Weak measurement

In quantum mechanics (and computation & information), weak measurements are a type of quantum measurement that results in an observer obtaining very little information about the system on average, but also disturbs the state very little.[1] From Busch's theorem[2] the system is necessarily disturbed by the measurement. In the literature weak measurements are also known as unsharp,[3] fuzzy,[3][4] dull, noisy,[5] approximate, and gentle[6] measurements. Additionally weak measurements are often confused with the distinct but related concept of the weak value.[7]

## Contents

* 1 History
* 2 Mathematics
* 2.1 Weak interaction and ancilla coupled measurement
* 2.2 Example Kraus operators
* 3 Information-gain–disturbance tradeoff
* 4 Applications
* 5 Further reading
* 6 References

## History

Weak measurements were first thought about in the context of weak continuous measurements of quantum systems[8] (i.e. quantum filtering and quantum trajectories). The physics of continuous quantum measurements is as follows. Consider using an ancilla, e.g. a field or a current, to probe a quantum system. The interaction between the system and the probe correlates the two systems. Typically the interaction only weakly correlates the system and ancilla (specifically, the interaction unitary need only to be expanded to first or second order in perturbation theory). By measuring the ancilla and then using quantum measurement theory, the state of the system conditioned on the results of the measurement can be determined. In order to obtain a strong measurement, many ancilla must be coupled and then measured. In the limit where there is a continuum of ancilla the measurement process becomes continuous in time. This process was described first by: Mensky;[9][10] Belavkin;[11][12] Barchielli, Lanz, Prosperi;[13] Barchielli;[14] Caves;[15][16] Caves and Milburn.[17] Later on Howard Carmichael[18] and Howard M. Wiseman[19] also made important contributions to the field.

The notion of a weak measurement is often misattributed to Aharonov, Albert and Vaidman.[7] In their article they consider an example of a weak measurement (and perhaps coin the phrase "weak measurement") and use it to motivate their definition of a weak value, which they defined there for the first time.

## Mathematics

There is no universally accepted definition of a weak measurement. One approach is to declare a weak measurement to be a generalized measurement where some or all of the Kraus operators are close to the identity.[20] The approach taken below is to interact two systems weakly and then measure one of them.[21] After detailing this approach we will illustrate it with examples.

### Weak interaction and ancilla coupled measurement

Consider a system that starts in the quantum state [math]\displaystyle{ |\psi\rangle }[/math] and an ancilla that starts in the state [math]\displaystyle{ |\phi\rangle }[/math], the combined initial state is [math]\displaystyle{ |\Psi\rangle = |\psi\rangle \otimes |\phi\rangle }[/math]. These two systems interact via the Hamiltonian [math]\displaystyle{ H = A \otimes B }[/math], which generates the time evolutions [math]\displaystyle{ U(t) = \exp[-ixtH] }[/math] (in units where [math]\displaystyle{ \hbar = 1 }[/math]), where [math]\displaystyle{ x }[/math] is the "interaction strength", which has units of inverse time. Assume a fixed interaction time [math]\displaystyle{ t = \Delta t }[/math] and that [math]\displaystyle{ \lambda = x \Delta t }[/math] is small, such that [math]\displaystyle{ \lambda^3 \approx 0 }[/math]. A series expansion of [math]\displaystyle{ U }[/math] in [math]\displaystyle{ \lambda }[/math] gives

[math]\displaystyle{ \begin{align} U &= I \otimes I - i\lambda H - \frac 1 2 \lambda^2 H^2 + O(\lambda^3) \\\ &\approx I \otimes I - i\lambda A \otimes B - \frac 1 2 \lambda^2 A^2 \otimes B^2. \end{align} }[/math]

Because it was only necessary to expand the unitary to a low order in perturbation theory, we call this is a weak interaction. Further, the fact that the unitary is predominately the identity operator, as [math]\displaystyle{ \lambda }[/math] and [math]\displaystyle{ \lambda^2 }[/math] are small, implies that the state after the interaction is not radically different from the initial state. The combined state of the system after interaction is

