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Advances in photonic quantum sensing

Abstract

Quantum sensing has become a broad field. It is generally related with the idea of using quantum resources to boost the performance of a number of practical tasks, including the radar-like detection of faint objects, the readout of information from optical memories, and the optical resolution of extremely close point-like sources. Here, we first focus on the basic tools behind quantum sensing, discussing the most recent and general formulations for the problems of quantum parameter estimation and hypothesis testing. With this basic background in hand, we then review emerging applications of quantum sensing in the photonic regime both from a theoretical and experimental point of view. Besides the state of the art, we also discuss open problems and potential next steps.

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Fig. 1: Protocols for quantum estimation and discrimination.
Fig. 2: Technological applications of quantum channel discrimination.
Fig. 3: Experimental demonstrations of quantum reading and quantum illumination.
Fig. 4: Proof-of-principle experiments demonstrating a quantum detection scheme able to measure a distance of two incoherent point-like sources better than the Rayleigh limit.

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References

  1. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).

    MATH  Google Scholar 

  2. Hayashi, M. Quantum Information Theory: Mathematical Foundation (Springer-Verlag, Berlin, Heidelberg, 2017).

  3. Watrous, J. The Theory of Quantum Information (Cambridge University Press, Cambridge, 2018).

    MATH  Google Scholar 

  4. Andersen, U. L., Neergaard-Nielsen, J. S., van Loock, P. & Furusawa, A. Hybrid discrete- and continuous-variable quantum information. Nat. Phys. 11, 713–719 (2015).

    ADS  Google Scholar 

  5. Weedbrook, C. et al. Gaussian quantum information. Rev. Mod. Phys. 84, 621–669 (2012).

    ADS  Google Scholar 

  6. Braunstein, S. L. & Van Loock, P. Quantum information with continuous variables. Rev. Mod. Phys. 77, 513–577 (2005).

    ADS  MathSciNet  MATH  Google Scholar 

  7. Adesso, G., Ragy, S. & Lee, A. R. Continuous variable quantum information: Gaussian states and beyond. Open Syst. Inf. Dyn. 21, 1440001 (2014).

    MathSciNet  MATH  Google Scholar 

  8. Serafini, A. Quantum Continuous Variables: A Primer of Theoretical Methods (Taylor & Francis, Oxford, 2017).

    MATH  Google Scholar 

  9. Degen, C. L., Reinhard, F. & Cappellaro, P. Quantum sensing. Rev. Mod. Phys. 89, 035002 (2017).

    ADS  MathSciNet  Google Scholar 

  10. Braunstein, S. L. & Caves, C. M. Statistical distance and the geometry of quantum states. Phys. Rev. Lett. 72, 3439–3443 (1994).

    ADS  MathSciNet  MATH  Google Scholar 

  11. Braunstein, S. L., Caves, C. M. & Milburn, G. J. Generalized uncertainty relations: theory, examples, and Lorentz invariance. Ann. Phys. 247, 135–173 (1996).

    ADS  MathSciNet  MATH  Google Scholar 

  12. Giovannetti, V., Lloyd, S. & Maccone, L. Quantum-enhanced measurements: beating the standard quantum limit. Science 306, 1330–1336 (2004).

    ADS  Google Scholar 

  13. Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology. Nat. Photon. 5, 222–229 (2011).

    ADS  Google Scholar 

  14. Braun, D. et al. Quantum enhanced measurements without entanglement. Rev. Mod. Phys. 90, 035006 (2018).

    ADS  Google Scholar 

  15. Helstrom, C. W. Quantum Detection and Estimation Theory (Academic, New York, 1976).

    MATH  Google Scholar 

  16. Barnett, S. M. & Croke, S. Quantum state discrimination. Adv. Opt. Photon. 1, 238–278 (2009).

    ADS  Google Scholar 

  17. Audenaert, K. M. R. et al. Discriminating states: the quantum Chernoff bound. Phys. Rev. Lett. 98, 160501 (2007).

  18. Pirandola, S. & Lloyd, S. Computable bounds for the discrimination of Gaussian states. Phys. Rev. A 78, 012331 (2008).

