rules of a quantum world
Post on 07-Dec-2014
365 Views
Preview:
DESCRIPTION
TRANSCRIPT
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?
top related