J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: understanding quantum weak values: basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).

[CrossRef]

L. J. Salazar-Serrano, D. Janner, N. Brunner, V. Pruneri, and J. P. Torres, “Measurement of sub-pulse-width temporal delays via spectral interference induced by weak value amplification,” Phys. Rev. A 89, 012126 (2014).

[CrossRef]

I. Shomroni, O. Bechler, S. Rosenblum, and B. Dayan, “Demonstration of weak measurement based on atomic spontaneous emission,” Phys. Rev. Lett. 111, 023604 (2013).

[CrossRef]

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111, 033604 (2013).

[CrossRef]

S. Wu and M. Zukowski, “Feasible optical weak measurements of complementary observables via a single Hamiltonian,” Phys. Rev. Lett. 108, 080403 (2012).

[CrossRef]

G. Puentes, N. Hermosa, and J. P. Torres, “Weak measurements with orbital-angular-momentum pointer states,” Phys. Rev. Lett. 109, 040401 (2012).

[CrossRef]

H. Kobayashi, G. Puentes, and Y. Shikano, “Extracting joint weak values from two-dimensional spatial displacements,” Phys. Rev. A 86, 053805 (2012).

[CrossRef]

A. G. Kofman, S. Ashhab, and F. Nori, “Nonperturbative theory of weak pre- and post-selected measurements,” Phys. Rep. 520, 43–133 (2012).

[CrossRef]

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. USA 108, 1256–1261 (2011).

[CrossRef]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).

[CrossRef]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).

[CrossRef]

A. Feizpour, X. Xing, and A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).

[CrossRef]

R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, P. Hyllus, L. Pezzé, and A. Smerzi, “Useful multiparticle entanglement and sub-shot-noise sensitivity in experimental phase estimation,” Phys. Rev. Lett. 107, 080504 (2011).

[CrossRef]

D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, “Continuous phase amplification with a Sagnac interferometer,” Phys. Rev. A 82, 011802(R) (2010).

[CrossRef]

N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: weak measurements or standard interferometry?” Phys. Rev. Lett. 105, 010405 (2010).

[CrossRef]

J. Dressel, S. Agarwal, and A. N. Jordan, “Contextual values of observables in quantum measurements,” Phys. Rev. Lett. 104, 240401 (2010).

[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).

[CrossRef]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A 80, 041803(R) (2009).

[CrossRef]

In this case, we refer to pure states. For a discussion of weak measurements in the context of mixed states; see A. Di Lorenzo and J. C. Egues, “Weak measurement: effect of the detector dynamics,” Phys. Rev. A 77, 042108 (2008).

[CrossRef]

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).

[CrossRef]

K. J. Resch, “Amplifying a tiny optical effect,” Science 319, 733–734 (2008).

[CrossRef]

R. Jozsa, “Complex weak values in quantum measurements,” Phys. Rev. A 76, 044103 (2007).

[CrossRef]

J. S. Lundeen and K. J. Resch, “Practical measurement of joint weak values and their connection to the annihilation operator,” Phys. Lett. A 334, 337–344 (2005).

[CrossRef]

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92, 043601 (2004).

[CrossRef]

N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).

[CrossRef]

K. Resch, J. Lundeen, and A. Steinberg, “Experimental realization of the quantum box problem,” Phys. Lett. A 324, 125–131 (2004).

[CrossRef]

A. M. Steinberg, “How much time does a tunneling particle spend in the barrier region?” Phys. Rev. Lett. 74, 2405–2409 (1995).

[CrossRef]

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).

[CrossRef]

Y. Aharonov and L. Vaidman, “Properties of a quantum system during the time interval between two measurements,” Phys. Rev. A 41, 11–20 (1990).

[CrossRef]

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).

[CrossRef]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).

[CrossRef]

J. Dressel, S. Agarwal, and A. N. Jordan, “Contextual values of observables in quantum measurements,” Phys. Rev. Lett. 104, 240401 (2010).

[CrossRef]

Y. Aharonov and L. Vaidman, “Properties of a quantum system during the time interval between two measurements,” Phys. Rev. A 41, 11–20 (1990).

[CrossRef]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. USA 108, 1256–1261 (2011).

