Quantum Measurements
and Back Action
(Spooky and Otherwise)
SM Girvin
Yale University
Thanks to Michel, Rob, Michael, Vijay, Aash, Simon, Dong, Claudia
1
for discussions and comments on Les Houches notes.
Quantum Back Action
is a Weird Thing
CNOT gate
control qubit
target qubit
c t c t
truth table
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0
Target is affected
but control is not
Are you sure???
2
Quantum Back Action
is a Weird Thing
CNOT gate
control qubit
0 −1
0 +1
target qubit
0 −1
0 −1
The control qubit is flipped from ← to
1 + Z1
1 − Z1
CNOT =
X2 +
I2
2
2
→
!!
3
It’s all about the measurement:
Stern Gerlach Experiment
Silver atom has magnetic moment
due to the electron ‘spin’
Silver atom is a qubit.
r r
V = µ gB
r
r
F = −∇V
Magnetic moment (spin) can point in any
direction and can be measured by passing
4
the atom through a magnetic field gradient.
Stern Gerlach Experiment:
Quantum Measurement
Silver atom has magnetic moment which can point
in any direction, and yet….
(Measured)
5
http://hyperphysics.phy-astr.gsu.edu
DISPERSIVE READOUT
qubit + readout pulses Qubit + resonator (MHD)
AMP dispersive shi8 χ
Readout amplitude 1
0
fr
f
width κ 90 Readout phase (deg) 1
θ
-­‐90 fr
0
f
MEASUREMENT HISTOGRAM
(data from Devoret lab)
(unlike Stern-Gerlach, the qubit is in nearly a pure state)
0
1
Counts
2000
-­‐2 0 2 Stern & Gerlach did not measure spin!
They entangled spin with position
and measured position.
(Measured)
8
http://hyperphysics.phy-astr.gsu.edu
More precisely, they used a spin-dependent force to
entangle spin with momentum and
waited for momentum to turn into position.
(Measured)
9
http://hyperphysics.phy-astr.gsu.edu
This is a measurement of EM modes
that were entangled with the qubit ‘spin’
0
1
Counts
2000
-­‐2 0 2 1D toy model with spin-dependent impulsive force
X
Bz
2
p
H ( p , x, t ) =
− hk0 xσ z δ (t )
2m
F ( x, t ) = + hk0 σ z δ (t )
[ H , σ z ] = 0 ⇒ QND
‘Quantum Non-Demolition’ (QND) Measurements
are Repeatable
Z
Z
First result is random, rest are repeats.
Z
Z
measurement measurement
Z
measurement
12
1D toy model with spin-dependent impulsive force
X
Bz
We will measure momentum just after impulse
rather than waiting for it to turn into position.
2
p
H ( p , x, t ) =
− hk0 xσ z δ (t )
2m
F ( x, t ) = + hk0 σ z δ (t )
2
p
H ( p , x, t ) =
− hk0 xσ z δ (t )
2m
Input product state
(
)
ψ ( x, t = 0 − ) = a ↑ + b ↓ Φ ( x )
Output entangled entangled state
(
ψ ( x, t = 0+ ) = aeik x Φ( x) ↑ + be −ik x Φ ( x) ↓
0
0
)
Output state in momentum basis
(
ψ [k , t = 0+ ] = aΦ[k − k0 ] ↑ + bΦ[k + k0 ] ↓
)
14
0.8
P(k |↓)
P(k |↑)
0.6
Strong measurement
0.4
0.2
-4
-2
2
4
k
(gaussian input packet)
0.8
P(k |↓)
P(k |↑)
0.6
Weak measurement
0.4
0.2
-4
-2
k
2
4
15
0.8
P(k |↓)
P(k |↑)
0.6
0.4
0.2
-4
-2
k
2
4
P(k |↑) is easy to understand
but what we need is:
P(↑| k )
16