    ADS  MathSciNet  MATH  Google Scholar 

  19. Hayashi, M. Discrimination of two channels by adaptive methods and its application to quantum system. IEEE Trans. Inf. Theory 55, 3807–3802 (2009).

    MathSciNet  MATH  Google Scholar 

  20. Harrow, A. W., Hassidim, A., Leung, D. W. & Watrous, J. Adaptive versus nonadaptive strategies for quantum channel discrimination. Phys. Rev. A 81, 032339 (2010).

    ADS  Google Scholar 

  21. Cooney, T., Mosonyi, M. & Wilde, M. M. Strong converse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication. Commun. Math. Phys. 344, 797–829 (2016).

    ADS  MathSciNet  MATH  Google Scholar 

  22. Giovannetti, V., Lloyd, S. & Maccone, L. Quantum metrology. Phys. Rev. Lett. 96, 010401 (2006).

    ADS  MathSciNet  Google Scholar 

  23. Pirandola, S. & Lupo, C. Ultimate precision of adaptive noise estimation. Phys. Rev. Lett. 118, 100502 (2017).

  24. Takeoka, M. & Wilde, M. M. Optimal estimation and discrimination of excess noise in thermal and amplifier channels. Preprint at https://arxiv.org/abs/arXiv:1611.09165 (2016).

  25. Zhou, S., Zhang, M., Preskill, J. & Jiang, L. Achieving the Heisenberg limit in quantum metrology using quantum error correction. Nat. Commun. 9, 78 (2018).

    ADS  Google Scholar 

  26. Demkowicz-Dobrzanski, R., Czajkowski, J. & Sekatski, P. Adaptive quantum metrology under general Markovian noise. Phys. Rev. X 7, 041009 (2017).

    Google Scholar 

  27. Cope, T. P. W. & Pirandola, S. Adaptive estimation and discrimination of Holevo-Werner channels. Quantum Meas. Quantum Metrol. 4, 44–52 (2017).

  28. Pirandola, S., Laurenza, R. & Lupo, C. Fundamental limits to quantum channel discrimination. Preprint at https://arxiv.org/abs/arXiv:1803.02834 (2018).

  29. Nielsen, M. A. & Chuang, I. L. Programmable quantum gate arrays. Phys. Rev. Lett. 79, 321–324 (1997).

    ADS  MathSciNet  MATH  Google Scholar 

  30. Ji, Z., Wang, G., Duan, R., Feng, Y. & Ying, M. Parameter estimation of quantum channels. IEEE Trans. Inf. Theory 54, 5172–5185 (2008).

    MathSciNet  MATH  Google Scholar 

  31. Demkowicz-Dobrzański, R. & Maccone, L. Using entanglement against noise in quantum metrology. Phys. Rev. Lett. 113, 250801 (2014).

    ADS  Google Scholar 

  32. Pirandola, S., Laurenza, R., Ottaviani, C. & Banchi, L. Fundamental limits of repeaterless quantum communications. Nat. Commun. 8, 15043 (2017).

    ADS  Google Scholar 

  33. Pirandola, S. et al. Theory of channel simulation and bounds for private communication. Quantum Sci. Technol. 3, 035009 (2018).

  34. Laurenza, R., Lupo, C., Spedalieri, G., Braunstein, S. L. & Pirandola, S. Channel simulation in quantum metrology. Quantum Meas. Quantum Metrol. 5, 1–12 (2018).

    ADS  Google Scholar 

  35. Pirandola, S. Quantum reading of a classical digital memory. Phys. Rev. Lett. 106, 090504 (2011).

    ADS  Google Scholar 

  36. Lupo, C., Pirandola, S., Giovannetti, V. & Mancini, S. Quantum reading capacity under thermal and correlated noise. Phys. Rev. A 87, 062310 (2013).

    ADS  Google Scholar 

  37. Spedalieri, G., Lupo, C., Mancini, S., Braunstein, S. L. & Pirandola, S. Quantum reading under a local energy constraint. Phys. Rev. A 86, 012315 (2012).

