INTRODUCTION TO QUANTUM RADAR Ricardo Gallego Torrom´e1 Department of Mathematics Faculty of Mathematics, Natural Sciences and Information Technologies University of Primorska, Koper, Slovenia Nadya Ben Bekhti-Winkel2 FHR, Fraunhofer-Institut f¨ur Hochfrequenzphysik und Radartechnik, Fraunhoferstr. 20, 53343 Wachtberg, Germany Peter Knott3 FHR, Fraunhofer-Institut f¨ur Hochfrequenzphysik und Radartechnik, Fraunhoferstr. 20, 53343 Wachtberg, Germany Chair of Radar Systems Engineering Institute of High Frequency Technology RWTH Aachen University Melatener Straße 25 52074 Aachen Germany Abstract. After a brief introduction to the notion of quantum entanglement and quantum correlations, several schemes for a quantum radar based upon the quantum illumination and others protocols are discussed. We review different concepts that have been introduced to overcome several of the inherent difficul- ties in the implementation of quantum generation and/or detection quantum sensing protocols for RADAR applications. Our review is an up-to date crit- ical presentation of the state of the art, with emphasis in the case by case assessment of the feasibility of the different concepts. We also aim that the review is accessible to non-experts in the field. Hence several appendixes and a technical glossary are included. Keywords: Quantum entanglement, quantum radar, quantum interferometric radar, quantum illumination arXiv:2006.14238v3 [quant-ph] 18 Jan 2021 Date: January 19, 2021. 1 Email: [email protected] 2Email: [email protected] 3 Email: [email protected] 2 QUANTUM RADAR Contents 1. Introduction 4 2. Quantum entanglement, non-classical correlations and types of quantum radars 5 2.1. Quantum entanglement 6 2.2. Entanglement in pure states 7 2.3. Entanglement in mixed states 8 2.4. Entanglement of states described by continuous variables 9 2.5. Discrimination of quantum operations and detection of targets 9 2.6. General approaches to quantum radar 10 3. QuantumInterferometricradarprotocols 12 3.1. Theoretical protocols for quantum interferometric radar and experimental demonstrations 12 3.2. Issues on the practical implementation of quantum interferometric radar 13 3.3. On the use of adaptive optics correction in quantum interferometric radar 14 3.4. Other possible approaches to quantum interferometric radar 15 4. Quantum illumination 15 4.1. Lloyd’s proposal on quantum illumination 16 4.2. The work of Lloyd and Shapiro on coherent state and quantum illumination 18 4.3. Gaussian quantum illumination 18 4.4. Receivers for Gaussian quantum illumination 20 4.5. Lopaeva et al. experiment 22 4.6. England et al experiment and direct photon detection methods 22 4.7. Experimental comparison of gaussian quantum illumination versus coherent illumination 24 4.8. Impracticability of gaussian optical quantum illumination for radar applications 24 4.9. Microwave quantum illumination 25 5. Issues in the implementation of quantum illumination for radar applications 26 5.1. Reception and process of the correlated signal/idler beams 26 5.2. Generation of the entangled beams at microwave frequencies 26 5.3. Lossesintroducedbythestorageoftheidlerbeam 27 5.4. The time-bandwidth problem 28 5.5. Rayleigh-fading targets 28 6. Hybrid quantum illumination protocols 28 6.1. Quantum radarprototypeof Changet. al. and experiments 28 6.2. Covariant matrix for classical correlated noise radar and quantum correlated noise radar 30 6.3. Quantum radar prototype of Barzahjeh et al. and experiments 31 6.4. JPA modulation and slow moving target tracking 32 6.5. Comparison of quantum noise radar and coherent light illumination 32 6.6. Discussion of the scope of Shapiro’s analysis 34 7. ThequantumradarprotocolofMacconeandRen 34 QUANTUM RADAR 3 7.1. Maccone-Renprotocolforquantumradar 35 7.2. Enhancement of sensitivity using quantum entanglement respect to non-entangled light 36 7.3. Practical issues implementing Maccone-Ren’s quantum radar protocol 36 8. Quantum illumination with multiple entangled photons 37 8.1. Lloyd’s quantum illumination using multiple entangled signals beams 37 8.2. Reduction of the required time band-width product 40 8.3. Determination of the range and transverse position using quantum illumination with two signal photon states 40 8.4. Potential issues with the implementation of the method 40 9. Other proposals for quantum radar 41 9.1. QuantumradarproposalofDurak,JamandDindar 41 9.2. Proof of principle experiment 42 9.3. Critical view of Durak et al. protocol 42 10. Conclusions and outlook 43 Appendix A. Glossary 46 Appendix B. Quantum states of relevance for quantum radar protocols 53 Appendix C. Enhancement of sensitivity in Lloyd’s quantum illumination: an illustrative example 57 Appendix D. Non-linear optical processes relevant