint quantum receiver attains

$$\xi_{\mathrm{QI}} \simeq \gamma.$$

Thus

$$\frac{\xi_{\mathrm{pair}}}{\xi_{\mathrm{coh}}} \simeq2, \qquad \frac{\xi_{\mathrm{QI}}}{\xi_{\mathrm{coh}}} \simeq4.$$

Expressed in decibels,

$$10\log_{10}2\simeq3.01\ \mathrm{dB}, \qquad 10\log_{10}4\simeq6.02\ \mathrm{dB}.$$

The logarithm applies to a ratio of error exponents. It does not mean that $P_e$ itself is smaller by 6 dB. Because $P_e$ is exponential in $M$, a fourfold exponent can yield a much larger error-probability ratio after many modes. Conversely, outside the asymptotic assumptions there need not be a fourfold ratio at all.

Exact coherent-state expression

For the ideal known-phase Gaussian model, the coherent-state Chernoff exponent before the bright-background approximation is

$$\xi_{\mathrm{coh}} = \kappa N_S \left( \sqrt{N_B+1}-\sqrt{N_B} \right)^2.$$

Since

$$\left( \sqrt{N_B+1}-\sqrt{N_B} \right)^2 \simeq \frac{1}{4N_B}$$

for $N_B\gg1$, this reduces to the expression above. An accessible comparison is phase-matched homodyne detection. Each return quadrature is Gaussian with variance $N_B+1/2$ and means separated by $\sqrt{2\kappa N_S}$. A midpoint threshold on the sum of $M$ measurements gives the exact error

$$P_{e,\rm hom}(M) =\frac12\operatorname{erfc}\!\left( \sqrt{\frac{M\kappa N_S}{4N_B+2}}\right).$$

Its exponent $\kappa N_S/(4N_B+2)$ approaches the optimal coherent exponent in bright background, but homodyne is not the optimal receiver at every $N_B$. The complementary error function also makes the nonconstant finite-$M$ prefactor explicit.

Quoting the 6 dB result at optical frequencies with negligible thermal occupancy, without recomputing the relevant bound, silently changes regime.

This is not the standard quantum limit

The Standard Quantum Limit is a matched independent-probe benchmark for an estimation problem, commonly expressed as a variance or root-mean-square error. Quantum illumination is usually a finite-separation hypothesis test, and its central figure of merit is an error probability, ROC, or asymptotic error exponent. Calling $\xi_{\mathrm{coh}}$ an SQL obscures the loss function and invites an invalid comparison between $1/\sqrt N$ precision scaling and exponential decision error. A reflectivity-estimation version can be formulated with Fisher information, but it is a different task from target absence versus presence.

Compare equal transmitted energy, mode bandwidth and duration, target and background models, priors or false-alarm constraints, and allowed receiver resources. A classical reference can use optimum quantum reception even though its transmitter has a positive $P$ representation. Beating a split-thermal source or coherent direct detection does not automatically beat that stronger reference. Idler storage, references, loss, and preparation failures must be included when comparing complete instruments.

Receiver Architectures

The source creates an opportunity; the receiver decides how much of it is realized.

Optical-parametric-amplifier receiver

A low-gain optical parametric amplifier mixes each return–idler pair. In one phase convention an output mode is

$$\hat c_m = \sqrt{G}\,\hat a_{I_m} + \sqrt{G-1}\,\hat a_{R_m}^{\dagger}, \qquad G-1\ll1.$$

Writing $C_{RI}=\langle a_Ra_I\rangle$, the target-present mean is

$$\langle c_m^\dagger c_m\rangle_1 =G N_S+(G-1)(N_B+\kappa N_S+1) +2\sqrt{G(G-1)}\operatorname{Re}C_{RI}.$$

Under $H_0$, both $C_{RI}$ and the $\kappa N_S$ term vanish. The last term is the useful phase-sensitive signal, but a weaker return-intensity term also remains. Photon counts accumulated over $M$ modes therefore shift between the hypotheses. With optimized gain in the canonical regime, this receiver reaches the 3 dB exponent advantage. It requires stable phase and good return–idler mode matching, and it remains suboptimal even with ideal components.