[math]\displaystyle{ |\Psi'\rangle = \left(I \otimes I - i\lambda A \otimes B - \frac 1 2 \lambda^2 A^2 \otimes B^2\right) |\Psi\rangle. }[/math]

Now we perform a measurement on the ancilla to find out about the system, this is known as an ancilla-coupled measurement. We will consider measurements in a basis [math]\displaystyle{ |q\rangle }[/math] (on the ancilla system) such that [math]\displaystyle{ \sum_q |q\rangle \langle q| = I }[/math]. The measurement's action on both systems is described by the action of the projectors [math]\displaystyle{ \Pi_q = I \otimes |q\rangle \langle q| }[/math] on the joint state [math]\displaystyle{ |\Psi'\rangle }[/math]. From quantum measurement theory we know the conditional state after the measurement is

[math]\displaystyle{ \begin{align} |\Psi_q\rangle &= \frac{\Pi_q |\Psi'\rangle}{\sqrt{\langle\Psi'| \Pi_q |\Psi'\rangle}} \\\ &= \frac{I \langle q|\phi\rangle - i\lambda A \langle q| B |\phi\rangle - \frac 1 2 \lambda^2 A^2 \langle q| B^2 |\phi\rangle}{\mathcal N} |\psi\rangle \otimes |q\rangle, \end{align} }[/math]

where [math]\displaystyle{ \mathcal N = \sqrt{\langle\Psi'| \Pi_q |\Psi'\rangle} }[/math] is a normalization factor for the wavefunction. Notice the ancilla system state records the outcome of the measurement. The object [math]\displaystyle{ M_q := I \langle q|\phi\rangle - i\lambda A \langle q| B |\phi\rangle - \frac 1 2 \lambda^2 A^2 \langle q| B^2 |\phi\rangle }[/math] is an operator on the system Hilbert space and is called a Kraus operator.

With respect to the Kraus operators the post-measurement state of the combined system is

[math]\displaystyle{ |\Psi_q\rangle = \frac{M_q |\psi\rangle}{\sqrt{\langle \psi|M_q^\dagger M_q|\psi\rangle}} \otimes |q\rangle. }[/math]

The objects [math]\displaystyle{ E_q = M_q^\dagger M_q }[/math] are elements of what is called a POVM and must obey [math]\displaystyle{ \sum_q E_q = I }[/math] so that the corresponding probabilities sum to unity: [math]\displaystyle{ \sum_q \Pr(q|\psi) = \sum_q \langle\psi| E_q |\psi\rangle = 1 }[/math]. As the ancilla system is no longer correlated with the primary system, it is simply recording the outcome of the measurement, we can trace over it. Doing so gives the conditional state of the primary system alone:

[math]\displaystyle{ |\psi_q\rangle = \frac{M_q |\psi\rangle}{\sqrt{\langle \psi|M_q^\dagger M_q|\psi\rangle}}, }[/math]

which we still label by the outcome of the measurement [math]\displaystyle{ q }[/math]. Indeed, these considerations allow one to derive a quantum trajectory.

### Example Kraus operators

We will use the canonical example of Gaussian Kraus operators given by Barchielli, Lanz, Prosperi;[13] and Caves and Milburn.[17] Take [math]\displaystyle{ H = x \otimes p }[/math], where the position and momentum on both systems have the usual Canonical commutation relation [math]\displaystyle{ [x, p] = i }[/math]. Take the initial wavefunction of the ancilla to have a Gaussian distribution

[math]\displaystyle{ |\Phi\rangle = \frac{1}{(2\pi\sigma^2)^{1/4}} \int dq' \exp[-q'^2/(4\sigma^2)] |q'\rangle. }[/math]

The position wavefunction of the ancilla is

[math]\displaystyle{ \Phi(q) = \langle q|\Phi\rangle = \frac{1}{(2\pi\sigma^2)^{1/4}} \exp[-q^2/(4\sigma^2)]. }[/math]

