Undoing a weak quantum measurement of a solid

Undoing a weak quantum measurement of a solid-state qubit
(Quantum Un-Demolition measurement)
Alexander N. Korotkov1 and Andrew N. Jordan2
1University
We propose an experiment which demonstrates the undoing
of a weak continuous measurement of a solid-state qubit, so
that any unknown initial state is fully restored. The undoing
procedure has only a finite probability of success because of
the non-unitary nature of quantum measurement, though it
is accompanied by a clear experimental indication of whether
or not the undoing has been successful. The probability of
success decreases with increasing strength of the
measurement, reaching zero for a traditional projective
measurement. Measurement undoing (``quantum undemolition'') may be interpreted as a kind of a quantum
eraser, in which the information obtained from the first
measurement is erased by the second measurement, which is
an essential part of the undoing procedure. The experiment
can be realized using quantum dot (charge) or
superconducting (phase) qubits.
PRL 97, 166805 (2006)
of California, Riverside and 2University of Rochester
H
e
ψ1
ul
successf
unsucces
sfu
ψ0 (still
unknown)
l
undoing
(information erasure)
ψ2
“Quantum Un-Demolition (QUD) measurement”
r (t)
| 1〉
| 1〉
×
| 0〉
H
Hˆ QB = (ε / 2) ( c c − c c ) + H ( c c + c c )
†
1 1
e
†
2 2
†
1 2
†
2 1
Assume “frozen” qubit: ε = H = 0
I(t)
Bayesian evolution due to measurement (Korotkov-1998)
1) Diagonal matrix elements of the density matrix
evolve according to the classical Bayes rule
2) Non-diagonal matrix elements evolve so that
the degree of purity ρij/[ρii ρjj]1/2 is conserved
ρ11 (τ ) =
ρ12 (τ )
ρ12 (0)
,
=
[ ρ12 (τ ) ρ 22 (τ )] 1/ 2 [ ρ12 (0) ρ22 (0)]1/ 2
where
I =
ρ22 (τ ) = 1 - ρ11 (τ )
1 τ
I ( t ) dt
τ ∫0
Alexander Korotkov
where
University of California, Riverside
where Tm = 2 S I /( ΔI )
(“measurement time”)
2
Second example
(phase qubit)
α | 0〉 + β | 1〉 →
PS ≤
α | 0〉 + e iφ β 1 − p | 1〉
Norm
1− p
ρ 00 (0) + (1 − p ) ρ11 (0)
University of California, Riverside
1) unitary transformation of N qubits
2) null-result measurement of a certain strength by a strongly
nonlinear QPC (tunneling only for state |11..1〉)
3) repeat 2N times, sequentially transforming the basis vectors
of the measurement operator into |11..1〉
p = 1 - e-Γt
Γ
(also reaches the upper bound for success probability)
H
Fourth example: Evolving charge qubit
e
Hˆ QB = (ε / 2) ( c1†c1 − c2†c2 ) + H ( c1†c2 + c2†c1 )
→
iφ
e α 1 − p | 0〉 + e β 1 − p | 1〉
= e iφ (α | 0〉 + β | 1〉 )
Norm
Alexander Korotkov
Probabilities of no-tunneling are 1 and exp(-Γt )=1-p
Third example: General undoing procedure
for entangled charge qubits
|1〉
|0〉
If no tunneling for both measurements,
then initial state is fully restored!
iφ
pi = (π S I / t )-1/ 2 exp[-( I - I i ) 2 t / S I ] dI
Alexander Korotkov
Second example: Undoing partial
measurement of a phase qubit
Start with an unknown state
Partial measurement of strength p
π-pulse (exchange |0> ↔ |1>)
One more measurement with
the same strength p
5) π-pulse
min ( p1 , p2 )
p1 ρ11 (0) + p2 ρ 22 (0)
Again same as before, so measurement
undoing for phase qubit is also optimal
University of California, Riverside
1)
2)
3)
4)
min Pr
Pr ( ρ (0))
Coincides with the pervious result, so the upper bound is reached,
therefore undoing strategy is optimal
(does not depend on initial state!)