Phase-conjugate receiver

A phase-conjugate receiver converts the return so that its phase-sensitive correlation with the idler becomes a phase-insensitive interference signal. The converted return and idler interfere on a balanced beam splitter, and the difference of output counts is the test statistic. Its ideal asymptotic exponent also reaches the 3 dB level.

Physical phase conjugation adds vacuum noise. A digital implementation that heterodynes the return and idler and conjugates one classical record is not the same measurement as lossless optical joint detection: separate heterodyne measurements have already added measurement noise and destroyed access to a later collective POVM.

Sum-frequency-generation receivers

Sum-frequency generation is the inverse nonlinear process to down-conversion. For many weak return–idler pairs with common phase-sensitive correlation, the interaction

$$\hat H_{\mathrm{SFG}} = \hbar g \sum_{m=1}^{M} \left( \hat b^\dagger \hat a_{R_m}\hat a_{I_m} + \hat b \hat a_{R_m}^\dagger\hat a_{I_m}^\dagger \right)$$

can convert pair correlations into a sum-frequency mode $\hat b$, for real $g$ in a fixed pump-phase convention. Initially, $d\langle b\rangle/dt=-ig\sum_m\langle a_{R_m}a_{I_m}\rangle$. For $M$ identical aligned pairs this is $-igM C_{RI}$, not a $\sqrt M$ amplitude at fixed $g$ and interaction time. A normalized sum $M^{-1/2}\sum_m a_{R_m}a_{I_m}$ instead has mean $\sqrt M C_{RI}$. Conversion time, depletion, and noise must be included to obtain receiver performance; initial coherent growth is not an $M^2$ information law. A staged SFG receiver can asymptotically attain the quantum Chernoff exponent in the ideal low-brightness model (Zhuang, Zhang, and Shapiro). Feedforward nulling between stages can approach the Helstrom limit in the low-brightness regime.

The ideal construction requires controlled weak nonlinear interactions, mode-resolved processing, idler storage, and adaptive photon counting; losses in those operations change the predicted error.

Correlation-to-displacement conversion

Correlation-to-displacement receivers conditionally map broadband return–idler correlation onto a coherent displacement of a selected mode. The final problem can then be attacked with coherent-state receivers such as Kennedy-like nulling or photon counting. This architecture is theoretically capable of optimal performance for several known-phase entanglement-assisted tasks and may reduce the nonlinear and memory burden of staged SFG.

Its practical performance still depends on conversion efficiency, amplifier noise, idler loss, phase knowledge, mode selection, and the final detector. Ideal noiseless amplification and actual microwave amplification are not interchangeable assumptions.

Coincidence and covariance receivers

Time-tagged photon-pair experiments often estimate a coincidence excess or a photon-number covariance,

$$C_{RI} = \langle \hat n_R\hat n_I\rangle - \langle \hat n_R\rangle \langle \hat n_I\rangle.$$

Such receivers can reject uncorrelated background and give accessible proof-of-principle demonstrations. Their performance depends on coincidence window, detector jitter, dark counts, multipair emission, accidental coincidences, and whether no-click records enter the likelihood. Discarding noncoincident trials without counting their acquisition cost can make a conditional statistic look better while reducing information per transmitted photon or per second.

Loss, Imperfections, and Nuisance Parameters

Idler loss is not round-trip loss

Signal attenuation is already represented by $\kappa$ and is the regime in which quantum illumination was designed to operate. Idler loss is different: it directly erases the local reference needed to read the return correlation. With return efficiency $\eta_R$, idler-memory efficiency $\eta_I$, mode overlap amplitude $\mu$, and Gaussian phase jitter of variance $\sigma_\phi^2$, a useful schematic correlation budget is

$$C_{RI}^{\mathrm{eff}} = \mu \sqrt{\eta_R\eta_I\kappa} \,e^{-\sigma_\phi^2/2} \sqrt{N_S(N_S+1)}.$$

This is an averaged correlation, not a complete noisy-state specification. The expression assumes independent, zero-mean phase jitter and uncorrelated loss environments. Observed phase tags can allow compensation; a shared unknown phase across many pairs produces correlated records. Noise variances and the full likelihood are still needed before converting $\lvert C_{RI}^{\mathrm{eff}}\rvert^2$ into a signal-to-noise or error ratio. A long idler delay can therefore erase the ideal factor-of-four advantage even though the outward signal path is supposed to be lossy. Any proposal for long range must specify how the idler is stored for the round-trip time without excessive attenuation, added noise, or mode drift.

