Reversible Weak Quantum
Measurements
09 December 2008
Presented by:
Adam Reiser
Anuj Das
Neil Kumar
George O’Quinn
Agenda
•
•
•
•
Strong and Weak Measurements
Hardy’s Paradox
Reversibility of Weak Measurements
Experiments and Possible Applications
Strong and Weak Measurements
Strong Measurement


First formulated 1943 by von Neumann
“Measurement” in quantum mechanics by default
refers to strong measurement
–
–

Associated with a physical observable
corresponding to a linear Hermitian operator
An irreversible projection of a superposition onto
one of its eigenstates
Example: the Stern-Gerlach apparatus
–
–
Intrinsic angular momentum assumes an
eigenstate upon interaction with a magnet
This interaction is a strong measurement
Stern-Gerlach Device
• Particles forced to assume an eigenstate (wave
collapses)
• Direction of beam indicates the eigenstate of spin
*(images available under GNU Free Documentation License)
Weak Measurement
•
•
•
•
Measurement is continuous
Partial wavefunction collapse can be undone
Underlying eigenstates are hidden
Demonstrated for both phase and charge qubits
Weak measurement of a charge qubit
• a quantum point contact is a narrow channel (approx. 10-6 m)
between two conducting regions that serves as an extremely
sensitive charge detector
Charge Qubits
• Strong measurement characterized by average
currents corresponding to qubit's eigenstates
• Weak measurement yields average current...
• Note: This is an average, not an eigenstate!
Undoing a weak measurement
Given a weak measurement r0
• The goal is to measure a new state ru(t) = -r0
• This fails if r(t) crosses origin
• Probability of success calculable
• Note that probability of reversal scales relative to
certainty of original measurement r0
• | r0 | >> 1: high confidence of measured state
Phase Qubits
• Similar to flux qubits –
Josephson junction +
SQUID
• Controlled by an
external flux øe
Detection of Quantum States
Lower barrier øe to initiate measurement
• Upper energy state tunnels with rate Γ
• Lower energy state remains
Weak Measurement (Phase)
• For Γ•t >> 1, collapse to one state occurs
• But if the barrier is raised after a finite time t ~ Γ-1
– If tunneling occurs, qubit is destroyed
– If not – phase accumulates, qubit remains
Φ℮
qubit
SQUID
Hardy’s Paradox
Hardy’s Paradox
Photon detectors
Outer arm
Inner arm
B
A
C
E
Inner arm
PD2
F
Quartz plates
2
D
1
Quartz
Outer arm
PD1
D1
• Double Interferometer Experiment
– Shared inner arm permits destructive interference
– Two emitters, two detectors, lots of splitters
The Paradox
If we have a coincidence detection at only one
output, there are three possibilities:
– Both particles traveled the outer path:
• But then we should have two output detections
– One traveled the outer path, one the inner path:
• Again, we should have two detections
– Both particles traveled the inner path:
• But then they should have annihilated through interference
– P[O1|C1C2] = P[O2|C1C2] = 1
– P[O1O2|C1C2] = 0
Possible Outcomes
C
C
E
F
N/2
B
B
E
F
N/2
N/2
What Does This Mean?
• We can’t tell the state of a qubit undergoing
unitary evolution
• Weak measurement allows us to take
measurements within the photon paths
Photon detectors
Outer arm
Inner arm
Inner arm
PD2
Quartz plates
2
Outer arm
1
PD1
Weak Measurement
Reversibility of Quantum
Measurements
Quantum vs. Classical Measurement
• In quantum physics, we seem to have a
significant difference from classical mechanics to
contend with because of measurements having
only certain probabilistic outcomes.
• Information about the current state can be
garnered from past measurements of identically
configured quantum states.
• However, information from future measurements
may tell a fundamentally different story.
• This makes quantum state description timeasymmetric.
Quantum States Defined by Future and
Past
• Until now, we have essentially been using the
familiar ket notation in order to describe
quantum states: |Ψ ›= U (t1,t) |a›
• Due to time-asymmetry, it appears we need
another state as well: ‹Φ| =‹b| U† (t,t2)
• We can generalize this as follows:
  ||  
i
i
i
i
• Note that ‹Φ||Ψ › is not an inner product!