    ADS  Google Scholar 

  38. Spedalieri, G. Cryptographic aspects of quantum reading. Entropy 17, 2218–2227 (2015).

    ADS  MathSciNet  Google Scholar 

  39. Pirandola, S., Lupo, C., Giovannetti, V., Mancini, S. & Braunstein, S. L. Quantum reading capacity. New J. Phys. 13, 113012 (2011).

    ADS  Google Scholar 

  40. Lupo, C. & Pirandola, S. Super-additivity and entanglement assistance in quantum reading. Quantum Inf. Comput. 17, 0611–0622 (2017).

  41. Guha, S. & Shapiro, J. H. Reading boundless error-free bits using a single photon. Phys. Rev. A 87, 062306 (2013).

    ADS  Google Scholar 

  42. Guha, S., Dutton, Z., Nair, R., Shapiro, J. H. & Yen, B. Information capacity of quantum reading. In Conference on Laser Science XXVII Paper LTuF2 (OSA, 2011).

  43. Das, S. & Wilde, M. M. Quantum reading capacity: general definition and bounds. Preprint at https://arxiv.org/abs/arXiv:1703.03706 (2017).

  44. Nair, R. Discriminating quantum-optical beam-splitter channels with number-diagonal signal states: applications to quantum reading and target detection. Phys. Rev. A 84, 032312 (2011).

    ADS  Google Scholar 

  45. Nair, R. & Yen, B. J. Optimal quantum states for image sensing in loss. Phys. Rev. Lett. 107, 193602 (2011).

    ADS  Google Scholar 

  46. Hirota, O. Error free quantum reading by quasi Bell state of entangled coherent states. Quantum Meas. Quantum Metrol. 4, 70–73 (2017).

  47. Prabhu Tej, J., Usha Devi, A. R. & Rajagopal, A. K. Quantum reading of digital memory with non-Gaussian entangled light. Phys. Rev. A 87, 052308 (2013).

    ADS  Google Scholar 

  48. Bisio, A., Dall’Arno, M. & D’Ariano, G. M. Tradeoff between energy and error in the discrimination of quantum-optical devices. Phys. Rev. A 84, 012310 (2011).

    ADS  Google Scholar 

  49. Dall’Arno, M. et al. Experimental implementation of unambiguous quantum reading. Phys. Rev. A 85, 012308 (2012).

  50. Invernizzi, C., Paris, M. G. A. & Pirandola, S. Optimal detection of losses by thermal probes. Phys. Rev. A 84, 022334 (2011).

    ADS  Google Scholar 

  51. Dall’Arno, M., Bisio, A. & D’Ariano, G. M. Ideal quantum reading of optical memories. Int. J. Quantum Inf. 10, 1241010 (2012).

    ADS  MathSciNet  MATH  Google Scholar 

  52. Wilde, M. M., Guha, S., Tan, S.-H., & Lloyd, S. Explicit capacity-achieving receivers for optical communication and quantum reading. In Proc. 2012 IEEE Int. Symposium on Information Theory 551–555 (IEEE, 2012).

  53. Roga, W. & Buono, D. & Illuminati, F. Device-independent quantum reading and noise-assisted quantum transmitters. New J. Phys. 17, 013031 (2015).

    ADS  Google Scholar 

  54. Lloyd, S. Enhanced sensitivity of photodetection via quantum illumination. Science 321, 1463–1465 (2008).

    ADS  Google Scholar 

  55. Tan, S.-H. et al. Quantum illumination with Gaussian states. Phys. Rev. Lett. 101, 253601 (2008).

    ADS  Google Scholar 

  56. Shapiro, J. H. & Lloyd, S. Quantum illumination versus coherent-state target detection. New J. Phys. 11, 063045 (2009).

    ADS  Google Scholar 

  57. Zhuang, Q., Zhang, Z. & Shapiro, J. H. Optimum mixed-state discrimination for noisy entanglement-enhanced sensing. Phys. Rev. Lett. 118, 040801 (2017).