Source impurity and brightness

At $N_S\ll1$, two-mode squeezing offers its largest correlation advantage per transmitted photon, but useful decisions require many modes. Increasing pump power raises $N_S$ and total flux but moves the source away from the regime in which the canonical exponent ratios are derived. In photon-pair language it also raises multipair events. Source purity, unwanted thermal population, spectral Schmidt number, pump phase noise, and collection loss must be included in the state delivered to the transmit and idler reference planes.

Unknown phase and Rayleigh fading

The 6 dB Gaussian result assumes a specular return with known phase. A rough target may produce speckle with random amplitude and nearly uniform phase. For one pair, the target-present state is then a mixture,

$$\bar\rho_1 = \int d\kappa\,d\phi\, p(\kappa,\phi)\rho_1(\kappa,\phi).$$

If the same unknown $\kappa,\phi$ persist throughout a pulse, however, the $M$-pair state is

$$\bar\rho_1^{(M)} =\int d\kappa\,d\phi\,p(\kappa,\phi) \rho_1(\kappa,\phi)^{\otimes M},$$

which generally differs from $\bar\rho_1^{\otimes M}$. The latter describes fresh independent fading in every mode. Averaging and taking tensor powers do not commute: common phase can leave relative information across the block even when each marginal loses its mean correlation. The cited Rayleigh-fading analysis uses a common fading coefficient over the return bandwidth and observation interval.

Uniform phase averaging gives

$$\int_0^{2\pi}\frac{d\phi}{2\pi} e^{i\phi} =0,$$

so the mean phase-sensitive correlation used by an OPA receiver vanishes, although its small return-intensity contribution need not vanish. The OPA’s known-phase 3 dB advantage can disappear. More collective receivers can retain an advantage for some fading models, but its scaling may become subexponential rather than a simple 6 dB exponent improvement. Phase tracking with a reference beam consumes power and changes the comparison.

Unknown range, angle, and Doppler

The basic protocol asks whether a target occupies one known mode. Radar and lidar usually search over delay, angle, and Doppler cells. If the return could occupy one of $K$ modes, the receiver faces a multiple-hypothesis problem,

$$H_0, H_{1,1}, H_{1,2}, \ldots, H_{1,K},$$

with a search penalty and a mode bank. Waveform ambiguity, sidelobes, multiple comparisons, moving-target phase, and the cost of a quantum memory must be included. A one-cell target-detection exponent is not a complete ranging or tracking analysis.

Thermal noise, clutter, and jamming

For a bosonic mode at temperature $T$,

$$N_{\mathrm{th}}(\omega,T) = \frac{1}{ \exp(\hbar\omega/k_{\mathrm B}T)-1 }.$$

Room-temperature microwave modes can have $N_{\mathrm{th}}\gg1$, while optical thermal occupancy is ordinarily negligible. Optical experiments often inject background light to access the bright-noise regime. Microwave systems must contend with cryogenic-to-room-temperature interfaces and amplifier noise before and after the target path.

Thermal Gaussian noise is not a universal model for terrain clutter, multipath, atmospheric fading, interference, or an adaptive jammer. Robustness to an entanglement-breaking thermal channel does not imply immunity to spoofing, interception, or adversarial waveform knowledge.

Worked comparison of error exponents

Consider an idealized known-phase test with

$$N_S=10^{-2}, \qquad \kappa=10^{-2}, \qquad N_B=20, \qquad M=2\times10^6.$$

The dimensionless accumulated scale is

$$M\gamma = \frac{M\kappa N_S}{N_B} =10.$$

Using only the leading asymptotic exponents gives

$$\begin{array}{c|c|c} \text{system} & M\xi & \tfrac12e^{-M\xi} \\ \hline \text{coherent, optimum} &2.5 &4.10\times10^{-2} \\ \text{OPA or phase-conjugate} &5 &3.37\times10^{-3} \\ \text{QI, optimum joint} &10 &2.27\times10^{-5} \end{array}$$

These are exponent-level estimates, not certified finite-$M$ errors. They show why saying “6 dB lower error probability” is wrong: the two approximate errors differ by orders of magnitude here, not by a fixed factor of four.