Measuring a Weak Value
• We use the following definition to find the weak
value of a measurement C:
 | C | 
Cw 
 | 
• Note: a weak measurement of a purely preselected system is its expectation value.

• Note: Cw may actually be well outside the
acceptable range of eigenvalues associated with C.
Pre-Selection and Post-Selection
Aharonov, Vaidman Experiment
Reversing a Measurement
• Involves using a device that
will only take a full
measurement (collapse state
to |1>) with probability p.
• If the device does not react,
the state partially collapses
to |0>.
• A pi-pulse, second
measurement, then another
pi-pulse can be used to
restore original state (w/
prob. 1-p).
Formal Quantum State Reconstruction
• A more complicated method of quantum state
reconstruction involves the creation of a
superoperator based off of a Gaussian
distribution of weakly measured states.
Experiments and Possible Applications
Using Weak Measurements
Realization of a Measurement of a "Weak Value”
Ritchie (1991)
• Performed a weak measurement using a
Gaussian beam of light polarized at 45 degrees,
and sent through a polarizing beam splitter
• The two beams were placed close enough
together to cause overlapping Gaussians
• Post-selected using a polarization filter close to
45 degrees
Quantum State Reconstruction via
Continuous Measurement
Silberfarb (2005)
• Reconstructed a quantum state using weak
continuous measurement of an ensemble average
• All members of the ensemble are evolved
identically in such a way as to map new
information onto the measured quantity
• This process provides enough information to
estimate the density matrix with minimal
disturbance to the system
Measurement of Quantum Weak Values of
Photon Polarization
Pryde (2005)
• Used a nondeterministic
entangling circuit to allow a
photon to make a weak
measurement of the polarization
of another photon
• Used pre- and post- selection to
eliminate extraneous results
• Non-classical interference
between two photons is
required to allow the weak
measurement to take place
Coherent State Evolution in a Superconducting
Qubit from Partial-Collapse Measurement
Katz (2006)
• Weak measurement with tomography
• Uses quantum-state tomography to investigate
state evolution
• A Josephson junction is used to produce the
superconducting phase qubit
• Measurement is carried out by lowering the
energy barrier to cause the probability of
tunneling by the |1> to increase
Reversal of the Weak Measurement of a
Quantum State in a Superconducting Phase
Qubit Katz (2008)
• The system is prepared as before, however two
steps are added
• A p-pulse is applied to swap the states
• Another weak measurement is performed
Direct Observation of Hardy’s Paradox by
Joint Weak Measurement with an Entangled
Photon Pair Yokota (2008)
• This experiment showed that Hardy’s paradox
exists, but does not solve the paradox
• Used photons replicate thought experiment
• Two interferometers were used, with inner arms
overlapped at the middle half beam splitter
• If the photons meet at the beam splitter they
interfere with each other
Inner arm
Outer arm
Photon detectors
Inner arm
PD2
Quartz plates
2
1
Quartz
Outer arm
PD1
Quantum Error Correction
• Traditional methods required extra qubits so that
strong measurements could be used without
destroying the original state
• Use of continuous feedback through weak
measurements could possibly be used to detect
defects
Possible Implications for Quantum
Cryptography
• The main aspect of security in quantum
cryptography is the complete collapse of the
quantum state if it is intercepted
• Weak measurements could possibly be used to
gain information about a system without
collapsing the quantum state
• Caveat: weak measurement and reversal would
introduce a slight delay that could be detected
References
•
•
•
•
•
•



Aharonov, Vaidman. The Two-State Vector Formalism: an Updated Review
Aharonov, Albert, Vaidman. How The Result of a Component of the Spin of
a Spin-1/2 Particle Can Turn Out to be 100.
Silberfarb, Jessen, Deutsch. Quantum State Reconstruction via Continuous
Measurement.
Korotkov, Jordan. Undoing a weak quantum measurement of a solid-state
qubit.
Ahnhert. Weak Measurement in Quantum Mechanics. Electronic Structure
Discussion Group.
Katz, et al. Reversal of the Weak Measurement of a Quantum State in a
Superconducting Phase Qubit. Phys. Rev. Lett. 101, 200401 (2008) –
Published November 10, 2008.
D. L. Christoph Bruder, Physics 1, 34 (2008).
A. N. Korotkov and A. N. Jordan, Phys. Rev. Lett. 97, 166805 (2006).
images provided under GFDL from commons.wikimedia.org