[CrossRef]

A. G. Kofman, S. Ashhab, and F. Nori, “Nonperturbative theory of weak pre- and post-selected measurements,” Phys. Rep. 520, 43–133 (2012).

[CrossRef]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).

[CrossRef]

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. USA 108, 1256–1261 (2011).

[CrossRef]

I. Shomroni, O. Bechler, S. Rosenblum, and B. Dayan, “Demonstration of weak measurement based on atomic spontaneous emission,” Phys. Rev. Lett. 111, 023604 (2013).

[CrossRef]

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: understanding quantum weak values: basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).

[CrossRef]

O. S. M. Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” arXiv:1312.2981 (2013).

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).

[CrossRef]

L. J. Salazar-Serrano, D. Janner, N. Brunner, V. Pruneri, and J. P. Torres, “Measurement of sub-pulse-width temporal delays via spectral interference induced by weak value amplification,” Phys. Rev. A 89, 012126 (2014).

[CrossRef]

N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: weak measurements or standard interferometry?” Phys. Rev. Lett. 105, 010405 (2010).

[CrossRef]

N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).

[CrossRef]

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92, 043601 (2004).

[CrossRef]

I. Shomroni, O. Bechler, S. Rosenblum, and B. Dayan, “Demonstration of weak measurement based on atomic spontaneous emission,” Phys. Rev. Lett. 111, 023604 (2013).

[CrossRef]

In this case, we refer to pure states. For a discussion of weak measurements in the context of mixed states; see A. Di Lorenzo and J. C. Egues, “Weak measurement: effect of the detector dynamics,” Phys. Rev. A 77, 042108 (2008).

[CrossRef]

D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, “Continuous phase amplification with a Sagnac interferometer,” Phys. Rev. A 82, 011802(R) (2010).

[CrossRef]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A 80, 041803(R) (2009).

[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).

[CrossRef]

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: understanding quantum weak values: basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).

[CrossRef]

J. Dressel, S. Agarwal, and A. N. Jordan, “Contextual values of observables in quantum measurements,” Phys. Rev. Lett. 104, 240401 (2010).

[CrossRef]

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).

[CrossRef]

In this case, we refer to pure states. For a discussion of weak measurements in the context of mixed states; see A. Di Lorenzo and J. C. Egues, “Weak measurement: effect of the detector dynamics,” Phys. Rev. A 77, 042108 (2008).

[CrossRef]

A. Feizpour, X. Xing, and A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).

[CrossRef]

M. Fox, Quantum Optics: An Introduction (Oxford University, 2006).

J. P. Gazeau, Coherent States in Quantum Mechanics (Wiley-VCH, 2009).

N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).

[CrossRef]

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).

[CrossRef]

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. USA 108, 1256–1261 (2011).

[CrossRef]

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111, 033604 (2013).

[CrossRef]

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987).

G. Puentes, N. Hermosa, and J. P. Torres, “Weak measurements with orbital-angular-momentum pointer states,” Phys. Rev. Lett. 109, 040401 (2012).

[CrossRef]

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92, 043601 (2004).

[CrossRef]

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).

[CrossRef]

D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, “Continuous phase amplification with a Sagnac interferometer,” Phys. Rev. A 82, 011802(R) (2010).

[CrossRef]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A 80, 041803(R) (2009).

[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).

[CrossRef]

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).

[CrossRef]

R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, P. Hyllus, L. Pezzé, and A. Smerzi, “Useful multiparticle entanglement and sub-shot-noise sensitivity in experimental phase estimation,” Phys. Rev. Lett. 107, 080504 (2011).

[CrossRef]

L. J. Salazar-Serrano, D. Janner, N. Brunner, V. Pruneri, and J. P. Torres, “Measurement of sub-pulse-width temporal delays via spectral interference induced by weak value amplification,” Phys. Rev. A 89, 012126 (2014).

[CrossRef]

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: understanding quantum weak values: basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).

[CrossRef]

J. Dressel, S. Agarwal, and A. N. Jordan, “Contextual values of observables in quantum measurements,” Phys. Rev. Lett. 104, 240401 (2010).

[CrossRef]

D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, “Continuous phase amplification with a Sagnac interferometer,” Phys. Rev. A 82, 011802(R) (2010).