The Kraus operators are (compared to the discussion above, we set [math]\displaystyle{ \lambda = 1 }[/math])

[math]\displaystyle{ \begin{align} M(q) &= \langle q| \exp[-ix \otimes p] |\Phi\rangle \\\ &= \frac{1}{(2\pi\sigma^2)^{1/4}} \exp[-(q - x)^2/(4\sigma^2)], \end{align} }[/math]

while the corresponding POVM elements are

[math]\displaystyle{ \begin{align} E(q) &= M_q^\dagger M_q \\\ &= \frac{1}{\sqrt{2\pi\sigma^2}} \exp[-(q - x)^2/(2\sigma^2)], \end{align} }[/math]

which obey [math]\displaystyle{ \int dq\, E(q) = I }[/math]. An alternative representation is often seen in the literature. Using the spectral representation of the position operator [math]\displaystyle{ x = \int x'dx' |x'\rangle \langle x'| }[/math], we can write

[math]\displaystyle{ \begin{align} M(q) &= \frac{1}{(2\pi\sigma^2)^{1/4}} \int dx' \exp[-(q - x')^2/(4\sigma^2)] |x'\rangle \langle x'|, \\\ E(q) &= \frac{1}{\sqrt{2\pi\sigma^2}} \int dx' \exp[-(q - x')^2/(2\sigma^2)] |x'\rangle \langle x'|. \end{align} }[/math]

Notice that [math]\displaystyle{ \lim_{\sigma \to 0} E(q) = |x = q\rangle \langle x = q| }[/math].[17] That is, in a particular limit these operators limit to a strong measurement of position; for other values of [math]\displaystyle{ \sigma }[/math] we refer to the measurement as finite-strength; and as [math]\displaystyle{ \sigma \to \infty }[/math], we say the measurement is weak.

## Information-gain–disturbance tradeoff

As stated above, Busch's theorem[2] prevents a free lunch: there can be no information gain without disturbance. However, the tradeoff between information gain and disturbance has been characterized by many authors, including Fuchs and Peres;[22] Fuchs;[23] Fuchs and Jacobs;[24] and Banaszek.[25]

Recently the information-gain–disturbance tradeoff relation has been examined in the context of what is called the "gentle-measurement lemma".[6][26]

## Applications

Since the early days it has been clear that the primary use of weak measurement would be for feedback control or adaptive measurements of quantum systems. Indeed, this motivated much of Belavkin's work, and an explicit example was given by Caves and Milburn. An early application of an adaptive weak measurements was that of Dolinar's receiver,[27] which has been realized experimentally.[28][29] Another interesting application of weak measurements is to use weak measurements followed by a unitary, possibly conditional on the weak measurement result, to synthesize other generalized measurements.[20] Wiseman and Milburn's book[21] is a good reference for many of the modern developments.

## Further reading

* Brun's article[1]
* Jacobs and Steck's article[30]
* Quantum Measurement Theory and its Applications, K. Jacobs (Cambridge Press, 2014) ISBN:9781107025486
* Quantum Measurement and Control, H. M. Wiseman and G. J. Milburn (Cambridge Press, 2009)[21]
* Tamir and Cohen's article[31]