    ADS  Google Scholar 

  58. Zhuang, Z., Zhang, Z. & Shapiro, J. H. Entanglement-enhanced Neyman–Pearson target detection using quantum illumination. J. Opt. Soc. Am. B 34, 1567–1572 (2017).

    ADS  Google Scholar 

  59. Barzanjeh, Sh. et al. Microwave quantum illumination. Phys. Rev. Lett. 114, 080503 (2015).

  60. Guha, S. & Erkmen, B. I. Gaussian-state quantum-illumination receivers for target detection. Phys. Rev. A 80, 052310 (2009).

    ADS  Google Scholar 

  61. Xiong, B., Li, X., Wang, X.-Y. & Zhou, L. Improve microwave quantum illumination via optical parametric amplifier. Ann. Phys. 385, 757–768 (2017).

    ADS  MathSciNet  MATH  Google Scholar 

  62. Sanz, M., Las Heras, U., Garca-Ripoll, J. J., Solano, E. & Di Candia, R. Quantum estimation methods for quantum illumination. Phys. Rev. Lett. 118, 070803 (2017).

    ADS  Google Scholar 

  63. Weedbrook, C., Pirandola, S., Thompson, J., Vedral, V. & Gu, M. How discord underlies the noise resilience of quantum illumination. New J. Phys. 18, 043027 (2016).

    ADS  Google Scholar 

  64. Ragy, S. et al. Quantifying the source of enhancement in experimental continuous variable quantum illumination. J. Opt. Soc. Am. B 31, 2045–2050 (2014).

    ADS  Google Scholar 

  65. Wilde, M. M., Tomamichel, M., Lloyd, S. & Berta, M. Gaussian hypothesis testing and quantum illumination. Phys. Rev. Lett. 119, 120501 (2017).

    ADS  Google Scholar 

  66. De Palma, G. & Borregaard, J. The minimum error probability of quantum illumination. Phys. Rev. A 98, 012101 (2018).

    ADS  Google Scholar 

  67. Lopaeva, E. D. et al. Experimental realization of quantum illumination. Phys. Rev. Lett. 110, 153603 (2013).

    ADS  Google Scholar 

  68. Meda, A. et al. Photon-number correlation for quantum enhanced imaging and sensing. J. Opt. 19, 094002 (2017).

    ADS  Google Scholar 

  69. Zhang, Z., Tengner, M., Zhong, T., Wong, F. N. C. & Shapiro, J. H. Entanglement’s benefit survives an entanglement-breaking channel. Phys. Rev. Lett. 111, 010501 (2013).

    ADS  Google Scholar 

  70. Zhang, Z., Mouradian, S., Wong, F. N. C. & Shapiro, J. H. Entanglement-enhanced sensing in a lossy and noisy environment. Phys. Rev. Lett. 114, 110506 (2015).

    ADS  Google Scholar 

  71. Las Heras, U. et al. Quantum illumination reveals phase-shift inducing cloaking. Sci. Rep. 7, 9333 (2017).

  72. Tsang, M., Nair, R. & Lu, X.-M. Quantum theory of superresolution for two incoherent optical point sources. Phys. Rev. X 6, 031033 (2016).

    Google Scholar 

  73. Lupo, C. & Pirandola, S. Ultimate precision bound of quantum and subwavelength imaging. Phys. Rev. Lett. 117, 190802 (2016).

    ADS  Google Scholar 

  74. Nair, R. & Tsang, M. Far-field superresolution of thermal electromagnetic sources at the quantum limit. Phys. Rev. Lett. 117, 190801 (2016).

    ADS  Google Scholar 

  75. Kerviche, R., Guha, S. & Ashok, A. Fundamental limit of resolving two point sources limited by an arbitrary point spread function. Preprint at https://arxiv.org/abs/1701.04913 (2017).

  76. Rehacek, J., Paúr, M., Stoklasa, B., Hradil, Z. & Sánchez-Soto, L. L. Optimal measurements for resolution beyond the Rayleigh limit. Opt. Lett. 42, 231–234 (2017).