[CrossRef]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A 80, 041803(R) (2009).

[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).

[CrossRef]

R. Jozsa, “Complex weak values in quantum measurements,” Phys. Rev. A 76, 044103 (2007).

[CrossRef]

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111, 033604 (2013).

[CrossRef]

H. Kobayashi, G. Puentes, and Y. Shikano, “Extracting joint weak values from two-dimensional spatial displacements,” Phys. Rev. A 86, 053805 (2012).

[CrossRef]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).

[CrossRef]

A. G. Kofman, S. Ashhab, and F. Nori, “Nonperturbative theory of weak pre- and post-selected measurements,” Phys. Rep. 520, 43–133 (2012).

[CrossRef]

R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, P. Hyllus, L. Pezzé, and A. Smerzi, “Useful multiparticle entanglement and sub-shot-noise sensitivity in experimental phase estimation,” Phys. Rev. Lett. 107, 080504 (2011).

[CrossRef]

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).

[CrossRef]

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. USA 108, 1256–1261 (2011).

[CrossRef]

N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).

[CrossRef]

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111, 033604 (2013).

[CrossRef]

O. S. M. Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” arXiv:1312.2981 (2013).

K. Resch, J. Lundeen, and A. Steinberg, “Experimental realization of the quantum box problem,” Phys. Lett. A 324, 125–131 (2004).

[CrossRef]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).

[CrossRef]

J. S. Lundeen and K. J. Resch, “Practical measurement of joint weak values and their connection to the annihilation operator,” Phys. Lett. A 334, 337–344 (2005).

[CrossRef]

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: understanding quantum weak values: basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).

[CrossRef]

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92, 043601 (2004).

[CrossRef]

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: understanding quantum weak values: basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).

[CrossRef]

O. S. M. Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” arXiv:1312.2981 (2013).

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).

[CrossRef]

A. G. Kofman, S. Ashhab, and F. Nori, “Nonperturbative theory of weak pre- and post-selected measurements,” Phys. Rep. 520, 43–133 (2012).

[CrossRef]

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. USA 108, 1256–1261 (2011).

[CrossRef]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).

[CrossRef]

R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, P. Hyllus, L. Pezzé, and A. Smerzi, “Useful multiparticle entanglement and sub-shot-noise sensitivity in experimental phase estimation,” Phys. Rev. Lett. 107, 080504 (2011).

[CrossRef]

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92, 043601 (2004).

[CrossRef]

L. J. Salazar-Serrano, D. Janner, N. Brunner, V. Pruneri, and J. P. Torres, “Measurement of sub-pulse-width temporal delays via spectral interference induced by weak value amplification,” Phys. Rev. A 89, 012126 (2014).

[CrossRef]

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. USA 108, 1256–1261 (2011).

[CrossRef]

G. Puentes, N. Hermosa, and J. P. Torres, “Weak measurements with orbital-angular-momentum pointer states,” Phys. Rev. Lett. 109, 040401 (2012).

[CrossRef]

H. Kobayashi, G. Puentes, and Y. Shikano, “Extracting joint weak values from two-dimensional spatial displacements,” Phys. Rev. A 86, 053805 (2012).

[CrossRef]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).

[CrossRef]

K. Resch, J. Lundeen, and A. Steinberg, “Experimental realization of the quantum box problem,” Phys. Lett. A 324, 125–131 (2004).

[CrossRef]

K. J. Resch, “Amplifying a tiny optical effect,” Science 319, 733–734 (2008).

[CrossRef]

J. S. Lundeen and K. J. Resch, “Practical measurement of joint weak values and their connection to the annihilation operator,” Phys. Lett. A 334, 337–344 (2005).

[CrossRef]

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).

[CrossRef]

O. S. M. Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” arXiv:1312.2981 (2013).

I. Shomroni, O. Bechler, S. Rosenblum, and B. Dayan, “Demonstration of weak measurement based on atomic spontaneous emission,” Phys. Rev. Lett. 111, 023604 (2013).

[CrossRef]

J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1985).