Practice on two continuous variables
P ( x , y ) = ψ ( x, y )
y
2
Y
x
P ( x, y ) = P ( x | y ) P ( y )
ψ ( x, Y )
2
P ( x, Y )
P( x | Y ) =
=
2
P
(
Y
)
ʹ′
ʹ′
dx
ψ
(
x
,
Y
)
∫
17
Density Matrix Equivalent
Reduced density matrix for spin
conditioned on measurement of momentum
k ρ k
k ρ k
ρk =
=
Tr k ρ k
P(k )
k ρ k
ρk
Full density matrix projected onto observed
momentum state
Reduced density matrix for spin given
observed value of momentum
18
Density Matrix Equivalent
Full state
ψ = a ↑ Φ↑ + b ↓ Φ↓
⎛ a Φ ↑ ⎞
= ⎜
⎟
⎜ b Φ ⎟
↓ ⎠
⎝
Full density matrix
⎛ aa * Φ ↑ Φ ↑
ρ = ψ ψ = ⎜
⎜ a * b Φ Φ
↓
↑
⎝
ab * Φ ↑ Φ ↓ ⎞
⎟
bb * Φ ↓ Φ ↓ ⎟⎠
19
Reduced density matrix for spin conditioned
on measurement of momentum
k ρ k
k ρ k
ρk =
=
Tr k ρ k
P(k )
Full density matrix
⎛ aa * Φ ↑ Φ ↑
ρ = ⎜
⎜ a * b Φ Φ
↓
↑
⎝
ab * Φ ↑ Φ ↓ ⎞
⎟
bb * Φ ↓ Φ ↓ ⎟⎠
Reduced density matrix for spin conditioned
on measurement of momentum
1 ⎛ aa * k Φ ↑ Φ ↑ k
ρk =
⎜
P(k ) ⎜⎝ a * b k Φ ↓ Φ ↑ k
ab * k Φ ↑ Φ ↓ k ⎞
⎟
bb * k Φ ↓ Φ ↓ k ⎟⎠
20
Reduced density matrix for spin conditioned
on measurement of momentum
1 ⎛ aa * k Φ ↑ Φ ↑ k
ρk =
⎜
P(k ) ⎜⎝ a * b k Φ ↓ Φ ↑ k
ab * k Φ ↑ Φ ↓ k ⎞
⎟
bb * k Φ ↓ Φ ↓ k ⎟⎠
Easy to verify conditional state is pure:
Det ρ k = 0
Tr ρ k = 1
⎛ 1 0 ⎞
⇒ ∃ exists a basis s.t. ρ k = ⎜
⎟
⎝ 0 0 ⎠
If we fully measure the state of the ‘bath’
then the conditional state remains pure!
21
Reduced density matrix for spin conditioned
on measurement of momentum
1 ⎛ aa * k Φ ↑ Φ ↑ k
ρk =
⎜
P(k ) ⎜⎝ a * b k Φ ↓ Φ ↑ k
ab * k Φ ↑ Φ ↓ k ⎞
⎟
bb * k Φ ↓ Φ ↓ k ⎟⎠
Easy to verify conditional state is pure:
Det ρ k = 0
Sanity check:
⎛
aa *
ρ = ∫ dkP(k ) ρ k = ⎜
⎜ a * b Φ Φ
↑
↓
⎝
ab * Φ ↓ Φ ↑ ⎞
⎟
⎟
bb *
⎠
Averaging over measurement results yields
measurement-induced dephasing
22
If the state is pure, there is a corresponding
wave function for the spin alone
ψk =
1
a ↑ k Φ↑ + b ↓ k Φ↓
P(k )
{
‘Spooky’ back action:
[H ,σ z ] = 0
}
and yet:
1
P(↑| k ) =
aa * k Φ ↑ Φ ↑ k ≠ aa *
P(k )
1
P(↓| k ) =
bb * k Φ ↓ Φ ↓ k ≠ bb *
P(k )
23
0.8
P(k |↓)
Gaussian packet
P(k |↑)
0.6
x2
1/4
⎛ 1 ⎞ − 4σ 02
Φ ( x) = ⎜
e
2 ⎟
⎝ 2πσ 0 ⎠
0.4
0.2
-4
k
-2
2
2
|a|
P(↑| k ) =
e
Z
+
4
2
0
1/4
⎛ 2σ ⎞ − ( k ± k0 )2 σ 02
Φ[k ± k0 ] = ⎜
⎟ e
⎝ π ⎠
k0 k
( Δk )2
1
2
(Δk ) =
k0 k
| b |2 − ( Δk )2
P(↓| k ) =
e
Z
2
Z ≡| a | e
+
4σ
2
0
k0 k
( Δk )2
2
−
k0 k
24 ( Δk )
+|b| e
2
0.8
P(k |↓)
P(k |↑)
0.6
S = − Tr ( ρ log ρ )
0.4
0.2
-4
k
-2
2
2
|a|
P(↑| k ) =
e
Z
2
+
k0 k
( Δk )2
k0 k
| b | − ( Δk )2
P(↓| k ) =
e
Z
4
(log base e)
Average Shannon entropy
reduction
(information gain)
for a weak measurement
2
0
k
I=
2(Δk ) 2
25
Summary so far:
•
•
•
•
Spin-dependent force entangles spin with momentum
Measurement of momentum improves knowledge of spin
σ z changes even though [ H , σ z ] = 0
‘Spooky’ back action drives qubit up and down in latitude
What happens if we measure x instead of k?
