* [42] L. Lin, W. Burleson, and C. Paar, “Moles: Malicious off-chip leakage enabled by side-channels,” in _2009 IEEE/ACM International Conference on Computer-Aided Design-Digest of Technical Papers_. IEEE, 2009, pp. 117–122.
  * [43] F. Tao, H. Zhang, A. Liu, and A. Y. Nee, “Digital twin in industry: State-of-the-art,” _IEEE Transactions on Industrial Informatics_ , vol. 15, no. 4, pp. 2405–2415, 2018.
  * [44] R. Bitton, T. Gluck, O. Stan, M. Inokuchi, Y. Ohta, Y. Yamada, T. Yagyu, Y. Elovici, and A. Shabtai, “Deriving a cost-effective digital twin of an ics to facilitate security evaluation,” in _European Symposium on Research in Computer Security_. Springer, 2018, pp. 533–554.
  * [45] X. Lou, Y. Guo, Y. Gao, K. Waedt, and M. Parekh, “An idea of using digital twin to perform the functional safety and cybersecurity analysis,” in _INFORMATIK 2019: 50 Jahre Gesellschaft für Informatik–Informatik für Gesellschaft (Workshop-Beiträge)_. Gesellschaft für Informatik eV, 2019.
  * [46] E. C. Balta, D. M. Tilbury, and K. Barton, “A digital twin framework for performance monitoring and anomaly detection in fused deposition modeling,” in _2019 IEEE 15th International Conference on Automation Science and Engineering (CASE)_. IEEE, 2019, pp. 823–829.
  * [47] M. Eckhart and A. Ekelhart, “A specification-based state replication approach for digital twins,” ser. CPS-SPC ’18. New York, NY, USA: Association for Computing Machinery, 2018, p. 36–47. [Online]. Available: https://doi.org/10.1145/3264888.3264892
  * [48] A. Saad, S. Faddel, T. Youssef, and O. A. Mohammed, “On the implementation of iot-based digital twin for networked microgrids resiliency against cyber attacks,” _IEEE transactions on smart grid_ , vol. 11, no. 6, pp. 5138–5150, 2020.
  * [49] C. Li, S. Mahadevan, Y. Ling, S. Choze, and L. Wang, “Dynamic bayesian network for aircraft wing health monitoring digital twin,” _Aiaa Journal_ , vol. 55, no. 3, pp. 930–941, 2017.
  * [50] J. Sleuters, Y. Li, J. Verriet, M. Velikova, and R. Doornbos, “A digital twin method for automated behavior analysis of large-scale distributed iot systems,” in _2019 14th Annual Conference System of Systems Engineering (SoSE)_. IEEE, 2019, pp. 7–12.
  * [51] H. Wang, S. Chen, M. S. U. I. Sami, F. Rahman, and M. Tehranipoor, “Digital twin with a perspective from manufacturing industry,” _Emerging Topics in Hardware Security_ , pp. 27–59, 2021.
  * [52] P. Jain, J. Poon, J. P. Singh, C. Spanos, S. R. Sanders, and S. K. Panda, “A digital twin approach for fault diagnosis in distributed photovoltaic systems,” _IEEE Transactions on Power Electronics_ , vol. 35, no. 1, pp. 940–956, 2019.
  * [53] Y. Xu, Y. Sun, X. Liu, and Y. Zheng, “A digital-twin-assisted fault diagnosis using deep transfer learning,” _IEEE Access_ , vol. 7, pp. 19 990–19 999, 2019.
  * [54] S. Kaewunruen and Q. Lian, “Digital twin aided sustainability-based lifecycle management for railway turnout systems,” _Journal of Cleaner Production_ , vol. 228, pp. 1537–1551, 2019.
  * [55] “Reliability,” in _Heterogeneous Integration Roadmap_ , P. Wesling, Ed. IEEE Electronics Packaging Society, 2021, ch. 24, pp. 1–30. [Online]. Available: https://eps.ieee.org/images/files/HIR_2021/ch24_rel.pdf
  * [56] R. G. Alves, G. Souza, R. F. Maia, A. L. H. Tran, C. Kamienski, J.-P. Soininen, P. T. Aquino, and F. Lima, “A digital twin for smart farming,” in _2019 IEEE Global Humanitarian Technology Conference (GHTC)_. IEEE, 2019, pp. 1–4.
  * [57] Y. Tchana, G. Ducellier, and S. Remy, “Designing a unique digital twin for linear infrastructures lifecycle management,” _Procedia CIRP_ , vol. 84, pp. 545–549, 2019.
  * [58] A. Pokhrel, V. Katta, and R. Colomo-Palacios, “Digital twin for cybersecurity incident prediction: A multivocal literature review,” in _Proceedings of the IEEE/ACM 42nd International Conference on Software Engineering Workshops_ , 2020, pp. 671–678.
  * [59] M. Rostami, F. Koushanfar, and R. Karri, “A primer on hardware security: Models, methods, and metrics,” _Proceedings of the IEEE_ , vol. 102, no. 8, pp. 1283–1295, 2014.
  * [60] N. Asadizanjani, M. T. Rahman, M. Tehranipoor, and M. H. Tehranipoor, _Physical Assurance: For Electronic Devices and Systems_. Springer, 2021.
  * [61] M. T. Rahman, Q. Shi, S. Tajik, H. Shen, D. L. Woodard, M. Tehranipoor, and N. Asadizanjani, “Physical inspection & attacks: New frontier in hardware security,” in _2018 IEEE 3rd International Verification and Security Workshop (IVSW)_. IEEE, 2018, pp. 93–102.
  * [62] S. E. Quadir, J. Chen, D. Forte, N. Asadizanjani, S. Shahbazmohamadi, L. Wang, J. Chandy, and M. Tehranipoor, “A survey on chip to system reverse engineering,” _ACM journal on emerging technologies in computing systems (JETC)_ , vol. 13, no. 1, pp. 1–34, 2016.
  * [63] U. J. Botero, R. Wilson, H. Lu, M. T. Rahman, M. A. Mallaiyan, F. Ganji, N. Asadizanjani, M. M. Tehranipoor, D. L. Woodard, and D. Forte, “Hardware trust and assurance through reverse engineering: A tutorial and outlook from image analysis and machine learning perspectives,” _ACM Journal on Emerging Technologies in Computing Systems (JETC)_ , vol. 17, no. 4, pp. 1–53, 2021.
  * [64] M. Sharma, “Amd hardware security tricks can be bypassed with a shock of electricity,” August 2021. [Online]. Available: https://www.techradar.com/news/amd-hardware-security-tricks-can-be-bypassed-with-a-shock-of-electricity
  * [65] D. Goodin, “Trusted platform module security defeated in 30 minutes, no soldering required,” August 2021. [Online]. Available: https://arstechnica.com/gadgets/2021/08/how-to-go-from-stolen-pc-to-network-intrusion-in-30-minutes/
  * [66] A. P. Fournaris, L. Pocero Fraile, and O. Koufopavlou, “Exploiting hardware vulnerabilities to attack embedded system devices: A survey of potent microarchitectural attacks,” _Electronics_ , vol. 6, no. 3, p. 52, 2017.
  * [67] S. T. King, J. Tucek, A. Cozzie, C. Grier, W. Jiang, and Y. Zhou, “Designing and implementing malicious hardware.” _Leet_ , vol. 8, pp. 1–8, 2008.
  * [68] D. Lee, D. Jung, I. T. Fang, C.-C. Tsai, and R. A. Popa, “An off-chip attack on hardware enclaves via the memory bus,” in _29th $\\{$USENIX$\\}$ Security Symposium ($\\{$USENIX$\\}$ Security 20)_, 2020.
  * [69] J. Grand, “Practical secure hardware design for embedded systems,” in _Proceedings of the 2004 Embedded Systems Conference, San Francisco, California_ , 2004.
  * [70] M. S. Rahman, A. Nahiyan, F. Rahman, S. Fazzari, K. Plaks, F. Farahmandi, D. Forte, and M. Tehranipoor, “Security assessment of dynamically obfuscated scan chain against oracle-guided attacks,” _ACM Transactions on Design Automation of Electronic Systems (TODAES)_ , vol. 26, no. 4, pp. 1–27, 2021.
  * [71] B. Ahmed, M. K. Bepary, N. Pundir, M. Borza, O. Raikhman, A. Garg, D. Donchin, A. Cron, M. A. Abdel-moneum, F. Farahmandi _et al._ , “Quantifiable assurance: From ips to platforms,” _Cryptology ePrint Archive_ , 2021.
  * [72] H. S. Choo, C. Y. Ooi, M. Inoue, N. Ismail, M. Moghbel, and C. H. Kok, “Register-transfer-level features for machine-learning-based hardware trojan detection,” _IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences_ , vol. 103, no. 2, pp. 502–509, 2020.
  * [73] M. He, J. Park, A. Nahiyan, A. Vassilev, Y. Jin, and M. Tehranipoor, “Rtl-psc: Automated power side-channel leakage assessment at register-transfer level,” in _2019 IEEE 37th VLSI Test Symposium (VTS)_. IEEE, 2019, pp. 1–6.
  * [74] F. Fallah, S. Devadas, and K. Keutzer, “Occom-efficient computation of observability-based code coverage metrics for functional verification,” _IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems_ , vol. 20, no. 8, pp. 1003–1015, 2001.
  * [75] Y. Serrestou, V. Beroulle, and C. Robach, “Functional verification of rtl designs driven by mutation testing metrics,” in _10th Euromicro Conference on Digital System Design Architectures, Methods and Tools (DSD 2007)_. IEEE, 2007, pp. 222–227.
  * [76] L. H. Goldstein and E. L. Thigpen, “Scoap: Sandia controllability/observability analysis program,” in _Proceedings of the 17th Design Automation Conference_ , 1980, pp. 190–196.
  * [77] A. Waksman, M. Suozzo, and S. Sethumadhavan, “Fanci: identification of stealthy malicious logic using boolean functional analysis,” in _Proceedings of the 2013 ACM SIGSAC conference on Computer & communications security_, 2013, pp. 697–708.
  * [78] R. S. Chakraborty, F. Wolff, S. Paul, C. Papachristou, and S. Bhunia, “Mero: A statistical approach for hardware trojan detection,” in _International Workshop on Cryptographic Hardware and Embedded Systems_. Springer, 2009, pp. 396–410.
  * [79] J. Cruz, F. Farahmandi, A. Ahmed, and P. Mishra, “Hardware trojan detection using atpg and model checking,” in _2018 31st international conference on VLSI design and 2018 17th international conference on embedded systems (VLSID)_. IEEE, 2018, pp. 91–96.
  * [80] A. Nahiyan and M. Tehranipoor, “Code coverage analysis for ip trust verification,” in _Hardware IP security and trust_. Springer, 2017, pp. 53–72.
  * [81] Q. Xu and S. Chen, “Fast thermal analysis for fixed-outline 3d floorplanning,” _Integration_ , vol. 59, pp. 157–167, 2017.
  * [82] J. Cong, G. Nataneli, M. Romesis, and J. R. Shinnerl, “An area-optimality study of floorplanning,” in _Proceedings of the 2004 international symposium on Physical design_ , 2004, pp. 78–83.
  * [83] H. Ma, J. He, Y. Liu, L. Liu, Y. Zhao, and Y. Jin, “Security-driven placement and routing tools for electromagnetic side-channel protection,” _IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems_ , vol. 40, no. 6, pp. 1077–1089, 2020.
  * [84] K. Xiao and M. Tehranipoor, “Bisa: Built-in self-authentication for preventing hardware trojan insertion,” in _2013 IEEE international symposium on hardware-oriented security and trust (HOST)_. IEEE, 2013, pp. 45–50.
  * [85] H. Salmani, M. Tehranipoor, and J. Plusquellic, “New design strategy for improving hardware trojan detection and reducing trojan activation time,” in _2009 IEEE International Workshop on Hardware-Oriented Security and Trust_. IEEE, 2009, pp. 66–73.
  * [86] “Ieee standard for verilog hardware description language,” _IEEE Std 1364-2005 (Revision of IEEE Std 1364-2001)_ , pp. 325–348, 2006.
  * [87] D. Forte, S. Bhunia, and M. M. Tehranipoor, _Hardware protection through obfuscation_. Springer, 2017.
  * [88] C. Mack, _Fundamental principles of optical lithography: the science of microfabrication_. John Wiley & Sons, 2008\.
  * [89] G. May and S. Sze, _Fundamentals of Semiconductor Fabrication_ , 1st ed. Wiley, 2003.
  * [90] S. Campbell, _Fabrication Engineering at the Micro and Nanoscale_ , 3rd ed. Oxford University Press, 2008.
  * [91] N. B. Cobb and E. Y. Sahouria, “Hierarchical gdsii-based fracturing and job deck system,” in _21st Annual BACUS Symposium on Photomask Technology_ , vol. 4562. International Society for Optics and Photonics, 2002, pp. 734–742.
  * [92] F. E. Abboud, M. Asturias, and M. Chandramouli, “Mask data processing in the era of multibeam writers,” _BACUS News_ , Jan 2015. [Online]. Available: https://spie.org/Documents/Membership/BacusNewsletters/BACUS-Newsletter-January-2015.pdf.