L. J. Salazar-Serrano, D. Janner, N. Brunner, V. Pruneri, and J. P. Torres, “Measurement of sub-pulse-width temporal delays via spectral interference induced by weak value amplification,” Phys. Rev. A 89, 012126 (2014).

[CrossRef]

N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).

[CrossRef]

R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, P. Hyllus, L. Pezzé, and A. Smerzi, “Useful multiparticle entanglement and sub-shot-noise sensitivity in experimental phase estimation,” Phys. Rev. Lett. 107, 080504 (2011).

[CrossRef]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).

[CrossRef]

H. Kobayashi, G. Puentes, and Y. Shikano, “Extracting joint weak values from two-dimensional spatial displacements,” Phys. Rev. A 86, 053805 (2012).

[CrossRef]

I. Shomroni, O. Bechler, S. Rosenblum, and B. Dayan, “Demonstration of weak measurement based on atomic spontaneous emission,” Phys. Rev. Lett. 111, 023604 (2013).

[CrossRef]

N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: weak measurements or standard interferometry?” Phys. Rev. Lett. 105, 010405 (2010).

[CrossRef]

R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, P. Hyllus, L. Pezzé, and A. Smerzi, “Useful multiparticle entanglement and sub-shot-noise sensitivity in experimental phase estimation,” Phys. Rev. Lett. 107, 080504 (2011).

[CrossRef]

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92, 043601 (2004).

[CrossRef]

D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, “Continuous phase amplification with a Sagnac interferometer,” Phys. Rev. A 82, 011802(R) (2010).

[CrossRef]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A 80, 041803(R) (2009).

[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).

[CrossRef]

K. Resch, J. Lundeen, and A. Steinberg, “Experimental realization of the quantum box problem,” Phys. Lett. A 324, 125–131 (2004).

[CrossRef]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).

[CrossRef]

A. Feizpour, X. Xing, and A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).

[CrossRef]

A. M. Steinberg, “How much time does a tunneling particle spend in the barrier region?” Phys. Rev. Lett. 74, 2405–2409 (1995).

[CrossRef]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).

[CrossRef]

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).

[CrossRef]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).

[CrossRef]

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).

[CrossRef]

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).

[CrossRef]

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111, 033604 (2013).

[CrossRef]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).

[CrossRef]

L. J. Salazar-Serrano, D. Janner, N. Brunner, V. Pruneri, and J. P. Torres, “Measurement of sub-pulse-width temporal delays via spectral interference induced by weak value amplification,” Phys. Rev. A 89, 012126 (2014).

[CrossRef]

G. Puentes, N. Hermosa, and J. P. Torres, “Weak measurements with orbital-angular-momentum pointer states,” Phys. Rev. Lett. 109, 040401 (2012).

[CrossRef]

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111, 033604 (2013).

[CrossRef]

Y. Aharonov and L. Vaidman, “Properties of a quantum system during the time interval between two measurements,” Phys. Rev. A 41, 11–20 (1990).

[CrossRef]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).

J. von Neuman, Mathematical Foundations of Quantum Mechanics (Princeton University, 1955).

N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).

[CrossRef]

R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, P. Hyllus, L. Pezzé, and A. Smerzi, “Useful multiparticle entanglement and sub-shot-noise sensitivity in experimental phase estimation,” Phys. Rev. Lett. 107, 080504 (2011).

[CrossRef]

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. USA 108, 1256–1261 (2011).

[CrossRef]

R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, P. Hyllus, L. Pezzé, and A. Smerzi, “Useful multiparticle entanglement and sub-shot-noise sensitivity in experimental phase estimation,” Phys. Rev. Lett. 107, 080504 (2011).

[CrossRef]

D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, “Continuous phase amplification with a Sagnac interferometer,” Phys. Rev. A 82, 011802(R) (2010).

[CrossRef]

S. Wu and M. Zukowski, “Feasible optical weak measurements of complementary observables via a single Hamiltonian,” Phys. Rev. Lett. 108, 080403 (2012).

[CrossRef]

A. Feizpour, X. Xing, and A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).

[CrossRef]

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111, 033604 (2013).

[CrossRef]

S. Wu and M. Zukowski, “Feasible optical weak measurements of complementary observables via a single Hamiltonian,” Phys. Rev. Lett. 108, 080403 (2012).