For a leading-envelope target of $10^{-3}$, define

$$L := \ln\!\left(\frac{1}{2P_e}\right) = \ln 500 \simeq6.215.$$

The corresponding leading mode requirements are

$$\begin{aligned} M_{\mathrm{coh}} &\simeq \frac{4N_BL}{\kappa N_S} \simeq4.97\times10^6, \\ M_{\mathrm{pair}} &\simeq \frac{2N_BL}{\kappa N_S} \simeq2.49\times10^6, \\ M_{\mathrm{QI}} &\simeq \frac{N_BL}{\kappa N_S} \simeq1.24\times10^6. \end{aligned}$$

The ideal joint receiver uses one quarter as many modes as the coherent benchmark at the same exponent target. Whether it finishes sooner depends on source bandwidth, idler storage, receiver cycle time, calibration, and accepted-shot fraction.

Leading envelopes $\tfrac12e^{-rx}$ with $x=M\kappa N_S/N_B$ and $r=1/4,1/2,1$ for coherent, pairwise, and optimum-QI reception. At envelope $10^{-3}$, the required $x$ values are approximately $24.9$, $12.4$, and $6.2$. These illustrate the asymptotic mode-count ratio; they are not exact finite-mode error probabilities. The assumptions are low brightness, weak reflection, bright background, known phase, and ideal reception.

Heralded trials and observation time

Suppose each fixed-duration cycle $T_c$ delivers $M_c$ useful pairs with probability $p$, the same under both hypotheses. The failure flag carries no target information, and successful cycles are independent. If a useful pair has Chernoff coefficient $Q=e^{-\xi}$, the full cycle has

$$Q_{\rm cycle}=(1-p)+pQ^{M_c}, \qquad \mathcal R_\xi= \frac{-\ln[1-p+p e^{-M_c\xi}]}{T_c}.$$

The direct-sum success and failure records give the first expression; the same Chernoff minimizer applies to the identical accepted modes. It applies to quantum states with optimum reception, or to classical outcome distributions with a specified receiver. For $M_c\xi\ll1$, $\mathcal R_\xi\simeq pM_c\xi/T_c$. Simply multiplying by the mean accepted fraction is not exact for a fixed number of attempted cycles: unusually many failures contribute to the average error. As $M_c\xi\to\infty$, the exponent per attempted cycle approaches $-\ln(1-p)$.

Stopping after a fixed number of successful cycles is a different design. Its expected duration is increased by $1/p$, and its exponent per expected duration is $pM_c\xi/T_c$; the completion time is then random. If acceptance depends on the target, include its informative flag in the joint model.

When comparing instruments, include photons transmitted on failed cycles, bandwidth, idler storage, receiver efficiency, and acquisition time. Pump, cryogenic, and local-oscillator power may matter for deployment even when a theoretical comparison fixes only photons incident on the sample.

Optical and Microwave Implementations

Optical implementations

Spontaneous parametric down-conversion naturally supplies broadband signal–idler pairs, low thermal source noise, mature photon counting, and optical nonlinear processing. Optical backgrounds are not normally bright thermal baths per mode, so experiments often inject broadband noise. Free-space optical targets introduce turbulence, rough-surface speckle, detector dark counts, daylight background, and mode-matching challenges. At long range, an optical idler can be delayed in fiber, but fiber attenuation and dispersion grow with round-trip time.

Microwave implementations

Microwave frequencies naturally provide bright room-temperature thermal backgrounds and are relevant to conventional radar bands. Josephson parametric devices can generate two-mode-squeezed microwave fields, but they operate cryogenically. Sending one mode into a room-temperature path while retaining and later measuring the idler requires low-loss coupling, amplification, isolation, and often a quantum memory or transduction stage.