  * [93] S. F. Schulze, P. LaCour, and P. D. Buck, “Gds-based mask data preparation flow: data volume containment by hierarchical data processing,” in _22nd Annual BACUS Symposium on Photomask Technology_ , vol. 4889. International Society for Optics and Photonics, 2002, pp. 104–114.
  * [94] M. J. Hampden-Smith and T. T. Kodas, “Chemical vapor deposition of metals: Part 1. an overview of cvd processes,” _Chemical Vapor Deposition_ , vol. 1, no. 1, pp. 8–23, 1995.
  * [95] W. L. Smith, A. Rosencwaig, and D. L. Willenborg, “Ion implant monitoring with thermal wave technology,” _Applied physics letters_ , vol. 47, no. 6, pp. 584–586, 1985.
  * [96] G. Neubauer, A. Erickson, C. C. Williams, J. J. Kopanski, M. Rodgers, and D. Adderton, “Two-dimensional scanning capacitance microscopy measurements of cross-sectioned very large scale integration test structures,” _Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures Processing, Measurement, and Phenomena_, vol. 14, no. 1, pp. 426–432, 1996.
  * [97] W. Vandervorst, P. Eyben, S. Callewaert, T. Hantschel, N. Duhayon, M. Xu, T. Trenkler, and T. Clarysse, “Towards routine, quantitative two-dimensional carrier profiling with scanning spreading resistance microscopy,” in _AIP Conference Proceedings_ , vol. 550, no. 1. American Institute of Physics, 2001, pp. 613–619.
  * [98] M. Bushnell and V. Agrawal, _Essentials of electronic testing for digital, memory and mixed-signal VLSI circuits_. Springer Science & Business Media, 2004, vol. 17.
  * [99] A. Chen and R. H.-Y. Lo, _Semiconductor Packaging: Materials Interaction and Reliability_. CRC Press, 2017.
  * [100] M. Nourani, M. Tehranipoor, and N. Ahmed, “Low-transition test pattern generation for bist-based applications,” _IEEE Transactions on Computers_ , vol. 57, no. 3, pp. 303–315, 2008.
  * [101] M. H. Tehranipour, N. Ahmed, and M. Nourani, “Testing soc interconnects for signal integrity using boundary scan,” in _Proceedings. 21st VLSI Test Symposium, 2003._ IEEE, 2003, pp. 158–163.
  * [102] M. Tehranipoor and F. Koushanfar, “A survey of hardware trojan taxonomy and detection,” _IEEE design & test of computers_, vol. 27, no. 1, pp. 10–25, 2010.
  * [103] V. C. Patil, A. Vijayakumar, and S. Kundu, “Manufacturer turned attacker: Dangers of stealthy trojans via threshold voltage manipulation,” in _2017 IEEE North Atlantic Test Workshop (NATW)_. IEEE, 2017, pp. 1–6.
  * [104] G. K. Contreras, M. T. Rahman, and M. Tehranipoor, “Secure split-test for preventing ic piracy by untrusted foundry and assembly,” in _2013 IEEE International symposium on defect and fault tolerance in VLSI and nanotechnology systems (DFTS)_. IEEE, 2013, pp. 196–203.
  * [105] K. Huang, J. M. Carulli, and Y. Makris, “Parametric counterfeit ic detection via support vector machines,” in _2012 IEEE International Symposium on Defect and Fault Tolerance in VLSI and Nanotechnology Systems (DFT)_. IEEE, 2012, pp. 7–12.
  * [106] X. Zhang, K. Xiao, and M. Tehranipoor, “Path-delay fingerprinting for identification of recovered ics,” in _2012 IEEE International symposium on defect and fault tolerance in VLSI and nanotechnology systems (DFT)_. IEEE, 2012, pp. 13–18.
  * [107] A. Nahiyan, M. Sadi, R. Vittal, G. Contreras, D. Forte, and M. Tehranipoor, “Hardware trojan detection through information flow security verification,” in _2017 IEEE International Test Conference (ITC)_. IEEE, 2017, pp. 1–10.
  * [108] W. Hu, D. Mu, J. Oberg, B. Mao, M. Tiwari, T. Sherwood, and R. Kastner, “Gate-level information flow tracking for security lattices,” _ACM Transactions on Design Automation of Electronic Systems (TODAES)_ , vol. 20, no. 1, pp. 1–25, 2014.
  * [109] Y. Liu, Y. Jin, A. Nosratinia, and Y. Makris, “Silicon demonstration of hardware trojan design and detection in wireless cryptographic ics,” _IEEE Transactions on Very Large Scale Integration (VLSI) Systems_ , vol. 25, no. 4, pp. 1506–1519, 2016.
  * [110] C. Kison, O. M. Awad, M. Fyrbiak, and C. Paar, “Security implications of intentional capacitive crosstalk,” _IEEE Transactions on Information Forensics and Security_ , vol. 14, no. 12, pp. 3246–3258, 2019.
  * [111] S. Rizvi, _Handbook of photomask manufacturing technology_. CRC Press, 2018.
  * [112] K. Takahashi, M. Osawa, M. Sato, H. Arimoto, K. Ogino, H. Hoshino, and Y. Machida, “Proximity effect correction using pattern shape modification and area density map,” _Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures Processing, Measurement, and Phenomena_, vol. 18, no. 6, pp. 3150–3157, 2000.
  * [113] K. Hasegawa, M. Yanagisawa, and N. Togawa, “Trojan-feature extraction at gate-level netlists and its application to hardware-trojan detection using random forest classifier,” in _2017 IEEE International Symposium on Circuits and Systems (ISCAS)_. IEEE, 2017, pp. 1–4.
  * [114] M. Sharma, “Standard test data format specification,” August 2021. [Online]. Available: http://www.kanwoda.com/wp-content/uploads/2015/05/std-spec.pdf
  * [115] U. Guin, D. Forte, and M. Tehranipoor, “Anti-counterfeit techniques: From design to resign,” in _2013 14th International workshop on microprocessor test and verification_. IEEE, 2013, pp. 89–94.
  * [116] D. Koller, N. Friedman, S. Džeroski, C. Sutton, A. McCallum, A. Pfeffer, P. Abbeel, M.-F. Wong, C. Meek, J. Neville _et al._ , _Introduction to statistical relational learning_. MIT press, 2007.
  * [117] H. Guo and W. Hsu, “A survey of algorithms for real-time bayesian network inference,” in _Join Workshop on Real Time Decision Support and Diagnosis Systems_ , 2002.
  * [118] S. H. Chen and C. A. Pollino, “Good practice in bayesian network modelling,” _Environmental Modelling & Software_, vol. 37, pp. 134–145, 2012.
  * [119] D. Heckerman, D. Geiger, and D. M. Chickering, “Learning bayesian networks: The combination of knowledge and statistical data,” _Machine learning_ , vol. 20, no. 3, pp. 197–243, 1995.
  * [120] L. R. Rabiner, “A tutorial on hidden markov models and selected applications in speech recognition,” _Proceedings of the IEEE_ , vol. 77, no. 2, pp. 257–286, 1989.
  * [121] L. Rabiner and B. Juang, “An introduction to hidden markov models,” _ieee assp magazine_ , vol. 3, no. 1, pp. 4–16, 1986.
  * [122] G. D. Forney, “The viterbi algorithm,” _Proceedings of the IEEE_ , vol. 61, no. 3, pp. 268–278, 1973.
  * [123] M. Richardson and P. Domingos, “Markov logic networks,” _Machine learning_ , vol. 62, no. 1-2, pp. 107–136, 2006.
  * [124] D. Lowd and P. Domingos, “Efficient weight learning for markov logic networks,” in _European conference on principles of data mining and knowledge discovery_. Springer, 2007, pp. 200–211.
  * [125] S. Skorobogatov and C. Woods, “Breakthrough silicon scanning discovers backdoor in military chip,” in _International Workshop on Cryptographic Hardware and Embedded Systems_. Springer, 2012, pp. 23–40.
  * [126] N. Laptev, S. Amizadeh, and I. Flint, “Generic and scalable framework for automated time-series anomaly detection,” in _Proceedings of the 21th ACM SIGKDD international conference on knowledge discovery and data mining_ , 2015, pp. 1939–1947.
  * [127] S. Shahbazmohamadi, D. Forte, and M. Tehranipoor, “Advanced physical inspection methods for counterfeit ic detection,” in _ISTFA 2014: Conference Proceedings from the 40th International Symposium for Testing and Failure Analysis_. ASM International, 2014, p. 55.
  * [128] M. Baygin, M. Karakose, A. Sarimaden, and A. Erhan, “Machine vision based defect detection approach using image processing,” in _2017 international artificial intelligence and data processing symposium (IDAP)_. Ieee, 2017, pp. 1–5.
  * [129] P. Ghosh and R. S. Chakraborty, “Recycled and remarked counterfeit integrated circuit detection by image-processing-based package texture and indent analysis,” _IEEE Transactions on Industrial Informatics_ , vol. 15, no. 4, pp. 1966–1974, 2018.
  * [130] B. Shakya, T. He, H. Salmani, D. Forte, S. Bhunia, and M. Tehranipoor, “Benchmarking of hardware trojans and maliciously affected circuits,” _Journal of Hardware and Systems Security_ , vol. 1, no. 1, pp. 85–102, 2017\.
  * [131] H. Salmani, M. Tehranipoor, and R. Karri, “On design vulnerability analysis and trust benchmarks development,” in _2013 IEEE 31st international conference on computer design (ICCD)_. IEEE, 2013, pp. 471–474.
  * [132] X. Meng, S. Kundu, A. K. Kanuparthi, and K. Basu, “Rtl-contest: Concolic testing on rtl for detecting security vulnerabilities,” _IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems_ , 2021\.
  * [133] A. Ardeshiricham, W. Hu, J. Marxen, and R. Kastner, “Register transfer level information flow tracking for provably secure hardware design,” in _Design, Automation & Test in Europe Conference & Exhibition (DATE), 2017_. IEEE, 2017, pp. 1691–1696.
  * [134] N. Vashistha, M. T. Rahman, H. Shen, D. L. Woodard, N. Asadizanjani, and M. Tehranipoor, “Detecting hardware trojans inserted by untrusted foundry using physical inspection and advanced image processing,” _Journal of Hardware and Systems Security_ , vol. 2, no. 4, pp. 333–344, 2018.
  * [135] B. G. Marcot, J. D. Steventon, G. D. Sutherland, and R. K. McCann, “Guidelines for developing and updating bayesian belief networks applied to ecological modeling and conservation,” _Canadian Journal of Forest Research_ , vol. 36, no. 12, pp. 3063–3074, 2006.
  * [136] K. Beedkar, L. D. Corro, and R. Gemulla, “Fully parallel inference in markov logic networks,” _Datenbanksysteme für Business, Technologie und Web (BTW) 2026_ , 2013.
  * [137] J. Shavlik and S. Natarajan, “Speeding up inference in markov logic networks by preprocessing to reduce the size of the resulting grounded network,” in _Twenty-First International Joint Conference on Artificial Intelligence_ , 2009.
  * [138] R. Daly and Q. Shen, “Learning bayesian network equivalence classes with ant colony optimization,” _Journal of Artificial Intelligence Research_ , vol. 35, pp. 391–447, 2009.
  * [139] A. P. D. Nath, S. Ray, A. Basak, and S. Bhunia, “System-on-chip security architecture and cad framework for hardware patch,” in _2018 23rd Asia and South Pacific Design Automation Conference (ASP-DAC)_. IEEE, 2018, pp. 733–738.
  * [140] T. Beyrouthy, A. Razafindraibe, L. Fesquet, M. Renaudin, S. Chaudhuri, S. Guilley, P. Hoogvorst, and J.-L. Danger, “A novel asynchronous e-fpga architecture for security applications,” in _2007 International Conference on Field-Programmable Technology_. IEEE, 2007, pp. 369–372.
  * [141] S. K. Saha and C. Bobda, “Fpga accelerated embedded system security through hardware isolation,” in _2020 Asian Hardware Oriented Security and Trust Symposium (AsianHOST)_. IEEE, 2020, pp. 1–6.
  * [142] L. Delshadtehrani, S. Canakci, B. Zhou, S. Eldridge, A. Joshi, and M. Egele, “Phmon: a programmable hardware monitor and its security use cases,” in _29th $\\{$USENIX$\\}$ Security Symposium ($\\{$USENIX$\\}$ Security 20)_, 2020, pp. 807–824.


