# Weak measurement

In quantum mechanics (and computation), weak measurements are a type of quantum measurement, where the measured system is very weakly coupled to the measuring device. After the measurement the measuring device pointer is shifted by what is called the "weak value", so that a pointer initially pointing at zero before the measurement would point at the weak value after the measurement. The system is not disturbed by the measurement. Although this may seem to contradict some basic aspects of quantum theory, the formalism lies within the boundaries of the theory and does not contradict any fundamental concept, in particular not Heisenberg's uncertainty principle.[1]

The idea of weak measurements and weak values, first developed by Yakir Aharonov, David Albert and Lev Vaidman, published in 1988,[2] is especially useful for gaining information about pre- and post-selected systems described by the two-state vector formalism.[3] This was the original reason that Aharonov et al. developed weak measurement. Since a "strong" perturbative measurement can both upset the outcome of the post-selection and tamper with all subsequent measurement, weak nonperturbative measurements may be used to learn about such systems during their evolution.

If $|\phi_i\rangle$ and $|\phi_f\rangle$ are the pre- and post-selected quantum mechanical states, the weak value of the observable Â is defined as

$A_w = \frac{\langle\phi_f|\hat{\mathbf{A}}|\phi_i\rangle}{\langle\phi_f|\phi_i\rangle}.$

The weak value of the observable becomes large when the post-selected state, $|\phi_f\rangle$, approaches being orthogonal to the pre-selected state, $|\phi_i\rangle$.[1][4][5] In this way, by properly choosing the two states, the weak value of the operator can be made arbitrarily large, and otherwise small effects can be amplified.[6][7]

Related to this, the research group of Aephraim Steinberg at the University of Toronto confirmed Hardy's paradox experimentally using joint weak measurement of the locations of entangled pairs of photons.[8][9] Independently, a team of physicists from Japan reported in December, 2008, and published in March, 2009, that they were able to use joint weak measurement to observe a photonic version of Hardy's paradox. In this version, two photons were used instead of a positron and an electron and relied not upon non-annihilation but on polarization degrees of freedom values measured.[10]

Building on weak measurements, Howard M. Wiseman proposed a weak value measurement of the velocity of a quantum particle at a precise position, which he termed its "naïvely observable velocity". In 2010, a first experimental observation of trajectories of a photon in a double-slit interferometer was reported, which displayed the qualitative features predicted in 2001 by Partha Ghose[11] for photons in the de Broglie-Bohm interpretation.[12][13]

In 2011, weak measurements of many photons prepared in the same pure state, followed by strong measurements of a complementary variable, were used to reconstruct the state in which the photons were prepared.[14]

• Stephen Parrott questions the meaning and usefulness of weak measurements, as described above.[3]
• Adrian Cho: Furtive Approach Rolls Back the Limits of Quantum Uncertainty, Science, 5 August 2011, vol. 333, no. 6043, pp. 690–693, doi:10.1126/science.333.6043.690

## References

1. ^ a b Boaz Tamir and Eliahu Cohen (2013). "Introduction to Weak Measurements and Weak Values". Quanta 2 (1): 7–17. doi:10.12743/quanta.v2i1.14.
2. ^ Yakir Aharonov, David Z. Albert, and Lev Vaidman (1988). "How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100". Physical Review Letters 60 (14): 1351–1354. Bibcode:1988PhRvL..60.1351A. doi:10.1103/PhysRevLett.60.1351.
3. ^ Yakir Aharonov and Lev Vaidman (2007). "The Two-State Vector Formalism of Quantum Mechanics: an Updated Review". Lecture Notes in Physics 734: 399–447. arXiv:quant-ph/0105101. doi:10.1007/978-3-540-73473-4_13.
4. ^ Bengt E. Y. Svensson (2013). "Pedagogical Review of Quantum Measurement Theory with an Emphasis on Weak Measurements". Quanta 2 (1): 18–49. doi:10.12743/quanta.v2i1.12.
5. ^ Dmitri Sokolovski (2013). "Are the Weak Measurements Really Measurements?". Quanta 2 (1): 50–57. doi:10.12743/quanta.v2i1.15.
6. ^ O. Hosten and P. Kwiat (2008). "Observation of the spin Hall effect of light via weak measurements". Science 319 (5864): 787–790. Bibcode:2008Sci...319..787H. doi:10.1126/science.1152697.
7. ^ P. Ben Dixon, David J. Starling, Andrew N. Jordan, and John C. Howell (2009). "Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification". Physical Review Letters 102 (17): 173601. arXiv:0906.4828. Bibcode:2009PhRvL.102q3601D. doi:10.1103/PhysRevLett.102.173601.
8. ^ J. S. Lundeen and A. M. Steinberg (2009). "Experimental Joint Weak Measurement on a Photon Pair as a Probe of Hardy’s Paradox". Physical Review Letters 102 (2): 020404. arXiv:0810.4229. Bibcode:2009PhRvL.102b0404L. doi:10.1103/PhysRevLett.102.020404.
9. ^ "Hardy's paradox confirmed experimentally". Perimeter Institute for Theoretical Physics. July 2, 2009. Retrieved June 8, 2013.
10. ^ K. Yokota, T. Yamamoto, M. Koashi, N. Imoto, "Direct observation of Hardy's paradox by joint weak measurement with an entangled photon pair", New J. Phys. 11, 033011 (2009) [1]
11. ^ Partha Ghose, A.S. Majumdar, S. Guhab, J. Sau: Bohmian trajectories for photons, Physics Letters A 290 (2001), pp. 205–213, 10 November 2001
12. ^ Sacha Kocsis, Sylvain Ravets, Boris Braverman, Krister Shalm, Aephraim M. Steinberg: Observing the trajectories of a single photon using weak measurement, 19th Australian Institute of Physics (AIP) Congress, 2010 [2]
13. ^ Sacha Kocsis, Boris Braverman, Sylvain Ravets, Martin J. Stevens, Richard P. Mirin, L. Krister Shalm, Aephraim M. Steinberg: Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer, Science, vol. 332 no. 6034 pp.&nbsp:1170-1173, 3 June 2011, doi:10.1126/science.1202218 (abstract)
14. ^ Jeff S. Lundeen, Brandon Sutherland, Aabid Patel, Corey Stewart, Charles Bamber: Direct measurement of the quantum wavefunction, Nature vol. 474, pp. 188–191, 9. June 2011, doi:10.1038/nature10120