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Weak measurement tomography

Jonathan Gross, Ninnat Dangniam, Chris Ferrie, Carl Caves

Center for Quantum Information and Control, University of New Mexico

Creative Commons License

Weak measurements

  • Measurements that give less than maximal information about the system
  • Can model with an ancilla and weakly correlating unitary

Random projective measurements

Minimal kind of standard measurement for tomography is a randomly chosen one-dimensional projective measurement (random ODOP)

Standard claims

  1. Many real measurements are weak
  2. Weak measurements are a resource for performing exotic measurements

Novel claims

  1. Weak measurements provide a fundamentally new way of doing tomography
  2. Weak measurements allow you to do tomography better than non-weak measurements
  3. Weak measurements give new insights into the fundamentals of quantum mechanics

Evaluation technique

Write down POVMs

  • Gives a very clean, complete description of a measurement's tomographic properties
  • Facilitates comparison to randomly chosen one-dimensional orthogonal projective measurement (random ODOP)

Das and Arvind

Das and Arvind, Phys. Rev. A 89, 062121 (2014)
Estimation of quantum states by weak and projective measurements

Claims:

  1. The measurement is something new beyond standard tomography
  2. The measurement performs better than standard tomography
  3. The insight is that weak measurements allow re-use of the system due to the small disturbance

Weak coupling

Measurement protocol

Analysis

Two perspectives:

  1. The weak measurements extract data prior to the projection
  2. The weak measurements modify the final measurement

K±(y)=12(I±σy)

K(j)(q)=q|U(j)|ϕ

K±(q1,q2)=K±(y)K(x)(q2)K(z)(q1)

POVM elements

E±(q1,q2)=K(z)(q1)K(x)(q2)E±(y)K(x)(q2)K(z)(q1)

E(n^)=G(n^)12(1+n^σ)

n^±(q1,q2)=n^(q1,q2)

G(n^)=G(n^)

Effective random ODOP

Equally weighted, orthogonal projectors: E±(q1,q2), E(q1,q2)

Bloch sphere distributions

G(n^), ϵ=1Δq2

Evaluation

The proposal has several problems:

  1. The measurement is not new
    • POVM is a random ODOP (i.e. standard tomography)
  2. The measurement is not better
    • Best measurement is projector sampled from Haar uniform distribution (random ODOP)
  3. The insight is misleading
    • Extra weak measurements don't extract more information, they generate a distribution

Direct state tomography (DST)

Lundeen et al., Nature 474, 188–191 (2011)

  1. The measurement uses a new technique inspired by weak-value procedures
  2. Simplicity of measurement makes it better for some systems
  3. The relationship between meter readings and probability amplitudes provides insight for the interpretation of the wavefunction

Direct state tomography (DST)

n|cm=1dωmn,ω=e2πi/d

n|𝜓σy|c0,n+iσz|c0,n+𝒪(φ2)

Maccone and Rusconi, Phys. Rev. A 89, 022122 (2014)
State estimation: A comparison between direct state measurement and tomography

Postselect on different state?

Z|cm=|cm1

Postselection throws away information

Z|n=ωn|n

ωjnn|ψσy|cj,n+iσz|cj,n+𝒪(φ2)

Z-measurement (imaginary part)

Measurement commutes with control

Un±=eiφ|nn|

Evaluation

  1. The weak-value-inspired procedure is only new for the real parts in dimensions higher than 2
    • All other measurements are random ODOPs
  2. It is not better than the same scheme without postselection
    • Eliminating postselection does not change experimental setup and leaves postprocessing virtually unchanged
  3. The insight is misleading since real and imaginary components cannot be determined until all measurements are made

Summary

  1. Random ODOPs already exhibit much of the new behavior seen in weak measurements
  2. For average fidelity, the best independent measurement is a random ODOP
  3. For tomography in general, the best thing to do is keep all the data
  4. Many of the insights of weak measurements have caveats that become clear when analyzed at the level of POVMs