# Extension of Laguerre polynomials with negative arguments II

T. N. Shorey and Sneh Bala Sinha  T. N. Shorey Sneh Bala Sinha
NIAS, Bangalore, IISc Bangalore,
560012, India. Department of Mathematics,
560012, India<EMAIL_ADDRESS><EMAIL_ADDRESS>;
<EMAIL_ADDRESS>
###### Abstract.

For $n\geq 3$ and $s\leq 92$, it is proved in [4] that, except for finitely
many pairs $(n,s),G_{1}(x)=G_{1}(x,n,s)$ is either irreducible or linear
factor times an irreducible polynomial. If $s\leq 30$, we determine here
explicitely the set of pairs $(n,s)$ in the above assertion. This implies a
new proof of the result of Nair and Shorey [3] that $G_{1}(x)$ is irreducible
for $s\leq 22$.

###### Key words and phrases:

###### Key words and phrases:

Generalised Laguerre polynomials, Irreducibility, Primes, Valuations.

###### 1991 Mathematics Subject Classification:

11A41, 11B25, 11N05, 11N13, 11C08, 11Z05

Acknowledgment: The first author is supported by INSA Senior Scientist award.
The second author is supported by NBHM postdoctoral fellowship with grant
number 0204/19/2017/R& D-II/10383 at IISc Bangalore.

##  1\. Introduction

This is a continuation of [4]. Therefore we shall follow the notations of [4]
but we shall recall here the key notations and key results from [4]. The
generalised Laguerre polynomial of degree $n$ with negative argument is

$L_{n}^{(\alpha)}(x)=\sum\limits_{j=0}^{n}\frac{(\alpha+n)\dots(\alpha+j+1)}{(n-j)!}\frac{(-x)^{j}}{j!}$

where $\alpha$ is negative. Then for $\alpha=-n-s-1$ where $s$ is a non-
negative integer, we have

$\displaystyle g(x):=g(x,n,s)=(-1)^{n}L_{n}^{(-n-s-1)}(x)$
$\displaystyle=\sum\limits_{j=0}^{n}a_{j}\frac{x^{j}}{j!}$

where $a_{j}={n+s-j\choose n-j}$ for $0\leq j\leq n$.

Thus $a_{n}=1$ and $a_{0}={n+s\choose n}=\frac{(n+1)\dots(n+s)}{s!}$ and

$G(x):=G(x,n,s)=\sum\limits_{j=0}^{n}\pi_{j}\frac{x^{j}}{j!}\qquad\text{
where}\quad\pi_{j}=b_{j}a_{j}$

such that $b_{j}\in\mathbb{Z}$ for $0\leq j\leq n$ with $|b_{0}|=1$,
$|b_{n}|=1$. For $k\geq 1$ we say we have $(n,k,s)$ if $G(x)=G(x,n,s)$ has a
factor of degree $k$ and we do not have $(n,k,s)$ if $G(x)$ has no factor of
degree $k$. Next we write

$g_{1}(x)=n!g(x),\quad G_{1}(x)=n!G(x).$

Schur proved that $G_{1}(x)$ with $s=0$ is irreducible. Therefore we always
assume that $s>0$.

##  2\. Lemmas

######  Lemma 1.

Let $k$ and $l$ be integers with $k>l\geq 0$. Suppose that
$h(x)=\sum\limits_{j=0}^{n}b_{j}x^{j}$ and $p$ prime such that $p\nmid b_{n}$
and $p\mid b_{j}$ for $0\leq j<n-l$ and the right most edge of the Newton
polygon for $h(x)$ with respect to $p$ has slope less than $\frac{1}{k}$. Then
for any $a_{0},a_{1},\dots,a_{n}\in\mathbb{Z}$ with $|a_{0}|=|a_{n}|=1$, the
polynomial $f(x)=\sum\limits_{j=0}^{n}a_{j}b_{j}x^{j}\in\mathbb{Z}[x]$ can not
have a factor with degree in the interval $[l+1,k]$.

The next result is Lemma 1 from [4].

######  Lemma 2.

Assume that $G_{1}(x)$ has a factor of degree $1$. Then

$n\leq s^{\pi(s)}.$

Further we state the following result from [4].

######  Lemma 3.

Let $n\geq 3$. Assume that $G_{1}(x)$ has a factor of degree $k\geq 2$. Then
$s>92$ unless

$\displaystyle(n,k,s)\in$
$\displaystyle\\{(4,2,7),(4,2,23),(9,2,19),(9,2,47),(16,2,14),(16,2,34),(16,2,89),(9,3,47),$
$\displaystyle(16,3,19),(10,5,4)\\}.$

As an immediate consequence of Lemma 3, we derive the following result.

######  Lemma 4.

Let $n\geq 3$ and $s\leq 92$. Except for finitely many triples

$\displaystyle(n,k,s)\in$
$\displaystyle\\{(4,2,7),(4,2,23),(9,2,19),(9,2,47),(16,2,14),(16,2,34),(16,2,89),(9,3,47),$
$\displaystyle(16,3,19),(10,5,4)\\},$

$G_{1}(x)$ is either irreducible or

(1) $G_{1}(x)=(x-\alpha)H_{1}(x)$

uniquely where $\alpha\in\mathbb{Z}$ and unique monic irreducible polynomial
$H_{1}(x)\in\mathbb{Z}[x]$.

###### Proof.

Let $s\leq 92$. Assume that $G_{1}(x)$ is reducible. Then we derive from Lemma
3 that either $(n,k,s)$ belongs to the finite set stated in Lemma 3 or
$G_{1}(x)$ has no factor of degree $k\geq 2$. Now the assertion follows
immediately. ∎

##  3\. Irreducibility of $G_{1}(x,2,s)$ for $s\in\\{3,7,15\\}$

We compute

(2) $G_{1}(x)=b_{2}x^{2}-2(1+s)b_{1}x+b_{0}\frac{(2+s)(1+s)}{2}$

where $|b_{0}|=|b_{2}|=1$. For the irreducibility of $G_{1}(x)$ it suffices to
show that the polynomials

$x^{2}\pm 2(1+s)b_{1}x\pm\frac{(2+s)(1+s)}{2}$

are irreducible. We prove

######  Lemma 5.

The polynomial (2) with $s=3$ and $s=15$ are irreducible for every
$b_{1}\in\mathbb{Z}$. Also the polynomial (2) with $s=7$ is irreducible for
every $b_{1}\in\mathbb{Z}$ except for $b_{1}=0$ where the polynomial is
$x^{2}-36$.

###### Proof.

The proof depends on a well known assertion that a quadratic polynomial is
irreducible if and only if its discriminant is not a square. We consider
$x^{2}-8b_{1}x+10$ obtained from (2) by putting $b_{0}=1=b_{2}$. Suppose it is
reducible. Then its discriminant $(8b_{1})^{2}-40=m^{2}$ for an integer $m\geq
0$. Thus $(8b_{1}-m,8b_{1}+m)\in\\{(1,40),(2,20),(4,10),(5,8)\\}$ and then
$16b_{1}\in\\{41,22,14,13\\}$. This is not possible since none of
$41,22,14,13$ is divisible by $16$. The assertion follows similarly for all
other cases. ∎

##  4\.  $G_{1}(x)$ divisible by a linear factor

For $s\leq 92$, we see from Lemma 4 except for finitely many cases, $G_{1}(x)$
is either irreducible or divisible by a linear factor. In this section, we
consider the case where $G_{1}(x)$ is divisible by a linear factor. Then we
derive from Lemma 3 that $n$ is bounded by a computable number depending only
on $s$. If $s$ is restricted to $30$, we prove a more precise assertion.