for quantum radar protocols 59 D.1. Three photon interactions 61 D.2. Four photons interactions 62 D.3. Methods of generation of entangled states for quantum radar 62 Acknowledgements 64 References 65 4 QUANTUM RADAR 1. Introduction After preliminary results of M. Sacchi on entanglement enhancement in entan- glement breaking quantum channels [76, 77] and the introduction of the quantum Chernoff bound [2, 69], S. Lloyd devised the protocol of quantum illumination [59] for discrete variable systems and Tan et al. [90] cite the corresponding version for continuous variable systems. Quantum illumination is an example of quantum pro- tocol where the enhancement in sensitivity derived by the use of quantum entangled states survives the effects of environmental noise. Indeed, the advantage of entan- glement respect to non-entangled sources is, surprisingly and counter-intuitively, larger when the system is immersed in noisy environments. This theoretical results ignited several investigations and demonstrations of quan- tum illumination, at the theoretical and experimental level. One of the goals in such research was and still remains, the realization of a quantum radar, a radar that ex- plodes quantum entanglement, quantum illumination being a pre-eminent, to target range detection with sensitivity beyond classical radar schemes. As a result of this intensive research activity, it became clear that the enhancement of quantum il- lumination over classical illumination was limited by theoretical and technological considerations and that the goal of the realization of a quantum radar will need to wait the resolution of several issues at the practical and theoretical levels and also, that the enhancement cannot not be as strong as initially was thought. Nev- ertheless, even the more modest theoretical advantages that quantum illumination had over classical illumination for target detection remained a strong motivation to develop the concept and more generally, to study quantum radar protocols and its practical and technological implementations. The major goal was the realization of a quantum radar, a goal which has not been reached yet. But other more modest potential applications has raised during these years. The present review aims to introduce to the non-expert in quantum radar the main achievements, concepts, methods and results of the field, but also to explain the limitations and problems that appear in the theory and in its experimental and practical technological implementations. The pre-requisites for understanding the material presented here is to have a basic acquaintance with the fundamental math- ematical tools of quantum mechanics (notion of Hilbert space and linear operators, scalar products,annihilation and creation operators,...). We expect that the review serves as a bridge between two communities. First, the quantum radar community, as a research community in the interface between quantum optics, quantum metrol- ogy and quantum sensing. For them we tried to present an accurate, complete and realistic survey of the field. Second, the classical radar community with interest to initiate research in the area of quantum radar. For them we have tried to present a basic and clear exposition of the field. We have approached the review with a minimal of formalism and we have developed several notions of quantum mechan- ics, quantum information and quantum optics in the appendix section and in the Glossary. In this way the reader familiar with the more technical notions can be dispensed from following their reading in the main part of the manuscript. The structure of this review is the following. In section 2, a succinct introduc- tion to quantum entanglement and several aspects of entanglement of relevance for quantum radar applications is provided. Then a general description of several QUANTUM RADAR 5 approaches to quantum radar is presented. Section 3 describes quantum interfero- metric radar. Aspects of quantum illumination are discussed in sections 4 and 5. Section 6 discusses experimental implementations of quantum illumination where the receiver uses classical digital techniques for the process of the signal. Section 7 is an introduction to a novel approach to quantum radar developed by L. Mac- cone and C. Ren [66]. Section 8 describes an innovative approach to quantum radar based in a combination of an idea extracted from Maccone-Ren protocol and Lloyd’s quantum illumination. Section 9 describes other new approaches to quan- tum radar, especially a recent proposal by Durak et al. that uses only classical concepts but illumination with entangled light [24]. We observe that several of the developments in the appendix are not easy to be found directly in the literature. This applies to exposition