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1
Quantum Measurements and Back Action
(Spooky and Otherwise)
SM Girvin Yale University
Thanks to Michel, Rob, Michael, Vijay, Aash, Simon, Dong, Claudia for discussions and comments on Les Houches notes.
Quantum Back Action is a Weird Thing
2
CNOT gate
target qubit
0 0 0 00 1 0 11 0 1 11 1 1 0
c c t t
control qubit
truth table Target is affected but control is not
Are you sure???
3
CNOT gate
target qubit
control qubit
Quantum Back Action is a Weird Thing
0 1−
0 1− 0 1−
0 1+
The control qubit is flipped from to →← !!
1 12 2
1 1CNOT 2 2Z ZX I+ −
= +
4
Stern Gerlach Experiment
Silver atom has magnetic moment due to the electron ‘spin’
Magnetic moment (spin) can point in any direction and can be measured by passing the atom through a magnetic field gradient.
Silver atom is a qubit.
It’s all about the measurement:
V BF V
µ==−∇
rrgr r
5
Stern Gerlach Experiment: Quantum Measurement
http://hyperphysics.phy-astr.gsu.edu
Silver atom has magnetic moment which can point in any direction, and yet….
(Measured)
DISPERSIVE READOUT
Readout phase (deg)
f
90
-‐90 fr
0
width κ
θ 1
Readout amplitude
f fr
01
dispersive shi8 χ
Qubit + resonator
qubit + readout pulses
AMP
(MHD)
10
Cou
nts
2000
0
2
-‐2
MEASUREMENT HISTOGRAM (data from Devoret lab)
(unlike Stern-Gerlach, the qubit is in nearly a pure state)
8
Stern & Gerlach did not measure spin!
They entangled spin with position and measured position.
http://hyperphysics.phy-astr.gsu.edu
(Measured)
9
More precisely, they used a spin-dependent force to entangle spin with momentum and
waited for momentum to turn into position.
http://hyperphysics.phy-astr.gsu.edu
(Measured)
10
Cou
nts
2000
0
2
-‐2
This is a measurement of EM modes that were entangled with the qubit ‘spin’
2
0
0
( , , ) ( )2
( , ) ( )
[ , ] 0 QND
z
z
z
pH p x t k x tm
F x t k t
H
σ δ
σ δ
σ
= −
= +
= ⇒
h
h
1D toy model with spin-dependent impulsive force
X zB
12
‘Quantum Non-Demolition’ (QND) Measurements are Repeatable
Z
Z
measurement
Z
measurement
Z
measurement
Z
First result is random, rest are repeats.
2
0
0
( , , ) ( )2
( , ) ( )
z
z
pH p x t k x tm
F x t k t
σ δ
σ δ
= −
= +
h
h
1D toy model with spin-dependent impulsive force
X zB
We will measure momentum just after impulse rather than waiting for it to turn into position.