######  Theorem 1.

Let $n\geq 2,s\leq 30$ and $G_{1}(x,2,7)\neq x^{2}-36$. Assume that
$G_{1}(x)=G_{1}(x,n,s)$ is divisible by a linear factor and

(3)
$(n,k,s)\not\in\\{(4,2,7),(4,2,23),(9,2,19),(16,2,14),(16,3,19),(10,5,4)\\}.$

Then $(n,s)\in X$ where

$\displaystyle X=$
$\displaystyle\\{(6,3),(4,5),(8,11),(72,11),(3,15),(10,15),(4,15),(12,15),(8,15),(16,17),$
$\displaystyle(272,17),(8,27),(16,29),(786600,25),(786600,26)\\}.$

###### Proof.

By definition, the assumption (3) is interpreted as $G_{1}(x)$ has no factor
of degree $2$ at $(n,s)\in\\{(4,7),(4,23),(9,19),(16,14)\\}$, no factor of
degree $3$ at $(n,s)=(16,19)$ and no factor of degree $5$ at $(n,s)=(10,4)$.
Assume that $G_{1}(x)$ is divisible by a linear factor. Then, as in Lemma 1 of
[1], we have

(4) $n=\prod\limits_{p|n}p^{\nu_{p}(n)}=\prod\limits_{p\leq s}p^{\nu_{p}(n)}$

where

(5) $p^{\nu_{p}(n)}\quad\text{for}\quad p\leq s$

and

(6) $p\mid\frac{(n+1)\dots(n+s)}{s!}\quad\text{for}\quad p\mid n.$

Denote by $T$ the set of all pairs $(n,s)$ satisfying (4), (5) and (6). By
applying Lemma 1 with $l=0,k=1$ to all pairs $(n,s)\in T$, we check that Lemma
1 does not hold for the following set $T_{1}$ of pairs $(n,s)$ given by

$\displaystyle\\{$
$\displaystyle(2,3),(6,3),(4,5),(2,7),(4,7),(8,11),(72,11),(8,13),(3,15),(2,15),(10,15),(4,15),(12,15),$
$\displaystyle(8,15),(16,17),(272,17),(16,19),(6,23),(4,23),(16,23),(16,24),(16,26),(8,27),(216,29),$
$\displaystyle(16,19),(786600,25),(786600,26)\\}.$

Denote by $T_{2}$ the pairs $(n,s)$ with $n=2$. These are excluded by Lemma 5.
Denote by $T_{3}$ the complement of $T_{2}\cup\\{(3,15)\\}$ in $T_{1}$. Then
all the pairs $(n,s)\in T_{1}$ satisfies $n\geq 4$. Therefore we derive (1)
uniquely for every $(n,s)\in T_{3}$ by Lemma 4. Denote by $T_{4}$ the set
obtained by applying Lemma 1 with $l=1$ and $k=[\frac{n}{2}]$ to $G_{1}(x)$
with $(n,s)\in T_{3}$. We calculate $T_{4}=X\setminus\\{(3,15)\\}$. Now the
assertion of Theorem 1 follows immediately. ∎

Now we give an application of Theorem 1 with $G_{1}(x)$ replaced by
$g_{1}(x)$. We prove

######  Corollary 1.

Let $s\leq 30$. If $g_{1}(x)$ is reducible, the

$(n,s)\in\\{(786600,25),(786600,26)\\}.$

This implies that $g_{1}(x)$ with $s\leq 24$ is irreducible which includes a
new proof of a result of Nair and Shorey [3]. We refer to [4] for a complete
account of results proved on the irreducibility of $g_{1}(x)$. The results of
Hajir and its refinement by Nair and Shorey and Jindal, Laishram and Sarma
depend on algebraic results of Hajir [2] on the Newton polygons. Our proof of
Corollary 1 is new in the sense that it does not use the above results of
Hajir [2] on Newton polygons.

###### Proof of Corollary.

Let $s\leq 30$ and $G_{1}(x)=g_{1}(x)$ be reducible. We compute that the
values of $g_{1}(x)$ at
$(n,s)\in\\{(4,7),(4,23),(9,19),(16,14),(16,19),(10,4)\\}$ are irreducible.
Now we derive from Lemma 3 that $g_{1}(x)$ is divisible by a linear factor and
we check $g_{1}(x)$ at $(n,s)=(2,7)$ is irreducible. Therefore the assumption
of Theorem 1 with $G_{1}$ replaced by $g_{1}$ are satisfied. Hence we conclude
$(n,s)\in X$ by Theorem 1. Now we compute $g_{1}(x)$ with $(n,s)\in X$ are
irreducible. This is a contradiction since $g_{1}(x)$ is divisible by a linear
factor. ∎

## References

  * [1] Filaseta M., Finch Carrie, Leidy, J. Russell , T. N. Shorey’s influence in the theory of irreducible polynomials. Diophantine equations, Tata Inst. Fund. Res. Stud. Math., Mumbai, 20, 77-102 (2008).
  * [2] Hajir F., Algebraic properties of a family of generalized Laguerre polynomials, Canad. J. Math., 61, 583–603 (2009).
  * [3] Nair Saranya G. and Shorey, T. N., Irreducibility of Laguerre polynomial $L_{n}^{(-n-s-1)}(x)$, Indagationes Mathematicae, 26, 615-625 (2015).
  * [4] Shorey, T. N. and Sinha Sneh Bala, Extension of Laguerre polynomials with negative arguments, Submitted (https://arxiv.org/abs/2103.02353 ).


# Gaussian beam quantum radar protocol

Lorenzo Maccone1, Yi Zheng2,3 and Changliang Ren4 $1.$ Dip. Fisica and INFN
Sez. Pavia, University of Pavia, via Bassi 6, I-27100 Pavia, Italy
$2.$CAS Key Laboratory of Quantum Information, University of Science and
Technology of China, Hefei 230026, People’s Republic of China
$3.$ CAS Center for Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei 230026, People’s Republic
of China
$4.$ Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control
of Ministry of Education, Key Laboratory for Matter Microstructure and
Function of Hunan Province, Department of Physics and Synergetic Innovation
Center for Quantum Effects and Applications, Hunan Normal University, Changsha
410081, People’s Republic of China

###### Abstract

We present an entangled quantum radar protocol. It consists in scanning the
sky with a thin Gaussian beam and measuring the travel time of the radiation
reflected from the target, as in conventional radars. Here the Gaussian beam
is composed of $N$ photons entangled in the frequency degrees of freedom. We
show that this provides a $\sqrt{N}$ quantum enhancement over the unentangled
case, as is usual in quantum metrology.

In this paper we introduce a radar protocol that can achieve quantum enhanced
ranging and target detection. Other quantum radar protocols others1 ; others2
; others3 ; others4  are typically based on the quantum illumination primitive
qillum1 ; qillum2 : they can only discriminate the target presence or absence
at a predetermined specific point, which implies that one must scan the whole
3d space in search of a target, which is impractical and time-consuming.
Recently, a protocol that claimed to achieve quantum enhanced ranging
capabilities was proposed in qradar . Unfortunately, its theoretical analysis
was flawed as it was based on an incorrect optical transfer function erratum .
This paper then, to our knowledge, presents the only known three dimensional
quantum enhanced radar protocol that can give quantum-enhanced ranging to the
target: we remind that radar stands for RAdio Detection And Ranging. A
protocol for enhanced ranging in the idealized one-dimensional case was
presented in qps , and this protocol is a sort of 3d extension of it. The
analysis presented here is agnostic to the wavelength used, so the same
protocol can be used also in the optical regime (lidar). It is also more
practical than the previous protocol qradar  since it does not require
wideband radiation entangled in the wave vector $\vec{k}$ which would require
large antennas or antenna arrays: the protocol presented here only employs
Gaussian thin beams that can be produced with small antennas (or lasers). The
beams are entangled only in the frequency degrees of freedom, which is more
practical. There is no quantum enhancement in the transversal direction,
although this feature can be added to our protocol, for example using the
techniques presented in fabre , namely by injecting squeezed vacuum in the
modes orthogonal to the Gaussian mode used by the protocol, or with similar
techniques.

As in the case of most other quantum metrology protocols review0 ; review1 ;
review2 ; review3 ; review4 , we show an enhancement in the precision of the
order of $\sqrt{N}$, where $N$ is the number of photons employed in the
ranging procedure. Namely, we show that this protocol can achieve the
Heisenberg bound in precision in the ideal noiseless situation. As usual, the
situation becomes extremely more complicated in the presence of noise, such as
loss of photons, but the usual general procedures and techniques to deal with
noise can be applied also in this case rafalguta ; davidov ; qec ; dep , e.g.
one can increase the robustness by reducing the entanglement (and hence the
precision gain) pranoise .

The outline follows: we start in Sec. I by showing how one can consistently
derive the correct transfer functions in quantum optics, based on the
quantization of the electromagnetic (EM) field which is reviewed (to set the
notation) in App. A. We then show in Sec. II how these techniques can be used
to give a quantum description of the usual classical radar protocols. This is
useful to show what are the ultimate bounds (due to quantum mechanics) that
can be achieved by these protocols in the absence of entanglement. Finally, in
Sec. III we introduce and analyze our proposed protocol, and show its
$\sqrt{N}$ enhancement.

##  I Quantization of optics through transfer functions

In this section we review how optical transfer functions can be consistently
quantized. The notation and the framework we employ is given in App. A. The
linearity and shift-independence form of the Helmholtz equation (19) implies
that its solutions $U_{\omega}(\vec{r})$ can be shifted in space, e.g. along
the $z$ axis saleh :

$\displaystyle U_{\omega}(\vec{r}\,^{\prime})=\int
d^{2}\vec{r}_{t}\>U_{\omega}(\vec{r}_{t},z)\;h_{\omega}(\vec{r}_{t}\,^{\prime},\vec{r}_{t},z^{\prime}-z)\;,$
(1)

where the $t$ index represents the two-dimensional transverse vector
${\vec{r}}_{t}=(x,y)$, and where $\vec{r}=(\vec{r}_{t},z)$ and $h_{\omega}$ is
the transfer function that takes the solution $U_{\omega}$ at the $xy$ plane
at position $z$ and, with a convolution, moves it to the $xy$ plane at
position $z^{\prime}$ (so the left-hand-side is independent of $z$). This
allows us to obtain the field at all positions starting from the boundary
values of the field in the plane $xy$ at position $z$. Of course, the general
solution of the field is given by (17), where one must sum $U_{\omega}$ over
all components $\omega$. Indeed, replacing Eq. (1) into (18), we obtain the
whole field $A({t},{\vec{r}}\,^{\prime})$ at position
${\vec{r}}\,^{\prime}=({\vec{r}}\,^{\prime}_{t},z^{\prime})$ from a field on
the plane $xy$ at position $z$ at time ${t}=0$ (boundary conditions):

$\displaystyle A^{+}({t},\vec{r}\,^{\prime})=\int
d\omega\>d^{2}\vec{r}_{t}U_{\omega}(\vec{r}_{t},z)\>h_{\omega}(\vec{r}_{t}\,^{\prime},\vec{r}_{t},d)e^{-i\omega{t}},$
(2)

where, for simplicity of notation, we consider only the positive-energy
component of the field $A^{+}$, namely only the first term in the integral of
(18).

