A paradigm for quantum-enhanced precision metrology is found in optical interferometry1, which is capable of sensing diverse physical quantities through measurement of a phase shift. When standard light sources are used, the precision of the phase determination is limited by shot noise, the origin of which can be traced to the random manner in which individual photons emerge from the interferometer. Quantum entanglement provides a means to exceed this limit2,3,4,5,6 with the celebrated example of N00N states7,8,9,10, which saturate the ultimate Heisenberg limit on precision11, but are extremely fragile to losses12,13,14. In contrast, we present experimental evidence that appropriately engineered quantum states15 outperform both standard and N00N states in the precision of phase estimation when losses are present. This shows that the quantum enhancement of metrology is possible even when decoherence is present, and that the strategy for realizing the enhancement is quite distinct from protecting quantum information encoded in light16,17.
Subscribe to Journal
Get full journal access for 1 year
only $15.58 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Hariharan, P. Optical Interferometry, 2nd edn (Elsevier, 2003).
Caves, C. M. Quantum-mechanical noise in an interferometer. Phys. Rev. D 23, 1693–1708 (1981).
Grangier, P., Slusher, R. E., Yurke, B. & LaPorta, A. Squeezed-light-enhanced polarization interferometer. Phys. Rev. Lett. 59, 2153–2156 (1987).
Xiao, M., Wu, L.-A. & Kimble, H. J. Precision measurement beyond the shot-noise limit. Phys. Rev. Lett. 59, 278–281 (1987).
Holland, M. J. & Burnett, K. Interferometric detection of optical phase shifts at the Heisenberg limit. Phys. Rev. Lett. 71, 1355–1358 (1993).
Sanders, B. C. & Milburn, G. J. Optimal quantum measurements for phase estimation. Phys. Rev. Lett. 75, 2944–2947 (1995).
Bollinger, J. J., Itano, W. M., Wineland, D. J. & Heinzen, D. J. Optimal frequency measurements with maximally correlated states. Phys. Rev. A 54, R4649–R4652 (1996).
Dowling, J. P. Correlated input-port, matter-wave interferometer: quantum-noise limits to the atom-laser gyroscope. Phys. Rev. A 57, 4736–4746 (1998).
Walther, P. et al. De Broglie wavelength of a non-local four-photon state. Nature 429, 158–161 (2004).
Mitchell, M. W., Lundeen, J. S. & Steinberg, A. M. Super-resolving phase measurements with a multiphoton entangled state. Nature 429, 161–164 (2004).
Giovannetti, V., Lloyd, S. & Maccone, L. Quantum-enhanced measurements: beating the standard quantum limit. Science 306, 1330–1336 (2004).
Huelga, S. F. et al. Improvement of frequency standards with quantum entanglement. Phys. Rev. Lett. 79, 3865–3868 (1997).
Rubin, M. A. & Kaushik, S. Loss-induced limits to phase measurement precision with maximally entangled states. Phys. Rev. A 75, 053805 (2007).
Gilbert, G., Hamrick, M. & Weinstein, Y. S. Use of maximally entangled N-photon states for practical quantum interferometry. J. Opt. Soc. Am. B 25, 1336–1340 (2008).
Dorner, U. et al. Optimal quantum phase estimation. Phys. Rev. Lett. 102, 040403 (2009).
Bourennane, M. et al. Decoherence-free quantum information processing with four-photon entangled states. Phys. Rev. Lett. 92, 107901 (2004).
Lu, C.-Y. et al. Experimental quantum coding against qubit loss error. Proc. Natl Acad. Sci. USA 105, 11050–11054 (2008).
Mandel, L., Sudarshan, E. C. G. & Wolf, E. Theory of photoelectric detection of light fluctuations. Proc. Phys. Soc. 84, 435–444 (1964).
Demkowicz-Dobrzański R. et al. Quantum phase estimation with lossy interferometers. Phys. Rev. A 80, 013825 (2009).
Helstrom, C. W. Quantum Detection and Estimation Theory (Academic Press, 1976).
Braunstein, S. L., Lane, A. S. & Caves, C. M. Maximum-likelihood analysis of multiple quantum phase measurements. Phys. Rev. Lett. 69, 2153–2156 (1992).
Huver, S. D., Wildfeuer, C. F. & Dowling J. P. Entangled Fock states for robust quantum optical metrology, imaging, and sensing. Phys. Rev. A 78, 063828 (2008).
Hong, C. K., Ou, Z. Y. & Mandel, L. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59, 2044–2046 (1987).
Rarity, J. G. et al. Two-photon interference in a Mach–Zehnder interferometer. Phys. Rev. Lett. 65, 1348–1351 (1990).
Grangier, P., Levenson, J. A. & Poizat, J.-Ph. Quantum non-demolition measurements in optics. Nature 396, 537–542 (1998).
Ralph, T. C., Langford, N. K., Bell, T. B. & White, A. G. Linear optical controlled-NOT gate in the coincidence basis. Phys. Rev. A 65, 062324 (2002).
O'Brien, J. L., Pryde, G. J., White, A. G., Ralph, T. C. & Branning, D. Demonstration of an all-optical quantum controlled-NOT gate. Nature 426, 264–267 (2003).
Kay, S. M. Fundamentals of Statistical Processing, Vol. I: Estimation Theory (Prentice Hall, 1993).
Delaubert, V. et al. Quantum limits in image processing. Europhys. Lett. 81, 44001 (2008).
Lamine, B., Fabre, C. & Treps, N. Quantum improvement of time transfer between remote clocks. Phys. Rev. Lett. 101, 123601 (2008).
The authors acknowledge insightful discussions with C.M. Caves, P.G. Kwiat and K.J. Resch. This work was supported by the EU 6th Framework Programme Integrated Project Qubit Applications (contract no. 015848), the Polish Ministry of Science and Higher Education (grant no. N N202 1489 33), the Engineering and Physical Sciences Research Council (EPSRC) (grant no. EP/C546237/1) and the Royal Society.
The authors declare no competing financial interests.
About this article
Cite this article
Kacprowicz, M., Demkowicz-Dobrzański, R., Wasilewski, W. et al. Experimental quantum-enhanced estimation of a lossy phase shift. Nature Photon 4, 357–360 (2010). https://doi.org/10.1038/nphoton.2010.39
Physical Review Applied (2020)
Super-resolved angular displacement estimation based upon a Sagnac interferometer and parity measurement
Optics Express (2020)
A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imaging
Physics Letters A (2020)
AVS Quantum Science (2020)
Physical Review Letters (2020)