Conventional microwave receivers are exceptionally mature. Their coherent transmit power, antenna aperture, low-noise amplification, waveform agility, and digital processing form a demanding baseline. A laboratory quantum advantage in a low-brightness channel is scientifically meaningful without yet establishing superior range, scan rate, robustness, or cost over a field radar.

Optical experiments with OPA reception (Zhang and colleagues) and microwave experiments with idler storage and joint measurement (Assouly and colleagues) provide examples of the detection mechanism under controlled channels. They test particular transmitters, receivers, and noise models. Their performance cannot be converted into a range or search-rate advantage without analyzing those additional tasks. The broader experimental questions are discussed in Quantum Metrology Frontiers.

Exercises

Exercise 1: Classical correlation bound

For a classical signal–idler state represented as a probability mixture of coherent states with complex amplitudes $\alpha_S$ and $\alpha_I$, prove

$$\bigl\lvert\langle\hat a_S\hat a_I\rangle\bigr\rvert^2 \leq N_SN_I.$$

Compare the result with a two-mode squeezed vacuum for $N_I=N_S$.

Solution

For a positive $P$ representation, normally ordered moments are classical averages. Therefore

$$\langle\hat a_S\hat a_I\rangle = \mathbb E(\alpha_S\alpha_I), \qquad N_S=\mathbb E\lvert\alpha_S\rvert^2, \qquad N_I=\mathbb E\lvert\alpha_I\rvert^2.$$

Cauchy–Schwarz gives

$$\bigl\lvert\mathbb E(\alpha_S\alpha_I)\bigr\rvert^2 \leq \mathbb E\lvert\alpha_S\rvert^2 \mathbb E\lvert\alpha_I\rvert^2 =N_SN_I.$$

For a two-mode squeezed vacuum with equal marginal brightness,

$$\bigl\lvert\langle\hat a_S\hat a_I\rangle\bigr\rvert^2 =N_S(N_S+1)>N_S^2$$

for every $N_S>0$. The fractional gap is largest at low brightness.

Exercise 2: Fixed-background convention

Show that choosing $\langle\hat b_1^\dagger\hat b_1\rangle=N_B/(1-\kappa)$ in the target-present channel gives return brightness $N_B+\kappa N_S$.

Solution

Independence and zero bath mean remove cross terms. Hence

$$\begin{aligned} \langle\hat a_R^\dagger\hat a_R\rangle_{H_1} &= \kappa \langle\hat a_S^\dagger\hat a_S\rangle + (1-\kappa) \langle\hat b_1^\dagger\hat b_1\rangle \\ &= \kappa N_S + (1-\kappa)\frac{N_B}{1-\kappa} \\ &=N_B+\kappa N_S. \end{aligned}$$

Thus the target does not create a change in the bath contribution itself; it adds only the attenuated signal energy.

Exercise 3: Decibels and required modes

Suppose two systems have exponents $\xi_A$ and $\xi_B=4\xi_A$. Compute the advantage in decibels and the asymptotic ratio of mode counts needed to reach the same target error.

Solution

The exponent advantage is

$$10\log_{10}\frac{\xi_B}{\xi_A} = 10\log_{10}4 \simeq6.02\ \mathrm{dB}.$$

At leading order $P_e\sim\tfrac12e^{-M\xi}$, so fixed $P_e$ requires $M\xi$ to be fixed. Therefore

$$\frac{M_B}{M_A} = \frac{\xi_A}{\xi_B} = \frac14.$$

This mode-count result, not a fixed fourfold probability ratio, is the direct meaning of the exponent comparison.

Exercise 4: Phase averaging

Under $H_1$, suppose the return phase is uniform on $[0,2\pi)$. Show that the ensemble-averaged phase-sensitive return–idler correlation vanishes. Why does this invalidate the known-phase OPA argument but not prove that every quantum receiver is useless?

Solution

Conditioned on $\phi$,

$$\langle\hat a_R\hat a_I\rangle_{\phi} = \sqrt{\kappa}\,e^{i\phi} \sqrt{N_S(N_S+1)}.$$

Averaging gives

$$\mathbb E_\phi \langle\hat a_R\hat a_I\rangle_\phi = \sqrt{\kappa N_S(N_S+1)} \int_0^{2\pi}\frac{d\phi}{2\pi}e^{i\phi} =0.$$

The OPA’s leading mean-count shift is linear in a chosen quadrature of this correlation, so averaging removes that contribution. The weaker intensity shift $(G-1)\kappa N_S$ remains; zero average correlation does not make the OPA output distributions identical. The mixed state $\bar\rho_1$ can still differ from $\rho_0$ through higher-order or phase-insensitive structure. A receiver designed for the mixture may retain some advantage, but the known-phase exponent cannot simply be reused.

