1. Strength in weakness: Foundations, trajectories and precision measurements
Quantum Foundations 2017 – Patna, India
Andrew N. Jordan
December 3-9, 2017
Funding gratefully
acknowledged:

2. Overview Quantum Foundations with WMs What is a weak measurement?
WM Leggett-Garg inequalities
Recent developments with Leggett-Garg inequalities.
Quantum Trajectories with Stochastic Path Integrals
Time sequences of weak measurements generates trajectories.
New approach using stochastic path integrals to find (a) global statistical characterization, (b) most likely paths, (c) correlation functions.
Comparisons to superconducting qubit experiments.
Time symmetry.
Precision measurements with WM
Using weak measurements + postselection as a filter.
Suppression of external noise sources, systematic noise, and correlated noise.

3. Flavors of Measurement
A projective (textbook) measurement is described as a system projector π 𝑗 , such that the probability of measurement outcome j on a system state ρ is 𝑃 𝑗 =Tr 𝜌 π 𝑗 , and π 𝑗 π 𝑘 = δ 𝑗𝑘 π 𝑗 .
A generalized measurement is described with a positive operator 𝐸 𝑗 , such that the probability of measurement outcome j on a system state ρ is 𝑃 𝑗 =Tr 𝜌 E 𝑗 , now however, since 𝐸 𝑗 is no longer a projector, the number of j’s can be (possibly much) larger than the dimension of the system.
A weak measurement is one such that E 𝑗 = 𝑝 𝑗 1+ 𝜖 𝑐 𝑗 𝐴 + …, so that 𝑃 𝑗 = 𝑝 𝑗 + 𝜖 𝑐 𝑗 Tr 𝜌 A + …, where 𝜖 is the “strength” of the measurement, 𝑐 𝑗 are constants, and A is a system operator. This is accomplished by weak coupling between a system and a meter.

4. Generalized Quantum Measurement
Measurement backaction on the state must be taken care of in all these cases. In the generalized measurement case, we have,
𝜌→ 𝜌′ 𝑗 = 𝑀 𝑗 𝜌 𝑀 𝑗 † Tr 𝑀 𝑗 𝜌 𝑀 𝑗 †
𝐸 𝑗 = 𝑀 𝑗 † 𝑀 𝑗
Quantum state update, analogous to Bayes rule, with ρ as the prior distribution, and 𝜌′ 𝑗 as the posterior distribution.
Simple description in terms of entanglement with a meter, with subsequent projection on the meter.

5. Weak value definition Aharanov, Albert, and Vaidman (1988)
Ingredients:
Pre-selection of system
Weak measurement w/ a meter
Post-selection of system
Measure (conditioned) meter shift.
Properties:
1) Time symmetric
2) Formally similar to the expectation value
3) Can exceed the eigenvalue range
4) Generally complex
“How the Result of Measurement of a Component of the Spin of a Spin- 1/2 Particle Can Turn Out to Be 100”

6. Stern-Gerlach z x Contains Preselection of spin state in ~ |𝑥
Weak splitting in z direction
Strong splitting in x direction
Postselection in x direction (or spin state)
Recording of the z deflection
Beam block

7. (Optical) Realization of the weak value
Polarization based
Interference based
Rev. Mod. Phys. 86, 307 (2014) 
Justin Dressel, Mehul Malik, Filippo M. Miatto, Andrew N. Jordan, Robert W. Boyd 

8. ISI Web of Science search for the topic “weak value”

9. Tests of quantum mechanics – Leggett-Garg inequality
Define LGI = <Q1 Q2 > + <Q2 Q3 > - <Q1 Q3 >
Macrorealism + noninvasive detector implies -3 < LGI < 1, violated by QM

10. Weak measurement version of Leggett-Garg inequality
Second measurement is now weak
I is continuous
Define LGI = <I1 I2 > + <I2 I3 > - <I1 I3 >
Macrorealism + noninvasive detector implies -3 < LGI < 1, violated by QM

11. Weak measurements and Foundations
System = electron on a quantum dot; Meter = current produced from a nearby quantum point contact.
Here, the weak value is also used as a test of quantum mechanics against “macrorealism” – theoretical use of the WV.
The weak measurement Leggett-Garg inequality uses time-correlation functions of current measurements. Result: A (generalized) weak value is strange if and only if the LGI is violated.