14
( )
( )
( )
0 0
0 0
( , 0 ) ( )
( , 0 ) ( ) ( )
[ , 0 ] [ ] [ ]
ik x ik x
x t a b x
x t ae x be x
k t a k k b k k
ψ
ψ
ψ
−
−+
+
= = ↑ + ↓ Φ
= = Φ ↑ + Φ ↓
= = Φ − ↑ + Φ + ↓
Input product state
Output entangled entangled state
Output state in momentum basis
2
0( , , ) ( )2
zpH p x t k x tm
σ δ= − h
15
-4 -2 2 4
0.2
0.4
0.6
0.8
( | )P k ↑( | )P k ↓
-4 -2 2 4
0.2
0.4
0.6
0.8
Strong measurement
Weak measurement
k
k
(gaussian input packet)
( | )P k ↓ ( | )P k ↑
16
-4 -2 2 4
0.2
0.4
0.6
0.8
( | )P k ↑( | )P k ↓
k
( | ) is easy to understandbut what we need is:
( | )
P k
P k
↑
↑
Practice on two continuous variables
17
2( , ) ( , )P x y x yψ=
x
y
Y
2
2
( , ) ( , )( | )( )( , )
x Y P x YP x YP Ydx x Y
ψ
ψ= =
ʹ′ ʹ′∫
( , ) ( | ) ( )P x y P x y P y=
Density Matrix Equivalent
18
Tr ( )k
k k k kk k P kρ ρ
ρρ
= =
Reduced density matrix for spin conditioned on measurement of momentum
k
k kρ
ρ
Full density matrix projected onto observed momentum state
Reduced density matrix for spin given observed value of momentum
Density Matrix Equivalent
19
Full state a
a bb
ψ ↑
↑ ↓
↓
⎛ ⎞Φ= ↑ Φ + ↓ Φ = ⎜ ⎟
⎜ ⎟Φ⎝ ⎠
* *
* *
aa ab
a b bbρ ψ ψ ↑ ↑ ↑ ↓
↓ ↑ ↓ ↓
⎛ ⎞Φ Φ Φ Φ= = ⎜ ⎟
⎜ ⎟Φ Φ Φ Φ⎝ ⎠
Full density matrix
20
Tr ( )k
k k k kk k P kρ ρ
ρρ
= =
Reduced density matrix for spin conditioned on measurement of momentum
* *
* *
aa ab
a b bbρ ↑ ↑ ↑ ↓
↓ ↑ ↓ ↓
⎛ ⎞Φ Φ Φ Φ= ⎜ ⎟⎜ ⎟Φ Φ Φ Φ⎝ ⎠
* *1( ) * *k
aa k k ab k kP k a b k k bb k k
ρ ↑ ↑ ↑ ↓
↓ ↑ ↓ ↓
⎛ ⎞Φ Φ Φ Φ= ⎜ ⎟
⎜ ⎟Φ Φ Φ Φ⎝ ⎠
Full density matrix
Reduced density matrix for spin conditioned on measurement of momentum
21
* *1( ) * *k
aa k k ab k kP k a b k k bb k k
ρ ↑ ↑ ↑ ↓
↓ ↑ ↓ ↓
⎛ ⎞Φ Φ Φ Φ= ⎜ ⎟
⎜ ⎟Φ Φ Φ Φ⎝ ⎠
Reduced density matrix for spin conditioned on measurement of momentum
Det 0 Tr 1
1 0 exists a basis s.t.
0 0
k
k
k
ρ
ρ
ρ
=
=
⎛ ⎞⇒ ∃ = ⎜ ⎟
⎝ ⎠
Easy to verify conditional state is pure:
If we fully measure the state of the ‘bath’ then the conditional state remains pure!
22
* *1( ) * *k
aa k k ab k kP k a b k k bb k k
ρ ↑ ↑ ↑ ↓
↓ ↑ ↓ ↓
⎛ ⎞Φ Φ Φ Φ= ⎜ ⎟
⎜ ⎟Φ Φ Φ Φ⎝ ⎠
Reduced density matrix for spin conditioned on measurement of momentum
Sanity check:
* *( )
* *k
aa abdkP k
a b bbρ ρ ↓ ↑
↑ ↓
⎛ ⎞Φ Φ= = ⎜ ⎟
⎜ ⎟Φ Φ⎝ ⎠∫
Averaging over measurement results yields measurement-induced dephasing
Det 0kρ =Easy to verify conditional state is pure:
If the state is pure, there is a corresponding wave function for the spin alone
23
{ }1( )k a k b kP k
ψ↑ ↓
= ↑ Φ + ↓ Φ
1( | ) * *( )
P k aa k k aaP k ↑ ↑
↑ = Φ Φ ≠
‘Spooky’ back action: and yet: [ , ] 0zH σ =
1( | ) * *( )
P k bb k k bbP k ↓ ↓
↓ = Φ Φ ≠
Gaussian packet
24
2 20 0
1/42( )0
02[ ] k kk k e σσπ
− ±⎛ ⎞Φ ± = ⎜ ⎟
⎝ ⎠-4 -2 2 4
0.2
0.4
0.6
0.8
( | )P k ↑( | )P k ↓
k
02
02
2( )
2( )
| |( | )
| |( | )
k kk
k kk
aP k eZbP k eZ
+Δ
−Δ
↑ =
↓ =
220
1( )4
kσ