The back action changes!
Qubit moved in longitude instead.
26
1D toy model with spin-dependent impulsive force
X
Bz
If we measure position just after the impulse, we
gain NO information about the momentum change.
We DO however learn the value of the magnetic field
that acted on the qubit.
2
p
H ( p , x, t ) =
− hk0 xσ z δ (t )
2m
F ( x, t ) = + hk0 σ z δ (t )
(
ψ ( x, t = 0+ ) = aeik x Φ( x) ↑ + be −ik x Φ ( x) ↓
0
0
)
− iϕ
⎛
e − iϕ ab * ⎞
aa
*
e
ab * ⎞
1 ⎛ aa *
2
ρx =
⎜ + iϕ
⎟ | Φ ( x) | = ⎜ + iϕ
⎟
PX ( x) ⎝ e a * b
bb * ⎠
bb * ⎠
⎝ e a * b
Measuring position gives no information about
momentum or spin but produces rotation of qubit:
p2
H ( p, x, t ) =
− hk0 xσ z δ (t )
2m
Non-spooky back action!
ϕ ≡ 2k0 x
Qubit in pure state.
28
− iϕ
⎛
e − iϕ ab * ⎞
aa
*
e
ab * ⎞
1 ⎛ aa *
2
ρx =
⎜ + iϕ
⎟ | Φ ( x) | = ⎜ + iϕ
⎟
PX ( x) ⎝ e a * b
bb * ⎠
bb * ⎠
⎝ e a * b
Measuring position gives no information about
momentum or spin but produces rotation of qubit
at constant latitude:
Non-spooky back action!
Qubit in pure state.
29
Same dephasing as before if average over x measurements.
DISPERSIVE READOUT is Exactly Analogous
qubit + readout pulses Qubit + resonator AMP dispersive shi8 χ
Readout amplitude 1
0
fr
f
width κ 90 Readout phase (deg) 1
θ
-­‐90 fr
0
f
Y
†
V = χ a a σ ≈ χ aX σ
↑
X
↓
z
a ≡ a +δa
(a real)
†
X ≡ δa +δa
Homodyne measurement of Y is analogous
to Stern-Gerlach measurement of momentum.
Spooky back action.
Homodyne measurement of X is analogous
to Stern-Gerlach measurement of position.
Non-spooky back action due to photon shot noise.
z
Y
†
↑
X
↓
{
ψ in = a ↑ + b ↓
+ iθ
z
V = χ a a σ ≈ χ aX σ
z
X ≡ δ a† + δ a
}α
− iθ
ψ out = a e α ↑ + b e α ↓
{
n ψ out = ae + inθ ↑ + be − inθ ↓
} nα
Each photon passing through the cavity
rotates the qubit by 2θ (but only if we measure n or X!!)
Pseudo-heterodyne
measurement
qubit + readout pulses vacuum port
Qubit + resonator X
Y
Back action has both
spooky and non-spooky
components.
Qubit is now entangled with two independent
oscillators (field modes). But still in a PURE
state if we measure one quadrature of each.
Pseudo-heterodyne
measurement
qubit + readout pulses vacuum port
Qubit + resonator X
Y
Back action has both
spooky and non-spooky
components.
Measurement efficiency is 50% but
‘added noise’ does not dephase if both
modes are fully measured.
Thanks
Many further details ‘soon’
in my
Les Houches notes.
Gerlach’s Postcard to Bohr
8 February 1922 ‘Attached [is] the experimental proof of directional
quantization. We congratulate [you] on the confirmation of your theory.’
(Historical note: they did not realize this was the discovery of electron spin.) 36
AIP Emilio Segrè Visual Archives.