12. Weak measurements and Foundations
Something interesting: The weaker the measurement (less coupling, less disturbance, etc.), the larger the weak value gets.
Suggests that the problem with LGI is not noninvasive measurability, but classical (macroscopic) realism assumption.

13. Tests of quantum mechanics and first solid state implementation of weak values
Groen, DiCarlo, et al Phys. Rev. Lett. 111, (2013)

14. More refined notions of LGIs
Ways to avoid the “clumsiness loophole”.
Concept of “No signaling in time” as analog of Bell nonlocality.
Precise ways to specify “non-invasive detector assumption”.
Alok: “Importantly, the violations of all formulations of LGIs can be achieved for any non-zero value of unsharpness parameter”.
Pusey: anomalous weak values imply contextuality. (Shows the probability of disturbance goes as the coupling^2, whereas a contextual theory needs a disturbance of order coupling or stronger).

15. Continuous Quantum Measurement
A continuous measurement is a time series of weak measurements where the measurement results are now effectively continuous, such that the measurement strength grows over time to become a projective measurement. Here 𝑟 0 , 𝑟 1 ,…, 𝑟 𝑛 are a sequence of continuous measurement results in time bins of size δt.
Allow also for Hamiltonian evolution as well as measurement dynamics.
The set of ρ’s is a quantum trajectory.

16. Example 1: Ballistic Electron Detector –Quantum Point Contact
It takes time to measure – there is background detector noise (electron shot noise) of spectral density 𝑆 𝐼 that must be averaged .
Measurement time – new time scale.

17. Example 2: Superconducting cavity QED– 3D Transmons
Paik et al., Phys. Rev. Lett. 107, (2011)
A microwave tone near the resonance frequency of the transmon produces a qubit state dependent phase shift that is amplified and read out in the reflected signal as a voltage.
Murch, Siddiqi, et al. Nature 2013

18. Quantum Trajectories in 3D Transmons
Murch, Siddiqi, et al. Nature 2013

19. Analogy
ANJ, Nature 502, 177 (2013)

20. QuantumTrajectory
A given stochastic current allows us to continuously update the quantum state of the qubit, ending in a final (red) state, starting from the (green) state
|+𝑥

21. Many Quantum Trajectories
Different detector outputs measured over the same time period will generally produce different final red states, all starting from the same green state.
Delta = 0
epsilon = 0.5
total time T = 3
tau = 5
\delta t (time step) = 0.02
initial state x+ state
post-selected state = theta = pi/4, phi = epsilon * T , r = 1

22. Many Quantum Trajectories
Different detector outputs measured over the same time period will generally produce different final red states, all starting from the same green state.

23. Many Quantum Trajectories
Different detector outputs measured over the same time period will generally produce different final red states, all starting from the same green state.

24. Postselection on the final state
Some symmetry may be restored to the problem when we recognize that since we begin the measurement in the same state every time, let us only consider the trajectories when we end in the same state every time!
Delta = 0
epsilon = 0.5
total time T = 3
tau = 5
\delta t (time step) = 0.02
initial state x+ state
post-selected state = theta = pi/4, phi = epsilon * T , r = 1

25. Postselection on the final state
Some symmetry may be restored to the problem when we recognize that since we begin the measurement in the same state every time, let us only consider the trajectories when we end in the same state every time!

26. Postselection on the final state
Some symmetry may be restored to the problem when we recognize that since we begin the measurement in the same state every time, let us only consider the trajectories when we end in the same state every time!
Delta = 0
epsilon = 0.5
total time T = 3
tau = 5
\delta t (time step) = 0.02
initial state x+ state
post-selected state = theta = pi/4, phi = epsilon * T , r = 1

27. The question
Given that we start at one state and end at the other after a time T, what is the most likely path connecting the two states?
Suggests an action principle.

28. Stochastic Path Integral Formalism
Ingredients into the new formalism:
The probability distribution functions of the detector output r(t), given known qubit states. (we parameterize the state as a vector q in an orthonormal operator basis)
We must impose the update rule for the quantum state with a set of auxiliary parameters { 𝑝 𝑗 }, where the dimension of p is the same as q, so we double our state space.
Apply the initial and final boundary conditions on q.
We then calculate the join probability distribution function of everything to find:
Detailed form of the “stochastic Hamiltonian” knows about the detector probability distribution and the state update.

