Weak measurement tomography

Jonathan Gross, Ninnat Dangniam, Chris Ferrie, Carl Caves

Center for Quantum Information and Control, University of New Mexico

Tomography

• Repeat measurement on each copy
• Adapt measurement based on previous outcomes
• Perform joint measurements

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 reasons for doing tomography with weak measurements

1. Many real measurements are weak
2. Weak measurements are a resource for performing exotic measurements
   • Oreshkov and Brun, Phys. Rev. Lett. 95, 110409 (2005)
     Weak measurements are universal

Less standard reasons for doing tomography with weak measurements

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 principles

1. Write down POVMs
   • Compare to randomly chosen one-dimensional orthogonal projective measurement (random ODOP)
2. Compare measurements using optimal estimator for a figure of merit (FOM)
   • Use average fidelity as the FOM: $$F={\max }_{\hat{\rho }}\int d\rho {\sum }_{\chi }p\left(\chi \right|\rho )f(\rho ,\hat{\rho }(\chi ))$$

Two schemes considered in this talk

• Das and Arvind, Phys. Rev. A 89, 062121 (2014)
  Estimation of quantum states by weak and projective measurements
• Lundeen et al., Nature 474, 188–191 (2011)
  Direct measurement of the quantum wavefunction

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
   • The measurement they compare to is a measurement of ${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$, which is a random ODOP
3. The insight is that weak measurements allow re-use of the system due to the small disturbance

Weak coupling

$$ϵ=\frac{1}{\Delta q^2}$$

Analysis

Two perspectives:

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

$$K_{\pm }^{(y)}=\frac{1}{2}(I\pm {\sigma }_y)$$

$$K^{(j)}(q)=⟨q|U^{(j)}|\varphi ⟩$$

$$K_{\mathrm{\pm }}(q_1,q_2)=K_{\pm }^{(y)}{K}^{(x)}(q_2)K^{(z)}(q_1)$$

POVM elements

$$E_{\pm }(q_1,q_2)=K^{(z)}(q_1{)}^{†}{K}^{(x)}(q_2{)}^{†}{E}_{\pm }^{(y)}{K}^{(x)}(q_2)K^{(z)}(q_1)$$

$$E(\hat{n})=G(\hat{n})\frac{1}{2}(1+\hat{n}\cdot \vec{\sigma })$$

$${\hat{n}}_{\pm }(q_1,q_2)=-{\hat{n}}_{\mp }(-q_1,-q_2)$$

$$G(\hat{n})=G(-\hat{n})$$

Equally weighted, orthogonal projectors: $E_{\pm }(q_1,q_2)$, $E_{\mp }(-q_1,-q_2)$

Projection onto the q-plane

$${\hat{n}}_{\pm }(q_1,q_2)$$

Original image: "Equirectangular projection SW" by Strebe - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons.

Q-plane distributions

$$G(q_1,q_2)$$

Bloch sphere distributions

$$G(\hat{n})$$

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)
Direct measurement of the quantum wavefunction

Claims:

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)

$$⟨n|c_m⟩=\frac{1}{\sqrt{d}}{\omega }^{mn},\omega =e^{2\pi i/d}$$

$$⟨n|𝜓⟩\propto {⟨{\sigma }_y⟩|}_{c_0,n}+i{⟨{\sigma }_z⟩|}_{c_0,n}+𝒪({\phi }^2)$$

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

Postselection throws away information

$$Z|c_m⟩=|c_{m\oplus 1}⟩$$

$$Z|n⟩={\omega }^n|n⟩$$

$${\omega }^{-jn}⟨n|\psi ⟩\propto {⟨{\sigma }_y⟩|}_{c_j,n}+i{⟨{\sigma }_z⟩|}_{c_j,n}+𝒪({\phi }^2)$$

Z-measurement (imaginary part)

Measurement commutes with control

$$U_{n\pm }=e^{\mp i\phi |n⟩⟨n|}$$

Y-measurement (real part) for qubits

POVMs for different coupling unitaries are not random ODOPs (for fixed $n$)

Y-measurement (real part) for qubits

If we only consider reconstructing qubits, flipping a coin to choose between $|0⟩$ and $|1⟩$ yields a POVM that is a random ODOP

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