[CrossRef]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).

[CrossRef]

K. Resch, J. Lundeen, and A. Steinberg, “Experimental realization of the quantum box problem,” Phys. Lett. A 324, 125–131 (2004).

[CrossRef]

J. S. Lundeen and K. J. Resch, “Practical measurement of joint weak values and their connection to the annihilation operator,” Phys. Lett. A 334, 337–344 (2005).

[CrossRef]

A. G. Kofman, S. Ashhab, and F. Nori, “Nonperturbative theory of weak pre- and post-selected measurements,” Phys. Rep. 520, 43–133 (2012).

[CrossRef]

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).

[CrossRef]

In this case, we refer to pure states. For a discussion of weak measurements in the context of mixed states; see A. Di Lorenzo and J. C. Egues, “Weak measurement: effect of the detector dynamics,” Phys. Rev. A 77, 042108 (2008).

[CrossRef]

D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, “Continuous phase amplification with a Sagnac interferometer,” Phys. Rev. A 82, 011802(R) (2010).

[CrossRef]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A 80, 041803(R) (2009).

[CrossRef]

H. Kobayashi, G. Puentes, and Y. Shikano, “Extracting joint weak values from two-dimensional spatial displacements,” Phys. Rev. A 86, 053805 (2012).

[CrossRef]

L. J. Salazar-Serrano, D. Janner, N. Brunner, V. Pruneri, and J. P. Torres, “Measurement of sub-pulse-width temporal delays via spectral interference induced by weak value amplification,” Phys. Rev. A 89, 012126 (2014).

[CrossRef]

R. Jozsa, “Complex weak values in quantum measurements,” Phys. Rev. A 76, 044103 (2007).

[CrossRef]

Y. Aharonov and L. Vaidman, “Properties of a quantum system during the time interval between two measurements,” Phys. Rev. A 41, 11–20 (1990).

[CrossRef]

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).

[CrossRef]

A. M. Steinberg, “How much time does a tunneling particle spend in the barrier region?” Phys. Rev. Lett. 74, 2405–2409 (1995).

[CrossRef]

J. Dressel, S. Agarwal, and A. N. Jordan, “Contextual values of observables in quantum measurements,” Phys. Rev. Lett. 104, 240401 (2010).

[CrossRef]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).

[CrossRef]

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92, 043601 (2004).

[CrossRef]

N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).

[CrossRef]

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).

[CrossRef]

G. Puentes, N. Hermosa, and J. P. Torres, “Weak measurements with orbital-angular-momentum pointer states,” Phys. Rev. Lett. 109, 040401 (2012).

[CrossRef]

I. Shomroni, O. Bechler, S. Rosenblum, and B. Dayan, “Demonstration of weak measurement based on atomic spontaneous emission,” Phys. Rev. Lett. 111, 023604 (2013).

[CrossRef]

A. Feizpour, X. Xing, and A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).

[CrossRef]

S. Wu and M. Zukowski, “Feasible optical weak measurements of complementary observables via a single Hamiltonian,” Phys. Rev. Lett. 108, 080403 (2012).

[CrossRef]

N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: weak measurements or standard interferometry?” Phys. Rev. Lett. 105, 010405 (2010).

[CrossRef]

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111, 033604 (2013).

[CrossRef]

R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, P. Hyllus, L. Pezzé, and A. Smerzi, “Useful multiparticle entanglement and sub-shot-noise sensitivity in experimental phase estimation,” Phys. Rev. Lett. 107, 080504 (2011).

[CrossRef]

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. USA 108, 1256–1261 (2011).

[CrossRef]

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: understanding quantum weak values: basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).

[CrossRef]

K. J. Resch, “Amplifying a tiny optical effect,” Science 319, 733–734 (2008).

[CrossRef]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).

[CrossRef]

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).

[CrossRef]

J. von Neuman, Mathematical Foundations of Quantum Mechanics (Princeton University, 1955).

J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1985).

J. P. Gazeau, Coherent States in Quantum Mechanics (Wiley-VCH, 2009).

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987).

M. Fox, Quantum Optics: An Introduction (Oxford University, 2006).

O. S. M. Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” arXiv:1312.2981 (2013).