Δ =
0 02 22 2( ) ( )| | | |
k k k kk kZ a e b e
+ −Δ Δ≡ +
2
20
1/44
20
1( )2
x
x e σ
πσ
−⎛ ⎞Φ = ⎜ ⎟
⎝ ⎠
25
-4 -2 2 4
0.2
0.4
0.6
0.8
( | )P k ↑( | )P k ↓
k
02
02
2( )
2( )
| |( | )
| |( | )
k kk
k kk
aP k eZbP k eZ
+Δ
−Δ
↑ =
↓ =
Average Shannon entropy reduction
(information gain) for a weak measurement
20
22( )kIk
=Δ
( ) Tr logS ρ ρ= −
(log base e)
Summary so far:
26
• Spin-dependent force entangles spin with momentum • Measurement of momentum improves knowledge of spin • changes even though • ‘Spooky’ back action drives qubit up and down in latitude
zσ [ , ] 0zH σ =
What happens if we measure x instead of k?
The back action changes!
Qubit moved in longitude instead.
2
0
0
( , , ) ( )2
( , ) ( )
z
z
pH p x t k x tm
F x t k t
σ δ
σ δ
= −
= +
h
h
1D toy model with spin-dependent impulsive force
X zB
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.
28
( )0 0( , 0 ) ( ) ( )ik x ik xx t ae x be xψ −+= = Φ ↑ + Φ ↓
2* * * *1 | ( ) |( ) * * * *
i i
x i iX
aa e ab aa e abx
P x e a b bb e a b bb
ϕ ϕ
ϕ ϕρ
− −
+ +
⎛ ⎞ ⎛ ⎞= Φ =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
Measuring position gives no information about momentum or spin but produces rotation of qubit:
02k xϕ ≡2
0( , , ) ( )2
zpH p x t k x tm
σ δ= −h
Non-spooky back action! Qubit in pure state.
29
2* * * *1 | ( ) |( ) * * * *
i i
x i iX
aa e ab aa e abx
P x e a b bb e a b bb
ϕ ϕ
ϕ ϕρ
− −
+ +
⎛ ⎞ ⎛ ⎞= Φ =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
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.
Same dephasing as before if average over x measurements.
DISPERSIVE READOUT is Exactly Analogous
Readout phase (deg)
f
90
-‐90 fr
0
width κ
θ 1
Readout amplitude
f fr
01
dispersive shi8 χ
Qubit + resonator
qubit + readout pulses
AMP
†
†
( real)
z zV a a aXa a a aX a a
χ σ χ σ
δ
δ δ
= ≈
≡ +
≡ +
X
Y
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 zV a a aXX a a
χ σ χ σ
δ δ
= ≈
≡ +X
Y
↑
↓
{ }in
outi i
a b
a e b eθ θ
ψ α
ψ α α+ −
= ↑ + ↓
= ↑ + ↓
{ }outin inn ae be nθ θψ α+ −= ↑ + ↓
Each photon passing through the cavity rotates the qubit by (but only if we measure n or X!!) 2θ
Pseudo-heterodyne measurement
Qubit + resonator
qubit + readout pulses
vacuum port
X
Y
Qubit is now entangled with two independent oscillators (field modes). But still in a PURE state if we measure one quadrature of each.
Back action has both spooky and non-spooky components.
Pseudo-heterodyne measurement
Qubit + resonator
qubit + readout pulses
vacuum port
X
Y
Measurement efficiency is 50% but ‘added noise’ does not dephase if both modes are fully measured.
Back action has both spooky and non-spooky components.
Thanks
Many further details ‘soon’ in my
Les Houches notes.
36
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.) AIP Emilio Segrè Visual Archives.