1. The document extends Ehrenfest's theorem to weak values by showing that the semiclassical evaluation of weak values agrees with classical trajectories within the semiclassical accuracy. 2. When the contribution from a unique classical trajectory dominates without quantum interference, the weak value is equal to the observable evaluated along that classical trajectory plus higher order corrections. 3. This suggests that some "anomalous" weak values outside the observable's eigenvalues can be understood classically without invoking quantum effects like interference or entanglement.

1. 1 / 13
2010-03-15
Ref. AT, PLA 297, p.307 (2002).
(QMKEK3), KEK
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2. Introduction 2 / 13
Weak values (Aharonov, Albert and Vaidman 1988)
For a quantum ensemble specified by
|ψ at t = t (preparation)
,
ψ | at t = t (postselection)
the weak value of the observable A at time t ∈ [t , t ], is defined as follows:
ˆ
ψ (t)| A |ψ (t)
ˆ
W[A, t] ≡
ˆ
ψ (t)|ψ (t)
where U(t , t ) is the time evolution operator, ψ (t)| ≡ ψ |U(t , t) and
ˆ ˆ
|ψ (t) ≡ U(t, t )|ψ .
ˆ
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3. Main result 3 / 13
An extension of Ehrenfest theorem for weak values
When the classical trajectory (qt , pt ) that contribute to the stationary
phase evaluation of the path integral representation of Feynman kernel
i t
ψ |exp − H(t)dt |ψ
ˆ
← h
¯ t
is unique, the semiclassical evaluation of weak value is
ˆ
W[A, t] = A(qt , pt ) + O(h).
¯
N.B. The resultant semiclassical weak value can be regarded as classical
one, because neither quantum interference nor quantum entanglement is
involved.
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4. Examples 4 / 13
Quantum-classical correspondence for complex-valued
trajectories
In the classical limit h → 0, the weak values obey classical theory, i.e.,
¯
(W[ˆ, t], W[ˆ, t]) = (qt , pt ) + O(h)
q p ¯
obserbable quantities classical trajectory
This quantum-classical correspondence is applicable to both real-valued
and complex-valued trajectories without discrimination.
Thus, the complex-valued trajectories that appear in semiclassical method
to describe classically forbidden time evolution (e.g. tunnelling and
nonadiabatic transitions) are not the artifact of the approximate theory,
but are observable by weak measurements.
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5. Examples 5 / 13
Why the Result of a Measurement of a Component of the
Spin of a Spin-1/2 Particle Turn Out to be 100
cf. AAV 1988
We examine semiclassical spin-coherent state path integral for the
Feynman kernel θ , φ |U(t , t )|θ , φ with U(t , t ) = ˆ The classical
ˆ ˆ 1.
trajectory (θ (t), φ (t)) obeys the classical equation of motion and
Klauder’s boundary condition (1978)
exp(i φ ) tan(θ /2) = exp(i φ (t )) tan(θ (t )/2)
exp(−i φ ) tan(θ /2) = exp(−i φ (t )) tan(θ (t )/2).
ˆ
Note that the trajectory is uniquely determined for U(t , t ) = ˆ The
1.
motion of spin determined by the classical trajectories agrees with the
corresponding weak values.
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6. Examples 6 / 13
AAV 1988 examined the following quantum ensemble
θ = π /2, φ = π |, t , |θ = 2α , φ = 0 , t
where |θ , φ is a spin-coherent state, (θ , φ ) specifies the orientation of the
spin and α ∈ [0, π /2].
W[σx ] = −1: This is due to the final state θ = π /2, φ = π |.
ˆ
W[σy ] = −i tan(α + π ):
ˆ 4
an example of complex-valued weak value.
W[σz ] = (cos α + sin α )/(cos α − sin α ):
ˆ
real-valued, but its magnitude is “anomalous”, i.e. greater than 1
The semiclassical prescription provides the exact result of AAV 1988.
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7. Summary 7 / 13
Summary
Weak values are evaluated with semiclassical method.
When the quantum fluctuation is negligible, the complex-valued
classical trajectories that appear in the semiclassical method agree
with corresponding weak values, within the semiclassical accuracy.
This may be regarded as an extension of Ehrenfest theorem for weak
values.
This suggests some “anomalous” weak values are classical. Namely,
there is an explanation why weak values can take “anomalous” values
that lie outside the range of the eigenvalues of the corresponding
operators, without invoking on quantum interference nor
entanglement, which are typical sources of quantum effect.
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8. Semiclassical evaluation 8 / 13
A generating functional
With
t
ˆ ≡ ψ |exp − i
Z (ζ (·), A) H(t) − Aζ (t) dt |ψ ,
ˆ ˆ
← h
¯ t
an exact formula for the weak value is
δ ln Z (ζ (·), A)
ˆ
W[A, t] = − i h
ˆ ¯
δ ζ (t)
ζ (·)≡0
Feynman path integral representation of Z will be evaluated by
semiclassical method.
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9. Semiclassical evaluation 9 / 13
Single-trajectory assumption
In the stationary phase evaluation of Z , it is assumed that interferences
among the contributions from classical trajectories do not appear, i.e.,
there appears only one classical trajectory, which is denoted by (qt , pt ).
Namely, Z is assumed to be in the following form
Z E exp(iF /h)
¯
where E and F is an amplitude factor and classical action. This implies
δF
ˆ
W[A, t] = + O(h).
¯
δ ζ (t) ζ (·)≡0
Now it is straightforward to obtain the main result.
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10. Semiclassical evaluation 10 / 13
The limitation of the present result
The single-trajectory assumption fails once there appears the quantum
interference among multiple classical trajectories. As a result, it become
impossible to observe each individual trajectory.
The scenario of the breakdown
1. In short time scales (until so-called Ehrenfest time), the
single-trajectory assumption is valid.
2. As the system evolves, the influence of caustics, which are the
bifurcation points of classical trajectories, become significant. The
caustics induces the divergence of semiclassical amplitude factor E .
2
Accordingly, the “weak variance” W A − W[A]
ˆ ˆ = O(h) diverges.
¯
3. After the anomalous fluctuation of the weak variance, the interference
among classical trajectories become prominent in general.
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11. Motivation 11 / 13
Ehrenfest’s theorem
The (conventional) expectation values E [·] of quantum system obey the
equations that resemble classical ones:
d ∂H d ∂H
E [ˆ] = E
q (ˆ, p ) ,
q ˆ E [ˆ] = E −
p (ˆ, p )
q ˆ
dt ∂p dt ∂q
When the quantum fluctuation is small, this provides a quantum-classical
correspondence.
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12. Motivation 12 / 13
Complex-valued classical trajectories in the semiclassical
method
Ψ(x) V(x)
The complex extensions of phase space or
time enable the semiclassical method to Complex Trajectory
describe classically-forbidden phenomena Real Trajectory
(e.g. tunnelling).
x
Mathematical origin: e.g., complex saddle points of Feynman path
integrals
e iS[x(·)]/h Dx(·)
¯
∑ Ak e iS[xk (·)]/h
¯
k
where xk (·) satisfies the classical eq. of motion.
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13. Motivation 13 / 13
The conventional status of complex-valued trajectories
1. The complex-valued classical trajectories are useful theoretical tools.
They are metaphors, expressed in terms of classical notions, of classically
forbidden phenomena.
2. It is impossible to observe the complex-valued classical trajectories,
because they do not correspond to any eigenvalues of observables. Also, it
is impossible to make any correspondence between complex trajectories
and statistical expectation values. Thus, the complex-valued classical
trajectories are just artifacts of approximate theories.
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