Weak
measurement tomography
Jonathan Gross, Ninnat
Dangniam, Chris Ferrie, Carl Caves
Center for Quantum Information and Control,
University of New Mexico
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
- Many real measurements are weak
- Weak measurements are a resource for performing exotic measurements
Novel claims
- Weak measurements provide a fundamentally new way of doing tomography
- Weak measurements allow you to do tomography better than non-weak measurements
- 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:
- The measurement is something new beyond
standard
tomography
- The measurement performs better than
standard
tomography
- The insight is that weak measurements allow re-use of the system due to the small disturbance
Weak coupling
Measurement protocol
Analysis
Two perspectives:
- The weak measurements extract data prior to the projection
- The weak measurements modify the final measurement
POVM elements
Effective random ODOP
Equally weighted, orthogonal projectors: ,
Bloch sphere distributions
,
Evaluation
The proposal has several problems:
-
The measurement is not new
- POVM is a random ODOP (i.e.
standard
tomography)
-
The measurement is not better
- Best measurement is projector sampled from Haar uniform distribution (random ODOP)
-
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)
- The measurement uses a new technique inspired by weak-value procedures
- Simplicity of measurement makes it better for some systems
- The relationship between meter readings and probability
amplitudes provides insight for the interpretation of the wavefunction
Postselect on different state?
Postselection throws away information
Z-measurement (imaginary part)
Measurement commutes with control
Evaluation
-
The weak-value-inspired procedure is only new for the real parts in dimensions higher than 2
- All other measurements are random ODOPs
-
It is not better than the same scheme without postselection
- Eliminating postselection does not change experimental setup and leaves postprocessing virtually unchanged
-
The insight is misleading since real and imaginary components cannot be determined until all measurements are made
Summary
- Random ODOPs already exhibit much of the
new
behavior seen in weak measurements
- For average fidelity, the best independent measurement is a random ODOP
- For tomography in general, the best thing to do is keep all the data
- Many of the insights of weak measurements have caveats that become clear when analyzed at the level of POVMs