Weak measurement
Encyclopedia
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
.
The idea of weak measurements and weak values, first developed by Yakir Aharonov
, David Albert
and Lev Vaidman
, published in 1988,
is especially useful for gaining information about pre- and post-selected systems described by the two state vector formalism. 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
and 
are the pre- and post-selected quantum mechanical states, the weak value of the observable  is defined as
The weak value of the observable becomes large when the post-selected state,
, approaches being orthogonal to the pre-selected state, 
. 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.
Note that the theory of weak measurement allows Hardy's paradox
to be explained.
In Hardy's paradox a positron and an electron go down both arms of each of their interferometers. If they meet in the overlapping arms, they should annihilate each other. But, strangely, they are
still registered as arriving at the detectors.
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. 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.
Building on weak mesurements, Howard M. Wiseman
proposed a weak values 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
for photons in the de Broglie-Bohm interpretation.
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.
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
.
The idea of weak measurements and weak values, first developed by Yakir Aharonov
Yakir Aharonov
Yakir Aharonov is an Israeli physicist specializing in quantum physics. He is a Professor of Theoretical Physics and the James J. Farley Professor of Natural Philosophy at Chapman University in California. He is also a distinguished professor in Perimeter Institute.He also serves as a professor...
, David Albert
David Albert
David Z Albert, Ph.D., is Frederick E. Woodbridge Professor of Philosophy and Director of the M.A. Program in The Philosophical Foundations of Physics at Columbia University in New York. He received his B.S. in physics from Columbia College and his doctorate in theoretical physics from The...
and Lev Vaidman
Lev Vaidman
Lev Vaidman is an Israeli physicist working at Tel Aviv University, Israel. He is noted for the Elitzur–Vaidman bomb-testing problem in quantum mechanics.- External links :* * *...
, published in 1988,
is especially useful for gaining information about pre- and post-selected systems described by the two state vector formalism. 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
and are the pre- and post-selected quantum mechanical states, the weak value of the observable  is defined asThe weak value of the observable becomes large when the post-selected state,
, approaches being orthogonal to the pre-selected state, . 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.Note that the theory of weak measurement allows Hardy's paradox
Hardy's paradox
Hardy's paradox is a thought experiment in quantum mechanics devised by Lucien Hardy in which a particle and its antiparticle may interact without annihilating each other...
to be explained.
In Hardy's paradox a positron and an electron go down both arms of each of their interferometers. If they meet in the overlapping arms, they should annihilate each other. But, strangely, they are
still registered as arriving at the detectors.
Related to this, the research group of Aephraim Steinberg at the University of Toronto
University of Toronto
The University of Toronto is a public research university in Toronto, Ontario, Canada, situated on the grounds that surround Queen's Park. It was founded by royal charter in 1827 as King's College, the first institution of higher learning in Upper Canada...
confirmed Hardy's paradox experimentally using joint weak measurement’ of the locations of entangled pairs of photons. 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.
Building on weak mesurements, Howard M. Wiseman
Howard M. Wiseman
Howard Mark Wiseman is a theoretical quantum physicist notable for his work on quantum feedback control, quantum measurements, quantum information, fundamental questions in quantum mechanics, and open quantum systems.- Early life :...
proposed a weak values 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
Partha Ghose
Partha Ghose, born 1939, is an Indian physicist, author, anchorperson and professor at the S.N. Bose National Centre for Basic Sciences in Calcutta.Partha Ghose is quoted as one of India's best known popularizers of modern science.Together with K...
for photons in the de Broglie-Bohm interpretation.
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.
Further reading
- Discover Magazine article: "Back From the Future" A series of quantum experiments shows that measurements performed in the future can influence the present. http://discovermagazine.com/2010/apr/01-back-from-the-future
- Stephen Parrott questions the meaning and usefulness of weak measurements, as described above.http://www.math.umb.edu/~sp/papers.html
- Quantum physics first: Researchers observe single photons in two-slit interferometer experiment: http://www.physorg.com/news/2011-06-quantum-physics-photons-two-slit-interferometer.html
- 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