This transfer function formalism is developed in the classical case but it can
be transferred to the quantum case by first expressing the solutions
$U_{\omega}(\vec{r})$ in terms of plane waves
$e^{i{\vec{\kappa}}\cdot\vec{r}}$, and then associating to each plane wave an
amplitude $a({\vec{\kappa}})$ as done in the customary EM quantization (see
App. A). Namely,

$\displaystyle
A^{+}({t},\vec{r}\,^{\prime})=\\!\int\\!d^{2}\vec{r}_{t}\>d^{3}{\vec{\kappa}}\>h_{\omega_{\kappa}}(\vec{r}_{t}\,^{\prime},\vec{r}_{t},d)a({\vec{\kappa}})e^{-i(\omega_{k}t-{\vec{\kappa}}\cdot\vec{r})},$
(3)

where $\vec{r}=(\vec{r}_{t},z)$,
$\vec{r}\,^{\prime}=(\vec{r}\,^{\prime}_{t},z^{\prime})$, and the integral
over $\omega$ is contained in the integral over ${\vec{\kappa}}$, since
$\omega_{\kappa}=c\kappa$. [More rigorously, the integral over $\omega$ comes
from (2), whereas in the input field we are considering only the $\omega$
component $U_{\omega}({\vec{r}})$, so we need to integrate only on the
directions ${\vec{\kappa}}/\kappa$ as discussed below Eq. (20).] For example,
$\vec{r}$ may represent the object plane and $\vec{r}\,^{\prime}$ the image
plane in an imaging apparatus, whose transfer function is given by
$h_{\omega}$ qimaging . Eq. (3) is the main result of this section.

This is the field operator, so by itself it says nothing about the physics:
operators in quantum mechanics only acquire values when applied to states,
e.g. the probability $p({t},\vec{r})\propto|\langle 0|A^{+}|\psi\rangle|^{2}$.
Alternatively, we may be interested in other expectation values of the field
in state $|\psi\rangle$. The field degrees of freedom (including its boundary
conditions) are encoded into $|\psi\rangle$. E.g., for a single photon with
$\psi({\vec{\kappa}})\propto\alpha({\vec{\kappa}})$ [see Eq. (26)], we have

$\displaystyle\langle 0|A^{+}({t},{\vec{r}}\,^{\prime})|\psi\rangle$
$\displaystyle=\int
d^{3}{\vec{\kappa}}\>d^{3}{\vec{\kappa}}^{\prime}\>d^{2}{\vec{r}}_{t}\>h_{\omega^{\prime}}\>e^{-i(\omega^{\prime}{t}-{\vec{\kappa}}^{\prime}\cdot{\vec{r}})}\alpha({\vec{\kappa}})\langle
0|a({\vec{\kappa}}^{\prime})a^{\dagger}({\vec{\kappa}})|0\rangle$
$\displaystyle=\int
d^{3}{\vec{\kappa}}\>d^{2}{\vec{r}}_{t}\>h_{\omega}\>e^{-i(\omega{t}-{\vec{\kappa}}\cdot{\vec{r}})}\>\alpha({\vec{\kappa}})$
$\displaystyle=\int
d\omega\>d^{2}{\vec{r}}_{t}\>h_{\omega}({\vec{r}}_{t}\,^{\prime},{\vec{r}}_{t},d)\>U_{\omega}({\vec{r}})\;,$
(4)

where we used the fact that integrating
$\alpha({\vec{\kappa}})e^{i{\vec{\kappa}}\cdot{\vec{r}}}$ over the directions
of ${\vec{\kappa}}$, one obtains $U_{\omega}({\vec{r}})$ with
$\omega=c\kappa$, as is clear by the comparison between Eqs. (18) and (20).
This result is what one would expect from (1) by integrating over $\omega$
both members.

### Free field transfer function

The specific form of the function $h_{\omega}$ depends on what is present
between the two $xy$ planes at $z$ and $z^{\prime}$, and on the approximations
used. In the case of vacuum propagation with the Fresnel approximation, we get
(saleh , Eq. 4.1-14)

$\displaystyle
h_{\omega}(\vec{r}_{t}\,^{\prime},\vec{r}_{t},d)=\frac{i\kappa}{2\pi
d}e^{-i\kappa(\vec{r}_{t}-\vec{r}_{t}\,^{\prime})^{2}/d}\>e^{-i\kappa d}\;,$
(5)

with $d=z^{\prime}-z$ the distance between the two planes. While the Rayleigh-
Sommerfeld diffraction can give better results in some cases, in the regimes
we are interested, the Fresnel approximation that gives rise to (5) is
sufficient to our aims. We will be using Eq. (5) in the following.

##  II Quantum treatment of a classical radar/lidar protocol

A radar/lidar works by scanning the sky with a directional beam and measuring
the time it takes for it to be bounced back. The direction of the beam and the
time of flight suffice to do a full 3d localization of the target. In this
section we analyze a classical radar/lidar protocol using quantized light to
show what are the ultimate bounds imposed by quantum mechanics to such
classical (unentangled) protocols.

As directional beam, we consider a Gaussian beam. For simplicity we will
consider the target as a perfectly (or partially) reflecting mirror orthogonal
to the beam direction, of size larger than the beam waist at the target
location. In this way we are guaranteed that the beam that returns to the
antenna is still in a (possibly attenuated) Gaussian beam, see Fig. 1. The
case in which the target is smaller than the beam waist should also not be too
difficult: the returning beam will be a spherical wave originating at the
target.

Figure 1: Sketch of the quantum radar protocol. A Gaussian beam composed by
frequency-entangled photons bounces off the target and returns to the sender’s
location. By measuring the average photon round-trip time, the sender can
recover the target’s position with quantum enhanced accuracy.

The Gaussian beam for every frequency $\omega$ has an amplitude
$U_{\omega}(\vec{r})=\varphi(\omega)G_{\omega}({\vec{r}})$, where
$\varphi(\omega)$ is the spectral amplitude (the amplitude for each frequency
$\omega$ in the light) and $G_{\omega}$ is (saleh , Eq. 3.1-7)

$\displaystyle G_{\omega}(\vec{r})\propto\frac{1}{W(z)}e^{-\frac{\kappa
r_{t}^{2}}{2z_{0}W^{2}(z)}}e^{-i[\kappa z+\frac{\kappa
r_{t}^{2}}{2zR(z)}-\arctan(z/z_{0})]}\;$ (6)

where $W(z)\equiv\sqrt{1+z^{2}/z_{0}^{2}}$, $R(z)\equiv 1+z_{0}^{2}/z^{2}$,
$z_{0}$ is a (length) constant that, together with the direction of the $z$
axis, fully specifies $G_{\omega}({\vec{r}})$. It is possible to check that
this solution for fixed $z$ is propagated to an arbitrary $z^{\prime}$ through
the transfer function (5). The field $A$ is obtained by integrating
$U_{\omega}$ over $\omega$, as in Eq. (17):

$\displaystyle A({t},{\vec{r}})=\int
d\omega\>e^{-i\omega{t}}\>\varphi(\omega)\>G_{\omega}({\vec{r}})\;.$ (7)

We now consider a single photon in a Gaussian beam111It would be more
appropriate to use a coherent state (or a thermal state) to model a classical
beam, but since the photons in coherent states are completely uncorrelated
(Poissonian statistics), one can easily obtain the same arrival statistics as
a coherent state $|\alpha\rangle$ (or its thermal mixtures) by considering
what happens to $N=|\alpha|^{2}$ uncorrelated single photons. (Of course the
photon number statistics will be different!) . Since the light intensity
$|A(\vec{r})|^{2}$ at each point is directly proportional to the probability
of finding the photon there (as discussed above) saleh ; mandel , we can
choose $\tilde{\psi}(\vec{r})\propto A=\int
d\omega\>\varphi(\omega)\>G_{\omega}(\vec{r})$, using (7) with ${t}=0$
(because of the Heisenberg picture) and the proportionality constant chosen by
the normalization condition. Namely, the photon wavepacket has probability
amplitude proportional to $G_{\omega}({\vec{r}})$ for each frequency $\omega$,
and the probability amplitude of having frequency $\omega$ is given by
$\varphi(\omega)$. So the state is

$\displaystyle|\psi\rangle=\int
d^{3}\vec{\rho}\>\tilde{\psi}(\vec{\rho})\>a^{\dagger}(\vec{\rho})|0\rangle=\int
d^{3}{\vec{\kappa}}^{\prime}\>\tilde{G}({\vec{\kappa}}^{\prime})\,a^{\dagger}({\vec{\kappa}}^{\prime})|0\rangle\;,$
(8)

where $\tilde{G}({\vec{\kappa}})$ is the Fourier transform of
$\varphi(\omega)\>G_{\omega}(\vec{\rho})$, which clearly only contains the
frequency $\omega$ (the amplitude $\varphi$ is included in $\tilde{G}$). We
can write the field at the image plane, i.e. the detector position
${\vec{r}}\,^{\prime}$ in terms of the field at the target position
${\vec{r}}$ using (3), with the transfer function (5). Then, the probability
amplitude of finding the photon in ${t},{\vec{r}}\,^{\prime}$ is

