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

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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.

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( )

( )

( )

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

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-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 ↑

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-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

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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

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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

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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

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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

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* *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!

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* *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

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{ }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

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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 = ⎜ ⎟

⎝ ⎠

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-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:

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•  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.

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( )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.

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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.

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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.