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arXiv:0903.3876 · Physical Review A

Quantum target detection using entangled photons

A. R. Usha Devi, A. K. Rajagopal

Published Jun 19, 2009• 38 citations
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Abstract

We investigate performances of pure continuous variable states in discriminating thermal and identity channels by comparing their $M$-copy error-probability bounds. This offers us a simplified mathematical analysis for quantum target detection with slightly modified features: the object---if it is present---perfectly reflects the signal beam irradiating it, while thermal noise photons are returned to the receiver in its absence. This model facilitates us to obtain analytic results on error-probability bounds, i.e., the quantum Chernoff bound and the lower bound constructed from the Bhattacharya bound on $M$-copy discrimination error probabilities of some important quantum states, like photon number states, $\text{N}$-photon maximally entangled (N00N) states, coherent states and the entangled photons obtained from spontaneous parametric down conversion (SPDC). Comparing the $M$-copy error-bounds, we identify that path-entangled states indeed offer enhanced sensitivity than the photon number state system, when average signal photon number is small compared to the thermal noise level. However, in the high signal-to-noise scenario, N00N states fail to be advantageous than the photon number states. Entangled SPDC photon pairs too outperform conventional coherent state system in the low signal-to-noise case. On the other hand, conventional coherent state system surpasses the performance sensitivity offered by entangled photon pair, when the signal intensity is much above that of thermal noise. We find an analogous performance regime in the lossy target detection (where the target is modeled as a weakly reflecting object) in a high signal-to-noise scenario.