## References

1. ↑ 1.0 1.1 Todd A Brun (2002). "A simple model of quantum trajectories". Am. J. Phys. 70 (7): 719–737. doi:10.1119/1.1475328. Bibcode: 2002AmJPh..70..719B.
2. ↑ 2.0 2.1 Paul Busch (2009). J. Christian. ed. "No Information Without Disturbance": Quantum Limitations of Measurement. Invited contribution, "Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle: An International Conference in Honour of Abner Shimony", Perimeter Institute, Waterloo, Ontario, Canada, July 18–21, 2006. 73. Springer-Verlag, 2008. 229–256. doi:10.1007/978-1-4020-9107-0. ISBN 978-1-4020-9106-3.
3. ↑ 3.0 3.1 Stan Gudder (2005). "Non-disturbance for fuzzy quantum measurements". Fuzzy Sets and Systems. Fuzzy Sets and Systems, Volume 155, Issue 1, Pages 1-164 (1 October 2005) Measures and conditioning, Measures and conditioning 155: 18–25. doi:10.1016/j.fss.2005.05.009.
4. ↑ Asher Peres (1993). Quantum Theory, Concepts and Methods. Kluwer. p. 387. ISBN 978-0-7923-2549-9.
5. ↑ A. N. Korotkov (2009). Y. v. Nazarov. ed. Quantum Noise in Mesoscopic Physics. Quantum Noise in Mesoscopic Physics. Springer Netherlands. pp. 205–228. doi:10.1007/978-94-010-0089-5_10. ISBN 978-1-4020-1240-2. https://archive.org/details/quantumnoisemeso00btti.
6. ↑ 6.0 6.1 A. Winter (1999). "Coding Theorem and Strong Converse for Quantum Channels". IEEE Trans. Inf. Theory 45 (7): 2481–2485. doi:10.1109/18.796385.
7. ↑ 7.0 7.1 Yakir Aharonov; David Z. Albert; 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. doi:10.1103/PhysRevLett.60.1351. PMID 10038016. Bibcode: 1988PhRvL..60.1351A. https://semanticscholar.org/paper/224a2054222418f3b9e00383add02d8e2189a22a.
8. ↑ A. Clerk; M. Devoret; S. Girvin; F. Marquardt; R. Schoelkopf (2010). "Introduction to quantum noise, measurement, and amplification". Rev. Mod. Phys. 82 (2): 1155–1208. doi:10.1103/RevModPhys.82.1155. Bibcode: 2010RvMP...82.1155C.
9. ↑ M. B. Mensky (1979). "Quantum restrictions for continuous observation of an oscillator". Phys. Rev. D 20 (2): 384–387. doi:10.1103/PhysRevD.20.384. Bibcode: 1979PhRvD..20..384M.
10. ↑ M. B. Menskii (1979). "Quantum restrictions on the measurement of the parameters of motion of a macroscopic oscillator". Zhurnal Éksperimental'noĭ i Teoreticheskoĭ Fiziki 77 (4): 1326–1339. Bibcode: 1979JETP...50..667M. http://www.jetp.ac.ru/cgi-bin/e/index/r/77/4/p1326?a=list.
11. ↑ V. P. Belavkin (1980). "Quantum filtering of Markov signals with white quantum noise". Radiotechnika I Electronika 25: 1445–1453.
12. ↑ V. P. Belavkin (1992). "Quantum continual measurements and a posteriori collapse on CCR". Commun. Math. Phys. 146 (3): 611–635. doi:10.1007/bf02097018. Bibcode: 1992CMaPh.146..611B.
13. ↑ 13.0 13.1 A. Barchielli; L. Lanz; G. M. Prosperi (1982). "A model for the macroscopic description and continual observations in quantum mechanics". Il Nuovo Cimento B 72 (1): 79–121. doi:10.1007/BF02894935. Bibcode: 1982NCimB..72...79B.
14. ↑ A. Barchielli (1986). "Measurement theory and stochastic differential equations in quantum mechanics". Phys. Rev. A 34 (3): 1642–1649. doi:10.1103/PhysRevA.34.1642. PMID 9897442. Bibcode: 1986PhRvA..34.1642B.