    ADS  Google Scholar 

  77. Yang, F., Nair, R., Tsang, M., Simon, C. & Lvovsky, A. I. Fisher information for far-field linear optical superresolution via homodyne or heterodyne detection in a higher-order local oscillator mode. Phys. Rev. A 96, 063829 (2017).

    ADS  Google Scholar 

  78. Lu, X.-M., Krovi, H., Nair, R., Guha, S. & Shapiro, J. H. Quantum-optimal detection of one-versus-two incoherent optical sources with arbitrary separation. Preprint at https://arxiv.org/abs/1802.02300 (2018).

  79. Tang, Z. S., Durak, K. & Ling, A. Fault-tolerant and finite-error localization for point emitters within the diffraction limit. Opt. Express 24, 22004–22012 (2016).

    ADS  Google Scholar 

  80. Nair, R. & Tsang, M. Interferometric superlocalization of two incoherent optical point sources. Opt. Express 24, 3684–3701 (2016).

    ADS  Google Scholar 

  81. Yang, F., Taschilina, A., Moiseev, E. S., Simon, C. & Lvovsky, A. I. Far-field linear optical superresolution via heterodyne detection in a higher-order local oscillator mode. Optica 3, 1148–1152 (2016).

    Google Scholar 

  82. Tham, W. K., Ferretti, H. & Steinberg, A. M. Beating Rayleigh’s curse by imaging using phase information. Phys. Rev. Lett. 118, 070801 (2017).

    ADS  MathSciNet  Google Scholar 

  83. Paúr, M., Stoklasa, B., Hradil, Z., Sánchez-Soto, L. L. & Rehacek, J. Achieving the ultimate optical resolution. Optica 3, 1144–1147 (2016).

    Google Scholar 

  84. Gatto Monticone, D. et al. Beating the Abbe diffraction limit in confocal microscopy via nonclassical photon statistics. Phys. Rev. Lett. 113, 143602 (2014).

    ADS  Google Scholar 

  85. Treps, N. et al. Surpassing the standard quantum limit for optical imaging using nonclassical multimode light. Phys. Rev. Lett. 88, 203601 (2014).

    ADS  Google Scholar 

  86. Classen, A. et al. Superresolving imaging of arbitrary one-dimensional arrays of thermal light sources using multiphoton interference. Phys. Rev. Lett. 117, 253601 (2016).

    ADS  Google Scholar 

  87. Tsang, M. Quantum imaging beyond the diffraction limit by optical centroid measurements. Phys. Rev. Lett. 102, 253601 (2009).

    ADS  Google Scholar 

  88. Rozema, L. A. et al. Scalable spatial superresolution using entangled photons. Phys. Rev. Lett. 112, 223602 (2014).

    ADS  Google Scholar 

  89. Chiribella, G., D’Ariano, G. M. & Perinotti, P. Quantum circuit architecture. Phys. Rev. Lett. 101, 060401 (2008).

    ADS  Google Scholar 

  90. Bennett, C. H. et al. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993).

    ADS  MathSciNet  MATH  Google Scholar 

  91. Ishizaka, S. & Hiroshima, T. Asymptotic teleportation scheme as a universal programmable quantum processor. Phys. Rev. Lett. 101, 240501 (2008).

    ADS  Google Scholar 

  92. Pirandola, S., Eisert, J., Weedbrook, C., Furusawa, A. & Braunstein, S. L. Advances in quantum teleportation. Nat. Photon. 9, 641–652 (2015).

    ADS  Google Scholar 

  93. Adesso, G., Dell’Anno, F., Siena, S. D., Illuminati, F. & Souza, L. A. M. Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states. Phys. Rev. A 79, 040305(R) (2009).

    ADS  Google Scholar 

  94. Monras, A. & Paris, M. G. A. Optimal quantum estimation of loss in bosonic channels. Phys. Rev. Lett. 98, 160401 (2007).

    ADS  Google Scholar 

  95. Whittaker, R. et al. Absorption spectroscopy at the ultimate quantum limit from single-photon states. New J. Phys. 19, 023013 (2017).