$\displaystyle\langle
0|A^{+}({t},{\vec{r}}\,^{\prime})|\psi\rangle=\qquad\qquad\;$ (9)
$\displaystyle\int
d^{2}\vec{r}_{t}\>d^{3}{\vec{\kappa}}\>h_{\omega}(\vec{r}_{t}\,^{\prime},\vec{r}_{t},d)\>e^{-i(\omega_{k}t-{\vec{\kappa}}\cdot\vec{r})}\langle
0|a({\vec{\kappa}})\times\qquad$ $\displaystyle\int
d^{3}{\vec{\kappa}}^{\prime}\tilde{G}_{\omega^{\prime}}({\vec{\kappa}}^{\prime})\>a^{\dagger}({\vec{\kappa}}^{\prime})|0\rangle=\qquad$
$\displaystyle\int
d^{2}\vec{r}_{t}\>d^{3}{\vec{\kappa}}\>h_{\omega}(\vec{r}_{t}\,^{\prime},\vec{r}_{t},d)\>e^{-i(\omega{t}-{\vec{\kappa}}\cdot{\vec{r}})}\>\tilde{G}({\vec{\kappa}})\propto$
$\displaystyle\int
d^{2}\vec{r}_{t}\>d\omega\>h_{\omega}(\vec{r}_{t}\,^{\prime},\vec{r}_{t},d)\>e^{-i\omega{t}}\>\varphi(\omega)\>G_{\omega}({\vec{r}}_{t},z),$

where we used the commutator (22) in the second equality, and in the third we
used the far field condition $\kappa_{z}\sim\kappa\gg\kappa_{x},\kappa_{y}$ to
separate the integral over ${\vec{\kappa}}$ into a frequency and a transverse
part ${\vec{\kappa}}_{t}$: $d^{3}{\vec{\kappa}}\propto d\omega
d^{2}{\vec{\kappa}}_{t}$, so that the integral of
$e^{i{\vec{\kappa}}\cdot{\vec{r}}}\psi({\vec{\kappa}})$ over the transverse
part of ${\vec{\kappa}}$ gives the spatial field at frequency $\omega$, namely
$U_{\omega}({\vec{r}})=\varphi(\omega)G_{\omega}({\vec{r}}_{t},z)$ with
${\vec{r}}=({\vec{r}}_{t},z)$ [compare Eqs. (18) and (20)]. Eq. (9) is
compatible with what one would expect from the transfer function of the
classical amplitudes: see Eq. (1) when the time evolution of the output field
is added and both members are integrated over $\omega$. We now use the fact
that the free space transfer function (5) applied to a Gaussian beam (6)
translates it forward by a factor $d$, the distance between target and
receiver (namely, a Gaussian beam is transformed in a Gaussian beam thanks to
the hypothesis that the target is a partially reflecting mirror larger than
the beam waist). Then, (9) becomes

$\displaystyle\langle 0|A^{+}({t},{\vec{r}}\,^{\prime})|\psi\rangle=\int
d\omega\>e^{-i\omega{t}}\varphi(\omega)G_{\omega}({\vec{r}}_{t}\,^{\prime},z+d)\;.$
(10)

As expected, at the image plane at position $z^{\prime}$, it gives a pulse
that is delayed by the transit time to the target. To see this, consider the
expression (10) at the center of the image plane ${\vec{r}}_{t}\,^{\prime}=0$
where, from (6) we see that $G_{\omega}({\vec{r}}_{t}\,^{\prime}=0,z)\propto
e^{-i[\kappa z-\arctan(z/z_{0})]}$, so that (10) becomes

$\displaystyle\int
d\omega\>\>e^{-i\omega[{t}+(z+d)/c]-i\arctan((z+d)/z_{0})}\varphi(\omega)$
$\displaystyle=\tilde{\varphi}({t}+(z+d)/c)\>e^{-i\arctan((z+d)/z_{0})}\;,$
(11)

where $\tilde{\varphi}$ is the Fourier transform of $\varphi$. Eq. (11)
describes a pulse of spectral amplitude $\varphi(\omega)$ and temporal
amplitude $\tilde{\varphi}({t})$ that is delayed by an amount $(z+d)/c$, where
$d$ is the distance between target and receiver and $z=d$ is the position of
the target. By measuring the time of arrival of the photon, one can obtain
twice the distance $2d$ to the target, as expected for a radar. The
statistical error in this measurement is given by the width $\Delta\tau$ of
$\tilde{\varphi}(\tau)$, proportional to the inverse of the bandwidth
$\Delta\omega$ of $\varphi(\omega)$.

Now we could do the same calculation with a coherent state $|\alpha\rangle$
instead of a single photon state (8), with
$|\alpha\rangle=\bigotimes_{\vec{\kappa}}|\alpha({\vec{\kappa}})\rangle$ with
$|\alpha({\vec{\kappa}})\rangle$ eigenstates of $a({\vec{\kappa}})$:
$a({\vec{\kappa}})|\alpha({\vec{\kappa}})\rangle=\alpha({\vec{\kappa}})|\alpha({\vec{\kappa}})\rangle$.
This calculation should give exactly the same outcome as a classical field
amplitude $\alpha({\vec{\kappa}})$, see Eq. (20).

##  III Quantum radar/lidar protocol

We now show how one can obtain an increased localization precision by using
frequency-entangled light. For simplicity of notation, we will consider only
the case of $N=2$ entangled photons. This can then be extended to arbitrary
$N$.

For the $N$-photon state $|\psi_{N}\rangle$ of (24), the probability of
detecting them at ${t}_{1},{\vec{r}}_{1},\cdots,{t}_{N},{\vec{r}}_{N}$ is
mandel

$\displaystyle p\propto|\langle 0|A^{+}({t}_{1},{\vec{r}}_{1})\cdots
A^{+}({t}_{N},{\vec{r}}_{N})|\psi_{N}\rangle|^{2}\;.$ (12)

Consider the biphoton entangled state with wavefunction

$\displaystyle\tilde{\psi}_{2}({\vec{r}},\vec{\rho})\propto\int
d\omega\>\varphi(\omega)\>G_{\omega}({\vec{r}},{\vec{\rho}})\;,$ (13)

which gives the probability amplitude of finding the two photons at positions
${\vec{r}}=({\vec{r}}_{t},z)$ and ${\vec{\rho}}=({\vec{\rho}}_{t},\rho_{z})$
(in the Heisenberg picture there is no time evolution), and where

$\displaystyle\;G_{\omega}({\vec{r}},{\vec{\rho}})\equiv$ (14)
$\displaystyle\tfrac{1}{W(z+\rho_{z})}e^{-\frac{\kappa({\vec{r}}_{t}^{2}+{\vec{\rho}}_{t}^{2})}{2z_{0}W^{2}}}e^{-i[\kappa(z+\rho_{z})+\frac{\kappa({\vec{r}}_{t}^{2}+{\vec{\rho}}_{t}^{2})}{2zR}-\arctan\frac{z+\rho_{z}}{z_{0}}]},$

which represents two photons of identical frequency $\omega$ in a Gaussian
beam, see Eq. (6). Except for the multiplicative term $1/W$ and the arctan
term, $G_{\omega}$ is basically a product of two Gaussian beam single-photon
amplitudes. So we can reuse the calculations above for the single-photon
amplitude to find that the temporal amplitude at the center of the image plane
${\vec{r}}_{t}={\vec{\rho}}_{t}=0$ at the image plane position
$z=\rho_{z}=z^{\prime}$ is given by the analogous of (11):

$\displaystyle\langle
0|A^{+}({t}_{1},{\vec{r}})A^{+}({t}_{2},{\vec{\rho}})|\psi\rangle=\tilde{\varphi}({t}_{1}+{t}_{2}+2(z+d)/c)\>e^{i\theta}\;,$
(15)

with $\theta$ some irrelevant phase factor. From this, it is clear that the
time of arrival sum ${t}_{1}+{t}_{2}$ has an uncertainty $\Delta\tau$,the
width of $\tilde{\varphi}$. Which means that the average time of arrival
$({t}_{1}+{t}_{2})/2$ is estimated to be the correct value $d+z=2d$ with a
statistical error $\Delta\tau/2$. Instead, from (11) we saw that, using a
single photon state, one estimates the time of arrival with an uncertainty
$\Delta\tau$, so the average time of arrival of two photons will be estimated
with an uncertainty $\simeq\Delta\tau/\sqrt{2}$. The $\sqrt{2}$ enhancement in
precision is the $\sqrt{N}$ gain that one expects from entanglement in quantum
metrology.

The biphoton analysis done here can be straightforwardly extended to the case
of $N$ entangled photons in a Gaussian beam. Namely, a state with wavefunction

$\displaystyle\psi({\vec{r}}_{1},\cdots,{\vec{r}}_{N})\propto\int
d\omega\varphi(\omega)G_{\omega}({\vec{r}}_{1},\cdots,{\vec{r}}_{N})\;,$ (16)

where $G_{\omega}$ is a trivial generalization of (14). It gives a $\sqrt{N}$
enhancement in the average photon time of arrival, which translates into a
$\sqrt{N}$ precision enhancement in the longitudinal localization for each
point in the sky scanned by the $N$-photon Gaussian beam (16), when one
measures the average arrival time $\sum_{i}t_{i}/N$.

##  IV Conclusions

In conclusion we have presented a quantum radar protocol that uses
entanglement in the frequency/wavelength degrees of freedom to provide an
quantum enhancement equal to the square root $\sqrt{N}$ of the number $N$ of
entangled photons employed. We have shown in detail how the optical transfer
function formalism can be employed in the fully quantum regime we analyze
here.

##  Appendix A Quantization of the EM field

In this appendix we review the usual theory for the quantization of the
electromagnetic field. This is useful to set the notation we use in the paper,
and also to keep track of the specific roles that all the radiation degrees of
freedom play in our protocol. Specifically, it is useful to understand the
peculiar role of the frequency degree of freedom of the radiation that our
protocol hinges on.

###  A.1 Classical EM in the Coulomb gauge

Start from the Maxwell equations in vacuum in the Coulomb gauge for the scalar
and vector potentials $\Phi$ and $\vec{A}$: $\nabla^{2}\Phi({t},{\vec{r}})=0$,
$\square\vec{A}({t},\vec{r})=\tfrac{\partial}{\partial t}\vec{\nabla}\Phi$,
where $\vec{r}=(x,y,z)$ is the spatial position, and
$\square=\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial{t}^{2}}$ is
the d’Alamabertian with $\vec{\nabla}=(\frac{\partial}{\partial
x},\frac{\partial}{\partial y},\frac{\partial}{\partial z})$ in Cartesian
coordinates. Conventionally, since we are interested only in the quantization
of the electromagnetic waves, one chooses the specific solution $\Phi=0$,
which implies $\square\vec{A}({t},\vec{r})=0$. For simplicity of notation we
will consider scalar fields $A({t},{\vec{r}})$ from now on: the vectorial part
can be added by introducing two independent components, connected to the two
polarizations of the em field (there are two polarizations because, in the
Coulomb gauge, the potential $\vec{A}$ is transverse:
$\vec{\nabla}\cdot\vec{A}=0$). We separate the temporal and spatial degrees of
freedom by taking a Fourier transform over time:

$\displaystyle A({t},\vec{r})=\int_{-\infty}^{+\infty}d\omega\>e^{-i\omega
t}\>U_{\omega}(\vec{r})\;,$ (17)

where $U_{\omega}$ is the component at frequency $\omega$. Since $A$ is real,
we must have $U_{-\omega}=U^{*}_{\omega}$, this condition can be enforced
automatically if we separate the integral into a sum of two and change
variable in the second:

$\displaystyle A({t},\vec{r})=\int_{0}^{\infty}d\omega[e^{-i\omega
t}U_{\omega}(\vec{r})+e^{+i\omega t}U^{*}_{\omega}(\vec{r})]\;.$ (18)