Exercise 5: A correlation-loss budget

Take $\eta_R=0.8$, $\eta_I=0.5$, $\kappa=10^{-3}$, $N_S=0.02$, $\lvert\mu\rvert=0.9$, and $\sigma_\phi=0.3$. Estimate $\lvert C_{RI}^{\mathrm{eff}}\rvert$.

Solution

Using the schematic budget,

$$\begin{aligned} \lvert C_{RI}^{\mathrm{eff}}\rvert &= 0.9 \sqrt{(0.8)(0.5)(10^{-3})} e^{-0.3^2/2} \sqrt{(0.02)(1.02)} \\ &\simeq 2.46\times10^{-3}. \end{aligned}$$

The source correlation before these losses is $\sqrt{0.02\times1.02}\simeq0.143$. The small returned correlation illustrates why a receiver must coherently accumulate many matched modes.

Exercise 6: Thermal occupancy

Estimate the room-temperature thermal occupancy at $10\ \mathrm{GHz}$ and at an optical frequency of $200\ \mathrm{THz}$. Use $T=300\ \mathrm K$ and $k_{\mathrm B}T/h\simeq6.25\ \mathrm{THz}$.

Solution

Since $\hbar\omega=h\nu$,

$$N_{\mathrm{th}} = \frac{1}{e^{h\nu/k_{\mathrm B}T}-1}.$$

At $10\ \mathrm{GHz}$,

$$\frac{h\nu}{k_{\mathrm B}T} \simeq \frac{0.010}{6.25} =1.6\times10^{-3},$$

so $N_{\mathrm{th}}\simeq1/(1.6\times10^{-3})\simeq625$. At $200\ \mathrm{THz}$ the exponent is about $32$, giving $N_{\mathrm{th}}\simeq e^{-32}\sim10^{-14}$. Bright natural thermal noise is therefore intrinsic in the microwave regime but not usually in the optical regime.

Exercise 7: Unequal priors

A target is present with prior probability $\pi_1=10^{-3}$. A receiver has $P_{\mathrm{FA}}=10^{-2}$ and $P_{\mathrm M}=0.1$. Compute its equal-cost Bayes error. Which term dominates, and why can equal-prior optimization be misleading?

Solution

With $\pi_0=0.999$,

$$\begin{aligned} P_e &= (0.999)(10^{-2}) + (10^{-3})(0.1) \\ &=0.00999+0.00010 \\ &=0.01009. \end{aligned}$$

False alarms dominate because the target-absent hypothesis is overwhelmingly common. A receiver optimized for equal priors may choose a threshold with an unacceptable false-alarm rate even if its symmetric error is small. Here the receiver even loses to always declaring the target absent, whose Bayes error is $\pi_1=0.001$. Thus the low-prevalence example demands a different decision threshold, not merely a smaller equal-prior error. The ROC or a fixed-false-alarm comparison reveals the tradeoff.

Exercise 8: Failed trials change the error exponent

A cycle succeeds with probability $p=0.1$, independent of the target. Successful records have Chernoff coefficient $e^{-5}$, and a failed record contains no target information. Find the exponent per attempted cycle and compare it with multiplying the successful exponent by $p$.

Solution

The flagged outcome coefficient is $Q_{\rm cycle}=0.9+0.1e^{-5}$, so

$$\xi_{\rm cycle}=-\ln(0.9+0.1e^{-5})\simeq0.1046.$$

The naive product $p\times5=0.5$ is nearly five times too large. For $n$ attempts, the all-failure event alone has probability $0.9^n$ and leaves equal-prior error $1/2$, preventing an exponent above $-\ln0.9\simeq0.1054$. If instead one waits for a fixed number of successes, the error and random duration must be described using that different stopping rule.

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