    ADS  Google Scholar 

  96. Moreau, P.-A. et al. Demonstrating an absolute quantum advantage in direct absorption measurement. Sci. Rep. 7, 6256 (2017).

    ADS  Google Scholar 

  97. Losero, E., Berchera, I. R., Meda, A., Avella, A. & Genovese, M. Unbiased estimation of an optical loss at the ultimate quantum limit with twin-beams. Sci. Rep. 8, 7431 (2018).

    ADS  Google Scholar 

  98. Samantaray, N., Berchera, I. R., Meda, M. & Genovese, M. Realization of the first sub-shot-noise wide field microscope. Light Sci. Appl. 6, e17005 (2017).

    Google Scholar 

  99. Brida, G., Genovese, M. & Berchera, I. R. Experimental realization of sub-shot-noise quantum imaging. Nat. Photon. 4, 227–230 (2010).

    ADS  Google Scholar 

  100. Abadie, J. et al. A gravitational wave observatory operating beyond the quantum shot-noise limit. Nat. Phys. 7, 962–965 (2011).

  101. Schnabel, R., Mavalvala, N., McClelland, D. E. & Lam, P. K. Quantum metrology for gravitational wave astronomy. Nat. Commun. 1, 121 (2010).

    ADS  Google Scholar 

  102. Banaszek, K., Demkowicz-Dobrzański, R. & Walmsley, I. A. Quantum states made to measure. Nat. Photon. 3, 673–676 (2009).

    ADS  Google Scholar 

  103. Nagata, T., Okamoto, R., O’Brien, J. L., Sasaki, K. & Takeuchi, S. Beating the standard quantum limit with four-entangled photons. Science 316, 726–729 (2007).

    ADS  Google Scholar 

  104. Slussarenko, S. et al. Unconditional violation of the shot-noise limit in photonic quantum metrology. Nat. Photon. 11, 700–703 (2017).

    ADS  Google Scholar 

  105. Dorner, U. et al. Optimal quantum phase estimation. Phys. Rev. Lett. 102, 040403 (2009).

    ADS  Google Scholar 

  106. Kacprowicz, M., Demkowicz-Dobrzański, R., Wasilewski, W., Banaszek, K. & Walmsley, I. A. Experimental quantum-enhanced estimation of a lossy phase shift. Nat. Photon 4, 357–360 (2010).

    ADS  Google Scholar 

  107. Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free Heisenberg-limited phase estimation. Nature 450, 393–396 (2007).

    ADS  Google Scholar 

  108. Yonezawa, H. et al. Quantum-enhanced optical phase tracking. Science 337, 1514–1517 (2012).

    ADS  MathSciNet  MATH  Google Scholar 

  109. Xiang, G. Y., Higgins, B. L., Berry, D. W., Wiseman, H. M. & Pryde, G. J. Entanglement-enhanced measurement of a completely unknown optical phase. Nat. Photon. 5, 43–47 (2011).

    ADS  Google Scholar 

  110. Berni, A. A. et al. Ab initio quantum-enhanced optical phase estimation using real-time feedback control. Nat. Photon. 9, 577–581 (2015).

    ADS  Google Scholar 

  111. Fuchs, C. A. & van de Graaf, J. Cryptographic distinguishability measures for quantum-mechanical states. IEEE Trans. Inf. Theory 45, 1216–1227 (1999).

    MathSciNet  MATH  Google Scholar 

  112. Banchi, L., Braunstein, S. L. & Pirandola, S. Quantum fidelity for arbitrary Gaussian states. Phys. Rev. Lett. 115, 260501 (2015).

    ADS  Google Scholar 

  113. Bose, S., Rallan, L. & Vedral, V. Communication capacity of quantum computation. Phys. Rev. Lett. 85, 5448–5451 (2000).

    ADS  Google Scholar 

  114. Barzanjeh, Sh., Abdi, M., Milburn, G. J., Tombesi, P. & Vitali, D. Reversible optical-to-microwave quantum interface. Phys. Rev. Lett. 109, 130503 (2012).