DOI
10.1103/physreva.79.062320

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We investigate performances of pure continuous variable states in discriminating thermal and identity channels by comparing their MMM-copy error-probability bounds. This offers us a simplified mathematical analysis for quantum target detection with slightly modified features: the object---if it is present---perfectly reflects the signal beam irradiating it, while thermal noise photons are returned to the receiver in its absence. This model facilitates us to obtain analytic results on error-probability bounds, i.e., the quantum Chernoff bound and the lower bound constructed from the Bhattacharya bound on MMM-copy discrimination error probabilities of some important quantum states, like photon number states, N\text{N}N-photon maximally entangled (N00N) states, coherent states and the entangled photons obtained from spontaneous parametric down conversion (SPDC). Comparing the MMM-copy error-bounds, we identify that path-entangled states indeed offer enhanced sensitivity than the photon number state system, when average signal photon number is small compared to the thermal noise level. However, in the high signal-to-noise scenario, N00N states fail to be advantageous than the photon number states. Entangled SPDC photon pairs too outperform conventional coherent state system in the low signal-to-noise case. On the other hand, conventional coherent state system surpasses the performance sensitivity offered by entangled photon pair, when the signal intensity is much above that of thermal noise. We find an analogous performance regime in the lossy target detection (where the target is modeled as a weakly reflecting object) in a high signal-to-noise scenario.
Quantum illumination employs entangled states to detect a weakly reflecting target in a thermal bath. The performance of a given entangled state is evaluated from the minimum error probability in the asymptotic limit, which is compared against the optimal coherent state scheme. Using a relationship between the minimum error probability and quantum Fisher information, we compare the optimal performance of definite photon-number entangled states against that of coherent states under local repetitive strategies. When optimized under the constraint of a fixed total energy, we find that a coherent state outperforms the definite photon-number entangled states with the same signal energy.
This thesis presents three studies in quantum-enhanced sensing and target detection. The first study explores covert target detection using optical or microwave probes, establishing quantum-mechanical limits on the error probabilities of entanglement-assisted detection methods while maintaining the sender's covertness. It identifies the minimal energy required to preserve covertness and reduce error probabilities, compares two-mode squeezed vacuum probes and coherent states against these limits, and extends the analysis to discriminating thermal loss channels and non-adversarial quantum illumination. The second study focuses on phase-insensitive optical amplifiers, determining the quantum limit on the precision of gain estimation using multimode probes possibly entangled with ancillary systems. It finds that the average photon number and the number of input modes are interchangeable resources for achieving optimal gain sensing precision, contrasting classical probes with quantum probes and highlighting the advantages of the latter, even with single-photon inputs and inefficient photodetection. It also provides a closed-form expression for the energy-constrained Bures distance between two amplifier channels. The third study compares three probe states -- coherent state, two-mode squeezed vacuum (TMSV), and single-photon entangled state (SPES) -- in quantum-enhanced target detection, assessing their performance under signal energy constraints relevant to covert radar sensing. SPES is uniquely positioned as a practical probe due to its non-classical properties after thermal loss and ease of generation. Numerical analysis shows that at low signal energies, the error exponent of TMSV aligns with SPES, indicating comparable detection capabilities and that SPES surpasses the best classical state-the coherent state-in accuracy for certain signal strengths.
Quantum illumination employs entangled states to detect a weakly reflective target in a thermal bath. The performance of a given entangled state is evaluated from the minimum error probability in the asymptotic limit, which is compared against the optimal coherent state scheme. We derive an upper bound as well as a lower bound on the asymptotic minimum error probability, as functions of the quantum Fisher information. The upper bound can be achieved using a repetitive local strategy. This allows us to compare the optimal performance of definite-photon-number entangled states against that of the coherent states under local strategies. When optimized under the constraint of a fixed total energy, we find that a coherent state outperforms the definite-photon-number entangled state with the same signal energy.
Quantum illumination is one of the most basic ways to realize the quantum radar for detection of low-reflectivity targets embedded in a thermal-noise bath. It preserves a strong advantage over the classical counterparts even though its fragile quantum entanglement is completely destroyed by a large amount of noise and losses. In this paper, we describe and analyze a discrete-variable quantum illumination system with ancillary degrees of freedom of entangled photon pairs. Comparing the error probability bound, i.e., the quantum Chernoff bound, we identify that this system indeed offers higher accuracy than the single-photon illumination and quantum illumination with single degree of freedom of entangled photon pairs. Furthermore, the utilization of ancillary degrees of freedom exponentially enhances signal-to-noise ratio over quantum illumination without ancillary degrees of freedom. It means that this system reduces the necessary amount of signal photons to accurately determine whether the object is present or absent in the target region, which is absolutely crucial to the stealthiness of the radar system.
We consider the problem of distinguishing, with minimum probability of error, two optical beam-splitter channels with unequal complex-valued reflectivities using general quantum probe states entangled over MMM signal and ${M}^{\ensuremath{'}}$ idler mode pairs of which the signal modes are bounced off the beam splitter while the idler modes are retained losslessly. We obtain a lower bound on the output state fidelity valid for any pure input state. We define number-diagonal signal (NDS) states to be input states whose density operator in the signal modes is diagonal in the multimode number basis. For such input states, we derive series formulas for the optimal error probability, the output state fidelity, and the Chernoff-type upper bounds on the error probability. For the special cases of quantum reading of a classical digital memory and target detection (for which the reflectivities are real valued), we show that for a given input signal photon probability distribution, the fidelity is minimized by the NDS states with that distribution and that for a given average total signal energy Ns{N}_{s}Ns​, the fidelity is minimized by any multimode Fock state with Ns{N}_{s}Ns​ total signal photons. For reading of an ideal memory, it is shown that Fock state inputs minimize the Chernoff bound. For target detection under high-loss conditions, a no-go result showing the lack of appreciable quantum advantage over coherent state transmitters is derived. A comparison of the error probability performance for quantum reading of number state and two-mode squeezed vacuum state (or EPR state) transmitters relative to coherent state transmitters is presented for various values of the reflectances. While the nonclassical states in general perform better than the coherent state, the quantitative performance gains differ depending on the values of the reflectances. The experimental outlook for realizing nonclassical gains from number state transmitters with current technology at moderate to high values of the reflectances is argued to be good.
We demonstrate an optical receiver that achieves the quantum Chernoff bound for discriminating coherent states from thermal states in the multicopy scenario. In contrast, we find that repeated use of the receiver approaching the Helstrom bound for single-copy measurement is suboptimal in this multicopy case. Furthermore, for a large class of multicopy discrimination tasks between a pure and a mixed state, we prove that any Helstrom-bound achieving single-copy receiver is suboptimal by a factor of at least 2 in error-probability exponent compared to the multicopy quantum Chernoff bound. This behavior has a classical analog in the performance gap between soft-decision and hard-decision receivers for detecting a multicopy signal embedded in white Gaussian noise.