Rules of a Quantum World

Rules of a Quantum World

The Stern-Gerlach Experiment

N

S

Electron gun

Beam splits into two! Not a continuous

spread

Ignore horizontal

deflection as per Fleming’s Left

Hand Rule

½

½

Abstract Representation

Electron gunUP

DN

Arrow points to the North pole

½

½

Cascading Devices Z Z

Electron gun

Only UP

½

½

½

Z

-Z

Cascading Devices Z -Z

Electron gun

All DN!

½

½½

Z

-Z

Cascading Devices Z X

Electron gun

Half UP, Half DN!!

Arrow goes into the screen

½

½

¼

¼

Z

-Z

-X

X

Cascading Devices Z Xθ

Electron gun

At angle θ

½

½

Cos2 (θ/2)/2

Sin2 (θ/2)/2

Z

-Z

-X

X

θ

Cascading Devices Z X Z

Electron gun

½

½¼

¼

1/8

1/8

Down along Z reappears!

Z

-Z

-X

X

How do we model this behaviour?

Starting Point• Electrons must have an intrinsic state

• This state differs with orientation in 3d space

• states along different orientations are dependent

Describing State

p

1-p

Prob of being in the UP state

Prob of being in the DN state

p changes with the orientation

p

1-p

Transformations

• Tzx must be a Stochastic Transformation

– Non-negative entries

– Each column sums to 1

p

1-pTzx

Transforms state along Z axis to

state along X axis

q

1-q=

Stochastic Transformations

• Can two stochastic matrices multiply to yield an identity matrix?

– All matrix entries are non-negative

– So NO, unless each matrix is I!

Tzx

Transforms state along Z axis to state along X

axis and then transform back

=Txz I

Stochastic Transformations

ruled out

Revisiting the State Description

a

b

Can we allow for negative values

here?

How do we translate these

to probabilities?

a2 +b2 =1

Points on a unit circle

Transformations

• Tzx must be preserve Euclidean length

– (Tzx)T Tzx = I

Tzx

Transforms state along Z axis to

state along X axis

=a

b

a’

b’

Cosθ

Sinθ Cosθ

-Sinθ

For any θ

Explanations IZ

-Z

-X

X

θ

1

0 1

0

1/√2

1/√2

0

1Cos(θ/2)

Sin(θ/2) Cos(θ/2)

-Sin(θ/2)

Initial state along Z

TZZ

Initial state along Z

TZXθ

Explanations II

1

0 1

0

1/√2

1/√2

Initial state along Z

TZZ

Initial state along Z

TZX

Z

-Z

-X

X

Initial state along X

TXZ = Inverse

of TZX

0

11/√2

1/√2 1/√2

-1/√2

0

11/√2

-1/√2 1/√2

1/√2

Bringing in the Y Dimension

TYZTZX = TYX0

1

0

1

0

11/√2

1/√2 1/√2

-1/√2 a

b d

c=

+/- 1/√2

+/- 1/√2

Initial state along Y transformed to

state along X

All UPs along Y translate to equal

UPs and DNs along X

+/- 1/√2

+/- 1/√2All UPs along Y

translate to equal UPs and DNs

along XNOT POSSIBLE!!

Z

-Z

-X

X

Y-Y

Revisiting the State Description Yet Again

a

b

Can we allow for complex values

here?

How do we translate these

to probabilities?

|a|2 +|b|2 =a a + b b = 1

Complex conjugate

Revisiting Transformations

• Tzx must be preserve |a|2 +|b|2

– (Tzx)† Tzx = I

Transforms state along Z axis to

state along X axis

=a

b

a’

b’

eiεCosθ

ei(φ+ ε) Sinθ ei(ψ+ ε) Cosθ

-ei(ψ – φ+ ε) Sinθ

For any θ,ψ,ε

TYX

Conjugate Transpose

Bringing in the Y Dimension

TYZTZX = TYX0

1

0

1

0

11/√2

1/√2 1/√2

-1/√2=

eiφ’ 1/√2

eiφ’’ 1/√2

Initial state along Y transformed to

state along X

All UPs along Y translate to equal

UPs and DNs along X

eiφ 1/√2

1/√2All UPs along Y

translate to equal UPs and DNs

along XΦ=π/2, Φ’’=-π/4,

Φ’=π/4!!

1/√2

eiφ 1/√2 1/√2

-e-iφ 1/√2

Z

-Z

-X

X

Y-Y

The Final Transformations

1/√2

1/√2 1/√2

-1/√2

TZX

1/√2

i/√2 1/√2

i/√2

TYZ

1/√2

1/√2 1/√2

-1/√2

TYX=TZXTYZ=1/√2

i/√2 1/√2

i/√2

e-iπ/4/√2

eiπ/4/√2 eiπ/4/√2

-e-iπ/4/√2

Can you write the transformation from Z to a general direction in 3D space?

Summary• State vector v has complex entries and satisfies

– |v|2 = v†v = Σ |vi|2 = 1

• vi’s are called Amplitudes

• Transformations T satisfy T†T = I

– T’s are called Unitary Transformations

• When we measure a system in state v

– We get i with Probability |vi|2

Contrast with Classical States

• Take 2 bits, so state vector [p1 p2 p3 p4] corresponding to 00, 01, 10, 11 resp.

• Suppose you replace the first bit by an AND of the 2 bits with prob p and by an OR with prob 1-p?

– Show this can be written as a stochastic transformation.

Classical

Our Two Worlds

Σ vi = 1 , 0<=vi<=1

T is stochastic (non-neg, col sums 1)

|v|2 = v†v = Σ |vi|2 = 1

T is Unitary T†T = I

QuantumMeasurement

What does this mean for computation?