1. Rules of a Quantum World

2. The Stern-Gerlach Experiment
N ½
Electron gun
½
S
Ignore
horizontal Beam splits into
deflection as per two! Not a
Fleming’s Left continuous
Hand Rule spread

3. Abstract Representation
½
UP
Electron gun
DN
½
Arrow points to
the North pole

4. Cascading Devices Z Z Z
½ ½
Electron gun
½ -Z
Only UP

5. Z
Cascading Devices Z -Z
½
Electron gun -Z
½
½
All DN!

6. Z
Cascading Devices Z X X
Arrow goes into
the screen
¼ -X
½
Electron gun ¼ -Z
½
Half UP, Half DN!!

7. Z
Cascading Devices Z Xθ X
θ
At angle θ
-X
-Z
Cos2 (θ/2)/2
½
Electron gun Sin2 (θ/2)/2
½

8. Z
X
Cascading Devices Z X Z
-X
-Z
1/8
¼
½ 1/8
Electron gun ¼
½
Down along Z
reappears!

9. How do we model this behaviour?

10. Starting Point
• Electrons must have an intrinsic state
• This state differs with orientation in 3d space
• states along different orientations are dependent

11. Describing State
Prob of being in
the UP state
p p
1-p 1-p
Prob of being in
the DN state
p changes with
the orientation

12. Transformations
p q
• Tzx must be a Stochastic
Tzx =
Transformation
1-p 1-q
– Non-negative entries
Transforms state – Each column sums to 1
along Z axis to
state along X axis

13. Stochastic Transformations
• Can two stochastic
matrices multiply to
Txz Tzx = I yield an identity
matrix?
– All matrix entries are non-
negative
Transforms state along
– So NO, unless each matrix is I!
Z axis to state along X
axis and then transform
back Stochastic
Transformations
ruled out

14. Revisiting the State Description
Can we allow for
negative values
a here?
a2 +b2 =1
b
Points on a unit
How do we
circle
translate these
to probabilities?

15. Transformations
a
• Tzx must be preserve
a’
Tzx b
=
Euclidean length
b’
– (Tzx)T Tzx = I
Transforms state Cosθ -Sinθ
along Z axis to
state along X axis Sinθ Cosθ
For any θ

16. Z
Explanations I X
θ
-X
1 0 1/√2 -Z
0 1 1/√2 Cos(θ/2) -Sin(θ/2) 1
TZZ
Sin(θ/2) Cos(θ/2) 0
Initial state along
Z
Initial state along
Z
TZXθ

17. Z
X
Explanations II
-X
1 0 -Z
1/√2
0 1 1/√2 1/√2 -1/√2 1
TZZ 1/√2 1/√2 1
1/√2 1/√2 0 -1/√2 1/√2 0
Initial state along
Z TXZ = Inverse
TZX of TZX
Initial state along
Z Initial state along
X

18. Z
X
Bringing in the Y Dimension
-Y Y Initial state along
1 1
TZXTYZ = TYX
Y transformed to
state along X
0 0
-X
-Z
1/√2 -1/√2 a c 1 +/- 1/√2
=
1/√2 1/√2 b d 0 +/- 1/√2
All UPs along Y
All UPs along Y
translate to equal
translate to equal +/- 1/√2 UPs and DNs
UPs and DNs
along X
along X +/- 1/√2
NOT POSSIBLE!!

19. Revisiting the State Description
Yet Again
Can we allow for
complex values
a here?
Complex
b conjugate
|a|2 +|b|2 =a a + b b = 1
How do we
translate these
to probabilities?

20. Revisiting Transformations
Conjugate
Transpose
a
• Tzx must be preserve |a|2
a’
TYX b
=
b’ +|b|2
– (Tzx)† Tzx = I
Transforms state eiεCosθ -ei(ψ – φ+ ε) Sinθ
along Z axis to
state along X axis ei(φ+ ε) Sinθ ei(ψ+ ε) Cosθ
For any θ,ψ,ε

21. Z
X
Bringing in the Y Dimension
-Y Y Initial state along
1 1
TZXTYZ = TYX
Y transformed to
state along X
0 0
-X
-Z
1/√2 -1/√2 1/√2 -e-iφ 1/√2 1 eiφ’’ 1/√2
=
1/√2 1/√2 eiφ 1/√2 1/√2 0 eiφ’ 1/√2
All UPs along Y
All UPs along Y
translate to equal
translate to equal 1/√2 UPs and DNs
UPs and DNs
along X
along X eiφ 1/√2 Φ=π/2, Φ’’=-π/4,
Φ’=π/4!!

22. The Final Transformations
1/√2 -1/√2 1/√2 i/√2
TZX 1/√2 1/√2
TYZ i/√2 1/√2
1/√2 -1/√2 1/√2 i/√2
TYX=TZXTYZ= 1/√2 1/√2 i/√2 1/√2
e-iπ/4/√2 -e-iπ/4/√2
eiπ/4/√2 eiπ/4/√2
Can you write the transformation from Z to a general direction in 3D space?

23. 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

24. 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.

25. Our Two Worlds
Σ vi = 1 , 0<=vi<=1
T is stochastic (non-neg, col sums 1)
Classical
Measurement
Quantum
|v|2 = v†v = Σ |vi|2 = 1
T is Unitary T†T = I

26. What does this mean for
computation?