ρ11 (0) exp[- ( I - I1 )2 / 2 D]
ρ11 (0) exp[- ( I - I1 )2 / 2 D] + ρ22 (0) exp[- ( I - I 2 )2 / 2 D]
PS ≤
First example
(DQD+QPC)
Pav = 1 - erf[ t / 2Tm ]
Alexander Korotkov
First example: DQD qubit with no tunneling,
measured by QPC
PS ≤
General bound:
-|r |
e 0
PS = |r |
-|r |
e 0 ρ11 (0) + e 0 ρ 22 (0)
(Figure partially adopted from A. Jordan,
A. Korotkov, and M. Büttiker, PRL-2006
University of California, Riverside
Alexander Korotkov
University of California, Riverside
Conclusions
• Partial (incomplete, weak, etc.) quantum measurement can be
undone, though with a finite probability Ps, decreasing with
increasing strength of the measurement (Ps = 0 for orthodox case)
• Though somewhat similar to the quantum eraser, undoing idea
is actually quite different:
“Quantum Un-Demolition” (QUD) measurement
• Measurement undoing for single phase qubit is realizable now,
experiment with a charge qubit will hopefully be possible soon
(difficulty to use SET: need an ideal quantum detector)
Pav ≤ ∑ r min Pr
University of California, Riverside
Comparison of the general bound for
undoing success with examples
we can analyze classical probabilities (as if qubit is in some
certain, but unknown state); then simple diffusion with drift
University of California, Riverside
Alexander Korotkov
Our idea of measurement undoing is quite different:
we really extract quantum information and then erase it
(similar to Koashi-Ueda, PRL, 1999)
Alexander Korotkov
Trick: since non-diagonal matrix elements are not directly involved,
Tundo = Tm | r0 |
(to satisfy completeness,
eigenvalues cannot be >1)
min i pi
min Pr
=
PS ≤
Σ i pi ρ ii (0) Pr ( ρ (0))
Averaged (over r) probability of success:
University of California, Riverside
Averaged probability
of success (over result r0)
C × M r−1
(POVM formalism)
Pr(ρ(0)) – probability of result r for initial state ρ(0),
min(Pr) – probability of result r minimized over all
possible initial states
(same for an entangled qubit)
It may happen though that r = 0 never happens;
then undoing procedure is unsuccessful.
Probability of
successful undoing
Tr( M r ρ M r† )
pi – probability of the measurement result r for initial state |i 〉
Then any unknown initial state is fully restored.
Results:
M r ρ M r†
pi , pi = Tr( M r† M r | i 〉 〈 i |)
Probability of success:
Average time to wait
(similar to Koashi-Ueda, PRL-1999)
max(C ) = min i
Simple strategy: continue measuring until r(t) becomes zero!
=
| 0〉
Undoing measurement operator:
where r0 is the result of the measurement to be undone,
and ρ(0) is our knowledge about an unknown initial state;
in case of an entangled qubit ρ(0) is traced over other qubits
| 0〉
ρ→
Measurement operator Mr :
t
Alexander Korotkov
Interference fringes restored for two-detector
correlations (since “which-path” information
is erased)
General theory of quantum
measurement undoing
Probability of success
How to undo? One more measurement!
University of California, Riverside
Alexander Korotkov
r0
Detector
(QPC)
Measurement undoing
Evolution due to partial (weak, continuous, etc.) measurement
is non-unitary (though coherent if detector is good!), therefore
it is impossible to undo it by Hamiltonian dynamics.
| 1〉
Can be realized experimentally pretty soon!!!
University of California, Riverside
Qubit
(DQD)
I(t)
Pav = 1 - p
Such an experiment is only slightly more difficult than recent
experiment on partial collapse (N. Katz et al., 2006).
If r = 0, then no information and no evolution!
University of California, Riverside
Alexander Korotkov
r =1
First “accidental”
Undoing
measurement measurement
If undoing is successful, an unknown state is fully restored
(partially
collapsed)
r = 0.5
Jordan and Korotkov, 2006
Is it possible to undo partial quantum measurement?
(To restore a “precious” qubit accidentally measured)
Yes! (but with a finite probability)
1- p
ρ 00 (0) + (1 - p ) ρ11 (0)
For measurement strength p increasing to 1, success probability
decreases to zero (orthodox collapse), but still exact undoing
Jordan-Korotkov-Büttiker, PRL-06
Alexander Korotkov
=
Total (averaged) success probability:
r=0
ΔI t
[ I ( t ') dt ' - I0 t ]
SI ∫ 0
e
ρ 00 (0) + e-Γ t ρ11 (0)
where ρ(0) is the density matrix of the initial state (either averaged
unknown state or an entangled state traced over all other qubits)
where measurement result r(t) is
It is impossible to undo “orthodox” quantum
measurement (for an unknown initial state)
weak (partial)
(unknown) measurement
r = -0.5
r = -1
Measurement undoing for DQD-QPC system
The problem
ψ0
PS =
ρ11 ( t ) ρ11 (0)
=
exp[2 r ( t )]
ρ 22 ( t ) ρ 22 (0)
ρ12 ( t )
= const
ρ11 ( t ) ρ 22 ( t )
r(t ) =
Quantum erasers in optics
Quantum eraser proposal by Scully and Drühl, PRA (1982)
Success probability if no tunneling during first measurement:
-Γ t
I(t)
University of California, Riverside
Alexander Korotkov
Probability of successful measurement
undoing for phase qubit
Graphical representation of the evolution
(now non-zero H and ε, qubit evolves during measurement)
I(t)
1) Bayesian equations to calculate measurement operator
2) unitary operation, measurement by QPC, unitary operation
Alexander Korotkov
University of California, Riverside
Alexander Korotkov
University of California, Riverside