15. ↑ Carlton M. Caves (1986). "Quantum mechanics of measurements distributed in time. A path-integral formulation". Phys. Rev. D 33 (6): 1643–1665. doi:10.1103/PhysRevD.33.1643. PMID 9956814. Bibcode: 1986PhRvD..33.1643C.
16. ↑ Carlton M. Caves (1987). "Quantum mechanics of measurements distributed in time. II. Connections among formulations". Phys. Rev. D 35 (6): 1815–1830. doi:10.1103/PhysRevD.35.1815. PMID 9957858. Bibcode: 1987PhRvD..35.1815C.
17. ↑ 17.0 17.1 17.2 Carlton M. Caves; G. J. Milburn (1987). "Quantum-mechanical model for continuous position measurements". Phys. Rev. A 36 (12): 5543–5555. doi:10.1103/PhysRevA.36.5543. PMID 9898842. Bibcode: 1987PhRvA..36.5543C. https://espace.library.uq.edu.au/view/UQ:247705/UQ247705_OA.pdf.
18. ↑ Carmichael, Howard (1993). An open systems approach to quantum optics, Lecture Notes in Physics. Springer.
19. ↑ Wiseman's thesis
20. ↑ 20.0 20.1 O. Oreshkov; T. A. Brun (2005). "Weak Measurements Are Universal". Phys. Rev. Lett. 95 (11): 110409. doi:10.1103/PhysRevLett.95.110409. PMID 16196989. Bibcode: 2005PhRvL..95k0409O.
21. ↑ 21.0 21.1 21.2 Wiseman, Howard M.; Milburn, Gerard J. (2009). Quantum Measurement and Control. Cambridge; New York City: Cambridge University Press. pp. 460. ISBN 978-0-521-80442-4. https://archive.org/details/quantummeasureme00wise.
22. ↑ C. A. Fuchs; A. Peres (1996). "Quantum-state disturbance versus information gain: Uncertainty relations for quantum information". Phys. Rev. A 53 (4): 2038–2045. doi:10.1103/PhysRevA.53.2038. PMID 9913105. Bibcode: 1996PhRvA..53.2038F.
23. ↑ C. A. Fuchs (1996). Information Gain vs. State Disturbance in Quantum Theory. Bibcode: 1996quant.ph.11010F.
24. ↑ C. A. Fuchs; K. A. Jacobs (2001). "Information-tradeoff relations for finite-strength quantum measurements". Phys. Rev. A 63 (6): 062305. doi:10.1103/PhysRevA.63.062305. Bibcode: 2001PhRvA..63f2305F.
25. ↑ K. Banaszek (2006). "Quantum-state disturbance versus information gain: Uncertainty relations for quantum information". Open Syst. Inf. Dyn. 13: 1–16. doi:10.1007/s11080-006-7263-8.
26. ↑ T. Ogawa; H. Nagaoka (1999). "Strong Converse to the Quantum Channel Coding Theorem". IEEE Trans. Inf. Theory 45 (7): 2486–2489. doi:10.1109/18.796386. Bibcode: 2002quant.ph..8139O.
27. ↑ S. J. Dolinar (1973). "An optimum receiver for the binary coherent state quantum channel". MIT Res. Lab. Electron. Quart. Progr. Rep. 111: 115–120.
28. ↑ R. L. Cook; P. J. Martin; J. M. Geremia (2007). "Optical coherent state discrimination using a closed-loop quantum measurement". Nature 446 (11): 774–777. doi:10.1038/nature05655. PMID 17429395. Bibcode: 2007Natur.446..774C.
29. ↑ F. E. Becerra; J. Fan; G. Baumgartner; J. Goldhar; J. T. Kosloski; A. Migdall (2013). "Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination". Nature Photonics 7 (11): 147–152. doi:10.1038/nphoton.2012.316. Bibcode: 2013NaPho...7..147B.
30. ↑ K. Jacobs; D. A. Steck (2006). "A straightforward introduction to continuous quantum measurement". Contemporary Physics 47 (5): 279–303. doi:10.1080/00107510601101934. Bibcode: 2006ConPh..47..279J.
31. ↑ Boaz Tamir; Eliahu Cohen (2013). "Introduction to Weak Measurements and Weak Values". Quanta 2 (1): 7–17. doi:10.12743/quanta.v2i1.14.