29. Properties of SPI Formalism
One can show that the SPI is equivalent to a stochastic master equation.
Consequently, this gives us a new way to calculate any correlation function or average in terms of averages of A(q(t), r(t)) over q and r.
The path integral has a canonical structure, where p is a canonical momentum, and q is the coordinate. `Stochastic energy’ is a constant of motion.
Naturally suggests the answer to the most-likely path in terms of a saddle-point approximation to extremize the stochastic action. Our action principle!
Action extremum simply gives Hamilton’s equations as the dynamics!

30. The action principle in action – qubit measurement
Consider qubit Hamiltonian 𝐻= 𝜖 2 𝜎 𝑧 − ∆ 2 𝜎 𝑥
Write qubit state in Bloch vector components. is the measurement time.
We get a set of 3+3 ODEs, plus 1 constraint. We are solving quantum measurement with classical dynamics!
3+3 constants of motion allow us to impose both initial and final boundary conditions on the state!

31. Comparison with Experiments (Siddiqi group & Murch group)

32. Superconducting transmon qubit measured by off-resonant microwave tone.
Transmon architecture and measurement procedure

33. Quantum trajectories with drive on

34. Initial and final boundary conditions in undriven case
Add post-selection and compare theory and experiment
Histogram is of all trajectory data
Yellow dashed = theory
Magenta solid = experimental estimate
Shaded band = 1 sigma around the estimate.

35. The driven case
Nature 511, 570–573 (2014)

36. Further applications of the formalism
Quantum jumps and the Zeno effect
Phys. Rev. A 88, (2013)
Diagrammatic approach to correlation functions
Phys. Rev. A 92, (2015)
Fluorescence
Quantum Stud.: Math. Found. (2016) 3: 237

37. Further applications of the formalism
Joint measurement of noncommuting observables
arXiv:
Entanglement trajectory statistics of joint measurement
Phys. Rev. X 6, (2016)
“Quantum caustics” – possibility of multiple most likely paths.
Phys. Rev. A 96, (2017)

38. Continuous measurement and foundations: Arrow of time
Quantum trajectories are time reversal symmetric, but a statistical arrow of time emerges for long trajectories.
Phys. Rev. Lett. 119, (2017)

39. Weak values and precision measurements
What’s the goal? We want a way to measure very small shifts in system parameters.
AAV, 1988:

40. Impact of AAV; it is mainly metrology
217
Hosten and Kwiat, Science, 2008
Dixon, Starling, Jordan, Howell, PRL 2009
Source: Google Scholar

41. Precision Beam Deflection
Dixon, Starling, Jordan, Howell, PRL 2009
𝑘 is the transverse momentum kick = wavenumber times deflection angle.
560 femto radians

42. Deflection
How big is this angle? “A hair’s breadth at the moon’s distance” - A. Steinberg

43. Why does this work?
As we found with the first paper after this one, it is not from AAV amplification alone, which gives same fundamental precision, Phys Rev A 80, (R) (2009).
Post-selecting and amplification can suppress some types of instrumental and environmental noise:
Vibration, Jitter, and Turbulence effects: Phys. Rev. X 4, (2013), Phys. Rev. A 92, (2015)
Correlated Noise: By sampling only occasionally, with an amplified average, the resulting correlated noise is suppressed with respect to a direct measurement. Phys. Rev. A 96, (2017)
Systematic noise: Affects accuracy, rather than precision. Caused by e.g. imperfect calibration of measurement instruments or interference of the environment. Does NOT decrease with more averaging, so a smaller sample with amplified average suppresses systematic noise of the parameter.
Phys. Rev. A 94, (2016)

44. New ideas: Power recycling
New elements
Tricks you can play to reinject the rejected light back into the system many times to enhance SNR.
Phys. Rev. Lett. 114, (2015)
Phys. Rev. A 88, (2013)

45. Collaborators Irfan Siddiqi Kater Murch Areeya Chantasri Phil Lewalle
Pierre Rouchon
Benjamin Huard
Justin Dressel
Yakir Aharonov

46. Collaborators Kevin Lyons Shengshi Pang John Howell Julian Martinis
Courtney Byard
Paul Kwiat
Aephraim Steinberg
Jeff Tollaksen

47. Conclusions
Weak measurements can shed further light on foundations of quantum mechanics
New formal approach for continuous measurements: stochastic path integrals
Weak values can be helpful in precision measurements of small parameter shifts