For each component at frequency $\omega$, it is clear from (17) that $\square
A=0$ becomes the Helmholtz equation

$\displaystyle\nabla^{2}U_{\omega}(\vec{r})=\frac{\omega^{2}}{c^{2}}U_{\omega}(\vec{r}).\;$
(19)

A convenient222Because of the linearity of the Helmholtz equation, any
solution $U_{\omega}(\vec{r})$ can be expressed as a sum of plane waves
$U_{\omega}(\vec{r})=\int
d^{3}\kappa\>\alpha({\vec{\kappa}})\>e^{i\vec{\kappa}\cdot\vec{r}}$. solution
is in terms of plane waves $U_{\omega}(\vec{r})=\alpha(\vec{\kappa})e^{\pm
i\vec{\kappa}\cdot\vec{r}}$. The real and imaginary part (or the modulus and
phase) of $\alpha$ are the two integration constants, and the wave direction
${\vec{\kappa}}/\kappa$ parametrizes all the solutions, while we must choose
$|\vec{\kappa}|$ such that
$\omega=\vec{c}\cdot\vec{\kappa}=\pm|\vec{\kappa}|c$. The sign parametrizes
two classes of solutions: we must choose $+$ if the wave vector $\vec{\kappa}$
is parallel to the wave velocity $\vec{c}$, or $-$ if it is antiparallel. The
first case refers to the retarded waves, the second to the advanced waves
(jackson , sec. 6.4). We usually choose past boundary conditions, so we only
use retarded waves333If we were to choose future boundary conditions, we would
need to consider only advanced waves, and if we were to choose mixed future-
and-past boundary conditions einstein1909 , we would have to keep both
solutions, which, incidentally, is a real problem in quantum field theory, as
this leads to a non-Hamiltonian evolution of the electromagnetic field!
Advanced waves can be seen as propagating negative energy in the forward time
direction or positive energy in the negative time direction.. Summarizing, the
vacuum solution of the Maxwell equations is of the form

$\displaystyle
A({t},\vec{r})=\int_{{\mathbb{R}}^{3}}d^{3}\vec{\kappa}[\alpha(\vec{k})\>e^{-i(\omega_{\kappa}t-\vec{\kappa}\cdot\vec{r})}+\alpha^{*}(\vec{k})\>e^{i(\omega_{\kappa}t-\vec{\kappa}\cdot\vec{r})}]\;,$
(20)

where $\alpha(\vec{\kappa})$ is the positive-frequency amplitude444Note that
the Maxwell equations are solved also by negative frequency plane waves of the
type
$\alpha_{-}(\vec{\kappa})\>e^{-i(\omega_{\kappa}t+\vec{\kappa}\cdot\vec{r})}$
but, as discussed above, we will ignore these solutions, by choosing past
boundary conditions as is done usually. and $\omega_{\kappa}=|\vec{\kappa}|c$,
so that the integral over ${\vec{\kappa}}$ takes care of the integral over
$\omega$ in (18) (its modulus) and of the integral over the directions
${\vec{\kappa}}/\kappa$ that enumerate all plane waves.

###  A.2 Quantum em: observables

The energy of the electromagnetic field is $H=\frac{\epsilon_{0}}{2}\int
d^{3}\vec{r}[E^{2}({t},\vec{r})+c^{2}B^{2}({t},\vec{r})]$, where $\vec{E}$ and
$\vec{B}$ are the electric and magnetic fields. In terms of the amplitudes
$\alpha(\vec{\kappa})$, one can show that, in the Coulomb gauge, the energy is

$\displaystyle H=\frac{1}{2}\int
d^{3}\vec{\kappa}\>(P^{2}_{\vec{\kappa}}+\omega_{\kappa}^{2}X^{2}_{\vec{\kappa}})\;,$
(21)

with $P_{\vec{\kappa}}\propto
i(\alpha_{\vec{\kappa}}^{*}-\alpha_{\vec{\kappa}})$ and
$X_{\vec{\kappa}}\propto(\alpha_{\vec{\kappa}}^{*}+\alpha_{\vec{\kappa}})$.
Eq. (21) is the energy of a collection of independent (noninteracting)
harmonic oscillators (one for each value of $\vec{\kappa}$), so we can
quantize by considering $X$ and $P$ as “position” and “momentum” operators,
promoting the amplitudes $\alpha$ to operators $a$. Namely, we impose
$[X_{\vec{\kappa}},P_{\vec{\kappa}^{\prime}}]=i\delta(\vec{\kappa}-\vec{\kappa}^{\prime})$,
where the delta shows that they are independent oscillators for each
$\vec{\kappa}$. From the definitions of $X$ and $P$, this implies the
commutators

$\displaystyle[a(\vec{\kappa}),a^{\dagger}(\vec{\kappa}^{\prime})]=\delta(\vec{\kappa}-\vec{\kappa}^{\prime})\;,[a(\vec{\kappa}),a(\vec{\kappa}^{\prime})]=0\;.$
(22)

The quantization of the general solution of the Maxwell equations in the
Coulomb gauge $\square A=0$, is then

$\displaystyle A({t},\vec{r})=\int
d^{3}\vec{\kappa}\>[a(\vec{k})\>e^{-i(\omega_{\kappa}t-\vec{\kappa}\cdot\vec{r})}+a^{\dagger}(\vec{k})\>e^{i(\omega_{\kappa}t-\vec{\kappa}\cdot\vec{r})}]\;,$
(23)

namely Eq. (20) quantized. Importantly, since we are introducing the time
evolution in the operators, we are working in the Heisenberg picture (or in
the interaction picture with the free-field Hamiltonian to evolve the
operators). We are working with the vector potential field $A$, but the
electric field and magnetic fields are trivially obtained from it:
$\vec{E}=-\frac{\partial}{\partial t}\vec{A}-\vec{\nabla}\Phi$,
$\vec{B}=\vec{\nabla}\times\vec{A}$, which give expressions very similar to
(23), except for the fact that the derivatives introduce a minus sign between
the two terms of the right-hand-side (as $\vec{A}$, also $\vec{E}$ and
$\vec{B}$ have two independent components since they are transverse:
$\vec{\nabla}\cdot\vec{B}=0$ and, in vacuum, $\vec{\nabla}\cdot\vec{E}=0$).

The intensity of the field is proportional to the time averaged square
$\langle E^{2}\rangle_{t}$. This is basically equal to the average photon
number in the field. Indeed, from (20) and the fact that
$\vec{E}=-\tfrac{\partial}{\partial t}\vec{A}$ we have that, classically,
$E^{2}=-\omega(\alpha^{2}e^{i\phi}+(\alpha^{*})^{2}e^{-i\phi}-2|\alpha|^{2})$
for a classical field with only a single ${\vec{\kappa}}$, with
$\Phi=\omega{t}-{\vec{\kappa}}\cdot{\vec{r}}$. This almost matches the quantum
result one would get for a coherent state $|\alpha\rangle$ for which
$a|\alpha\rangle=\alpha|\alpha\rangle$ (which represents a classical field). A
coherent state has
$E^{2}=-\omega(\alpha^{2}e^{i\phi}+(\alpha^{*})^{2}e^{-i\phi}-2|\alpha|^{2}-1)$,
where the -1 term comes from the commutator $[a,a^{\dagger}]=1$ (valid for the
quantization of a field with single ${\vec{\kappa}}$ vector). The time average
removes the terms with the phase, leaving only the average photon number for a
coherent state, i.e. $\langle\alpha|a^{\dagger}a|\alpha\rangle=|\alpha|^{2}$.
So while $E^{2}$ does not coincide with the average photon number, the time-
averaged $E^{2}$ essentially does for classical fields. Similar considerations
apply also to states with fixed photon number we consider below, where
$\langle a^{2}\rangle=0$.

###  A.3 Quantum em: states

In the classical case, we can choose a specific form of the
$\alpha(\vec{\kappa})$ to obtain a specific solution of the Maxwell equations
(which can be done by choosing appropriate boundary conditions for the field).
In the quantum case, the $\alpha\to a(\vec{\kappa})$ are operators. There are
two ways to assign a value to them: (i) have them act on eigenstates of the
field (which implies that the field is in a state where there are no quantum
fluctuations of the field). From the form of Eq. (23), it is clear that the
eigenstates of the field are quadrature eigenstates for each ${\vec{\kappa}}$,
where the quadrature is
$Q_{\varphi}\equiv(a\>e^{-i\varphi}+a^{\dagger}e^{i\varphi})/\sqrt{2}$. These
eigenstates are unphysical as they are infinitely squeezed states with
infinite average energy $\hbar\omega_{\kappa}\langle
a^{\dagger}(\kappa)a(\kappa)\rangle$. (ii) we can calculate the field
expectation value on an arbitrary state $|\psi\rangle$ of the field (which
implies that we can calculate the average field, because there are quantum
fluctuations: measuring the field multiple times, we would get different
results).

How do we choose $|\psi\rangle$? It is the state of the em degrees of freedom.
Since these are given by a harmonic oscillator for each ${\vec{\kappa}}$, the
Hilbert space is a Fock space for each ${\vec{\kappa}}$, so the most general
state is given by

$\displaystyle|\psi\rangle={\textstyle\sum_{n}}\gamma_{n}|\psi_{n}\rangle,\mbox{
with }$ (24)
$\displaystyle|\psi_{n}\rangle=\\!\int\\!\\!d^{3}{\vec{\kappa}}_{1}\cdots
d^{3}{\vec{\kappa}}_{n}\>\psi_{n}({\vec{\kappa}}_{1},\cdots,{\vec{\kappa}}_{n})a^{\dagger}({\vec{\kappa}}_{1})\cdots
a^{\dagger}({\vec{\kappa}}_{n})|0\rangle,$

where $\gamma_{n}$ is the probability amplitude to have $n$ photons,
$\psi_{n}$ is the joint probability amplitude that they are in modes
${\vec{\kappa}}_{1},\cdots,{\vec{\kappa}}_{n}$, and $|0\rangle$ is the vacuum.
In this paper we only consider states with a single component $\gamma_{N}=1$.
In the Heisenberg picture, the state does not evolve in time. The wavefunction
normalization condition is

$\displaystyle\int d^{3}{\vec{\kappa}}_{1}\cdots
d^{3}{\vec{\kappa}}_{n}\>|\psi_{n}({\vec{\kappa}}_{1},\cdots,{\vec{\kappa}}_{n})|^{2}=1\
\forall n\;.$ (25)

More rigorously, the state (24) refers only to the situation in which all the
photons are in different modes (namely, the $\psi_{n}$ does not contain Dirac
$\delta$s over the ${\vec{\kappa}}_{i}$). The most general situation is to
have the $n$ photons distributed into $m\leqslant n$ modes ($m=1$ if all the
$n$ photons are in one mode, $m=n$ if there is one photon per mode, as above).
The indistinguishable nature of the photons implies that only $m$ of the
$\textstyle\left(\begin{matrix}m\cr n\end{matrix}\right)$ possibilities can be
tracked, namely we can only know that $n_{i}$ photons are in mode
${\vec{\kappa}}_{i}$ for $i=1,\cdots,m$. In this case, the wavefunction is
$\psi_{n}(n_{1},{\vec{\kappa}}_{1},\cdots,n_{m},{\vec{\kappa}}_{m})/\sqrt{n_{1}!\cdots
n_{m}!}$ with $\sum_{i}n_{i}=n$, where the factorials appear because the
$n_{i}$ photon Fock state in a mode is given by
$|n_{i}\rangle=(a^{\dagger})^{n_{i}}|0\rangle/\sqrt{n_{i}!}$ and where
$\psi_{n}$ is the joint probability amplitude that the $n$ photons are
partitioned as $\\{n_{i}\\}$ and that their wave vectors are
$\\{{\vec{\kappa}}_{i}\\}$.