  115. Barzanjeh, Sh., Vitali, D., Tombesi, P. & Milburn, G. J. Entangling optical and microwave cavity modes by means of a nanomechanical resonator. Phys. Rev. A 84, 042342 (2011).

  116. Spedalieri, G. & Braunstein, S. L. Asymmetric quantum hypothesis testing with Gaussian states. Phys. Rev. A 90, 052307 (2014).

    ADS  Google Scholar 

  117. Berta, M., Hirche, C., Kaur, E. & Wilde, M. M. Amortized channel divergence for asymptotic quantum channel discrimination. Preprint at https://arxiv.org/abs/arXiv:1808.01498 (2018).

  118. Pirandola, S., Mancini, S., Lloyd, S. & Braunstein, S. L. Continuous-variable quantum cryptography using two-way quantum communication. Nat. Phys. 4, 726–730 (2008).

    ADS  Google Scholar 

  119. Novotny, L. & Hecht, B. Principles of Nano-Optics (Cambridge University Press, Cambridge, 2006).

  120. Pendry, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966–3969 (2000).

    ADS  Google Scholar 

  121. Liu, Z., Lee, H., Yi, X., Sun, C. & Zhang, X. Far-field optical hyperlens magnifying sub-diffraction-limited objects. Science 315, 1686 (2007).

    ADS  Google Scholar 

  122. Smolyaninov, I. I., Hung, Y.-J. & Davis, C. C. Magnifying superlens in the visible frequency range. Science 315, 1699–1701 (2007).

    ADS  Google Scholar 

  123. Hell, S. W. & Wichmann, J. Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy. Opt. Lett. 19, 780–782 (1994).

    ADS  Google Scholar 

  124. Hell, S. W. Far-field optical nanoscopy. Science 316, 1153–1158 (2007).

    ADS  Google Scholar 

  125. Betzig, E. et al. Imaging intracellular fluorescent proteins at nanometer resolution. Science 313, 1642–1645 (2006).

    ADS  Google Scholar 

  126. Small, A. & Stahlheber, S. Fluorophore localization algorithms for super-resolution microscopy. Nat. Methods 11, 267–279 (2014).

    Google Scholar 

  127. Tsai, M. J. & Dunn, K. P. Performance Limitations on Parameter Estimation of Closely Spaced Optical Targets Using Shot-Noise Detector Model Technical Report ADA073462 (Lincoln Laboratory, MIT, 1979).

  128. Bettens, E. et al. Model-based two-object resolution from observations having counting statistics. Ultramicroscopy 77, 37–48 (1999).

    Google Scholar 

  129. Ram, S., Ward, E. S. & Ober, R. J. Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy. Proc. Natl Acad. Sci. USA 103, 4457–4462 (2006).

    ADS  Google Scholar 

  130. Zhuang, Q., Zhang, Z. & Shapiro, J. H. Entanglement-enhanced lidars for simultaneous range and velocity measurements. Phys. Rev. A 96, 040304(R) (2017).

    ADS  Google Scholar 

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Acknowledgements

The authors would like to thank U. L. Andersen, L. Banchi, Sh. Barzanjeh, J. Borregaard, S. L. Braunstein, V. Giovannetti, S. Guha, C. Lupo, A. Lvovsky, M. Miková, M. Tsang and Z. Zhang for feedback. S.P. would like to specifically thank J. H. Shapiro and A. Farina for discussions on the experimental challenges related with a quantum radar, and R. Nair for the feedback on the experimental challenges in optical super-resolution. S.P. thanks support from the EPSRC via the ‘UK Quantum Communications Hub’ (EP/M013472/1). T.G. would like to acknowledge support from the Danish Research Council for Independent Research (Sapere Aude 4184-00338B) as well as the Innovation Fund Denmark (Qubiz) and the Danish National Research Foundation (Center for Macroscopic Quantum States, bigQ DNRF142).

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Pirandola, S., Bardhan, B.R., Gehring, T. et al. Advances in photonic quantum sensing. Nature Photon 12, 724–733 (2018). https://doi.org/10.1038/s41566-018-0301-6

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