For single photon states, only the term $n=1$ of (24) survives. The spatial
dependence of the wavefunction can be obtained by taking the Fourier transform
$\tilde{\psi}$ of $\psi\equiv\psi_{1}$:

$\displaystyle\int
d^{3}{\vec{\kappa}}\>\psi({\vec{\kappa}})\>a^{\dagger}({\vec{\kappa}})|0\rangle=\int
d^{3}{\vec{r}}\>\tilde{\psi}({\vec{r}})\>a^{\dagger}({\vec{r}})|0\rangle\;,$
(26)

where $a({\vec{r}})\propto\int
d^{3}{\vec{\kappa}}\>a({\vec{\kappa}})\>e^{i{\vec{\kappa}}\cdot{\vec{r}}}$ is
the annihilator of a photon at position ${\vec{r}}$, so that
$\tilde{\psi}({\vec{r}})$ is the probability that the photon is in
${\vec{r}}$. (Note that, except in the limit discussed in the next subsection,
this is not in general equal to the probability amplitude of measuring the
photon at position ${\vec{r}}$, since there is a difference between the
position of the photon and of its energy, a well known problem in quantum
field theory, e.g. posphoton1 ; posphoton2 ; mandel . Indeed, as is clear from
the above analysis, the photon is obtained from the quantization of the vector
potential $A$, which is a gauge-dependent quantity, whereas its energy is,
clearly, a gauge-independent quantity.)

We can choose $\psi({\vec{\kappa}})={\cal N}\alpha({\vec{\kappa}})$, where
$\alpha({\vec{\kappa}})$ is the Fourier transform of the classical solution
$A({t},{\vec{r}})$ of (20) and $\cal N$ is a normalization for Eq. (25).
Indeed, as discussed below, $|A({t},{\vec{r}})|^{2}$ is the light intensity at
position ${t},{\vec{r}}$, so it is proportional to the probability of finding
the photon at such position, so $A$ is the probability amplitude, and its
Fourier transform $\alpha({\vec{\kappa}})$ is the probability amplitude in the
${\vec{\kappa}}$ space.

###  A.4 Quantum em: photodetection

It can be shown that for photodetectors with efficiency $\eta$, sufficiently
small temporal resolution $\tau$, and spatial resolution $\sigma$, the
probability of a photodetection at spacetime position $({t},\vec{r})$ is
mandel
$p({t},\vec{r})\propto\eta\tau\sigma\langle\psi|[A^{+}({t},\vec{r})]^{\dagger}A^{+}({t},\vec{r})|\psi\rangle$.
In the case in which the system state $|\psi\rangle$ contains a single photon,
we can use the fact that $a$ is the photon annihilator to simplify it to
$p\propto|\langle 0|A^{+}|\psi\rangle|^{2}$. To show this, consider
$|\psi\rangle=\int
d^{3}{\vec{\kappa}}^{\prime}\psi({\vec{\kappa}}^{\prime})a^{\dagger}({\vec{\kappa}}^{\prime})|0\rangle$,
with $\psi({\vec{\kappa}}^{\prime})$ the probability amplitude that the photon
has wave vector ${\vec{\kappa}}^{\prime}$ (so that its Fourier transform can
be interpreted as the probability amplitude that the photon is in position
$\vec{r}$). Then

$\displaystyle A^{+}|\psi\rangle=\int
d^{3}{\vec{\kappa}}\>d^{3}{\vec{\kappa}}^{\prime}\psi({\vec{\kappa}}^{\prime})a({\vec{\kappa}})e^{-i(\omega_{\kappa}{t}-{\vec{\kappa}}\cdot\vec{r})}a^{\dagger}({\vec{\kappa}}^{\prime})|0\rangle=$
$\displaystyle\tilde{\psi}(\vec{r}-\vec{c}t)|0\rangle\;,$ (27)

where $\vec{c}$ is the speed of light with the direction
${\vec{\kappa}}/\kappa$ of the beam. Eq. (27) follows from the commutator (22)
and the fact that $a|0\rangle=0$, and where $\tilde{\psi}$ is the Fourier
transform of $\psi$. [Note the use of the Heisenberg picture: the time
evolution is only in the operator $A^{+}$, not in the state.]

## Acknowledgments

This work received support from EU H2020 QuantERA ERA-NET Cofund in Quantum
Technologies, Quantum Information and Communication with High-dimensional
Encoding (QuICHE) under Grant Agree- ment 731473 and 101017733, from the U.S.
Department of Energy, Office of Science, National Quantum Information Science
Research Centers, Superconducting Quantum Materials and Systems Center (SQMS)
under Contract No. DE-AC02-07CH11359. L.M. acknowledges support from the PNRR
MUR Project PE0000023-NQSTI and from the National Research Centre for HPC, Big
Data and Quantum Computing, PNRR MUR Project CN0000013-ICSC. Y.Z. thanks Jin-
Shi Xu for the discussions. C.R. was supported by the National Natural Science
Foundation of China (Grant No.12075245, 12247105), the Natural Science
Foundation of Hunan Province (2021JJ10033).

## References

  * (1) M. Lanzagorta, J. Uhlmann, Quantum radar. San Rafael, California (USA): Morgan and Claypool,66 (2012).
  * (2) J. F. Smith III, Quantum interferometer and radar theory based on N00N, M and M or linear combinations of entangled states, in Proc. Quantum information and computation VIII, New York, 770201 (2010)
  * (3) M. Malik, O. S. Magana-Loaiza, and R. W. Boyd, Quantum-secured imaging, Appl. Phy. Lett. 101, 241103 (2012)
  * (4) D. Luong, C. W. Sandbo Chang, A. M. Vadiraj, A. Damini, C. M. Wilson, B. Balaji, Receiver Operating Characteristics for a Prototype Quantum Two-Mode Squeezing Radar, IEEE Trans. Aerosp. Electr. Sys. 56, 3 (2020) and arXiv:1903.00101 (2019).
  * (5) S. Lloyd, Enhanced Sensitivity of Photodetection via Quantum Illumination, Science 321, 1463 (2008).
  * (6) S.-H. Tan, B.I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, J. H. Shapiro, Quantum Illumination with Gaussian States, Phys. Rev. Lett. 101, 253601 (2008).
  * (7) L. Maccone, C. Ren, “Quantum Radar”, Phys. Rev. Lett. 124, 200503 (2020).
  * (8) L. Maccone, C. Ren, Erratum, submitted (2023).
  * (9) V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced positioning and clock synchronization, Nature 412, 417 (2001).
  * (10) S.M. Barnett, C. Fabre, A. Maitre, Ultimate quantum limits for resolution of beam displacements, Eur. Phys. J. D 22, 513 (2003).
  * (11) S. L. Braunstein, C. M. Caves, G. J. Milburn, Generalized uncertainty relations: Theory, examples, and Lorentz invariance, Ann. Phys. 247, 135-173 (1996); S. L. Braunstein, C. M. Caves, Statistical distance and the geometry of quantum states. Phys. Rev. Lett. 72, 3439 (1994).
  * (12) V. Giovannetti, S. Lloyd, L. Maccone, Quantum metrology, Phys. Rev. Lett. 96, 010401 (2006).
  * (13) V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced measurements: beating the standard quantum limit, Science 306, 1330 (2004); V. Giovannetti, S. Lloyd, L. Maccone, Advances in quantum metrology, Nature Phot. 5, 222 (2011).
  * (14) M. G. A. Paris, Quantum estimation for quantum technology, Int. J. Quantum Inf. 7, 125 (2009).
  * (15) R. Demkowicz-Dobrzan̆ski, M. Jarzyna, J. Kodyn̆ski, Quantum Limits in Optical Interferometry, Progress in Optics, 60, 345 (2015) also at arXiv:1405.7703 (2014).
  * (16) R. Demkowicz-Dobrzanski, J. Koldynski, M. Guta, The elusive Heisenberg limit in quantum-enhanced metrology, Nature Comm. 3, 1063 (2012).
  * (17) B. M. Escher, R. L. de Matos Filho, L. Davidovich, General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology, Nature Phys. 7, 406 (2011).
  * (18) E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, Phys. Rev. Lett. 112, 150802 (2014); W. Dur, M. Skotiniotis, F. Fröwis, and B. Kraus, Phys. Rev. Lett. 112, 080801 (2014); G. Arrad, Y. Vinkler, D. Aharonov and A. Retzker, Phys. Rev. Lett. 112, 150801 (2014).
  * (19) R. Demkowicz-Dobrzański, L. Maccone, Using entanglement against noise in quantum metrology, Phys. Rev. Lett. 113, 250801 (2014).
  * (20) V. Giovannetti, S. Lloyd, and L. Maccone, Positioning and clock synchronization through entanglement, Phys. Rev. A 65, 022309 (2002).
  * (21) B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics (John Wiley & Sons 1991).
  * (22) V. Giovannetti, S. Lloyd, L. Maccone, J. H. Shapiro, Sub-Rayleigh-diffraction-bound quantum imaging, Phys. Rev. A 79, 013827 (2009).
  * (23) L. Mandel and E. Wolf Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge, 1995).
  * (24) J.D. Jackson, Classical Electrodynamics, Third Edition, (John Wiley & Sons, Inc., 1999)
  * (25) A. Einstein, On the present status of the radiation problem, Physicalische Zeitscrhift 10, 185 (1909); English translation in The Collected Papers of A. Einstein, Vol. 2. (Princeton Univ. Press, 1989).
  * (26) M. G. Raymer and B. J. Smith; A. Muthukrishnan, M. O. Scully and M. S. Zubairy, in The Nature of Light: What is a Photon?, edited by C. Roychoudhuri, K. Creath, A. F. Kracklauer, Proceedings of SPIE Vol. 5866 (SPIE, Bellingham, WA, 2005); A. Muthukrishnan, M. O. Scully and M. S. Zubairy, Optics and Photonics News, 14, S-18 (2003).
  * (27) I. Bialynicki-Birula, Z. Bialynicka-Birula, Why photons cannot be sharply localized, Phys. Rev. A 79, 032112 (2009).