group theory in quantum mechanics lecture 3 analyzers ... · 1/20/2015 · tuesday, january 20,...
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Group Theory in Quantum MechanicsLecture 3 (1.20.15)
Analyzers, operators, and group axioms (Quantum Theory for Computer Age - Ch. 1-2 of Unit 1 )
(Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-3 of Ch. 1 )
Review: Axioms 1-4 and“Do-Nothing”vs“ Do-Something” analyzers
Abstraction of Axiom-4 to define projection and unitary operators Projection operators and resolution of identity
Unitary operators and matrices that do something (or “nothing”) Diagonal unitary operators Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker ⊗-products Axiom-4 similarity transformationMatrix representation of beam analyzers Non-unitary “killer” devices: Sorter-counter, filter Unitary “non-killer” devices: 1/2-wave plate, 1/4-wave plateHow analyzers “peek” and how that changes outcomes Peeking polarizers and coherence loss Classical Bayesian probability vs. Quantum probability Feynman 〈j⏐k〉-axioms compared to Group axioms
1Tuesday, January 20, 2015
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Review: Axioms 1-4 and“Do-Nothing”vs“ Do-Something” analyzers
Abstraction of Axiom-4 to define projection and unitary operators Projection operators and resolution of identity
Unitary operators and matrices that do something (or “nothing”) Diagonal unitary operators Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker ⊗-products Axiom-4 similarity transformation
Matrix representation of beam analyzers Non-unitary “killer” devices: Sorter-counter, filter Unitary “non-killer” devices: 1/2-wave plate, 1/4-wave plate
How analyzers “peek” and how that changes outcomes Peeking polarizers and coherence loss
2Tuesday, January 20, 2015
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(1) The probability axiomThe first axiom deals with physical interpretation of amplitudes . Axiom 1: The absolute square gives probability for occurrence in state-j of a system that started in state-k'=1',2',..,or n' from one sorter and then was forced to choose between states j=1,2,...,n by another sorter.
(3) The orthonormality or identity axiom The third axiom concerns the amplitude for "re measurement" by the same analyzer. Axiom 3: If identical analyzers are used twice or more the amplitude for a passed state-k is one, and for all others it is zero:
(2) The conjugation or inversion axiom (time reversal symmetry) The second axiom concerns going backwards through a sorter or the reversal of amplitudes. Axiom 2: The complex conjugate of an amplitude equals its reverse:
j k ' *
Feynman amplitude axioms 1-4Feynman-Dirac Interpretation of
〈j⏐k′ 〉=Amplitude of state-j afterstate-k′ is forced to choose
from available m-type states
j k '2= j k ' * j k '
j k '
j k ' *
= k ' j
j k '
j k = δ jk =
1 if: j=k0 if: j ≠ k
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪= j ' k '
(4) The completeness or closure axiomThe fourth axiom concerns the "Do-nothing" property of an ideal analyzer, that is, a sorter followed by an "unsorter" or "put-back-togetherer" as sketched above.Axiom 4. Ideal sorting followed by ideal recombination of amplitudes has no effect:
j" m ' =
k=1
n∑ j" k k m '
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x'-polarized light
y'-polarized light x'
y
xy'Θout -polarized light
(a)“Do-Nothing”Analyzer
(b)Simulation setting ofinput
polarization2Θin =βin=200°
tilt of analyzer
analyzer activity angle Ω(Ω=0 means do-nothing)
Ω
inputpolarization
Θin =βin/2=100°
Θanalyzer=-30°
Θout =Θin
in
No change if analyzerdoes nothing
analyzerβ
analyzerΘ = -30°
=2Θ
Θin=100°
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x'-polarized light
y'-polarized light x'
y
xy'Θout -polarized light
(a)“Do-Nothing”Analyzer
(b)Simulation setting ofinput
polarization2Θin =βin=200°
tilt of analyzer
analyzer activity angle Ω(Ω=0 means do-nothing)
Ω
inputpolarization
Θin =βin/2=100°
Θanalyzer=-30°
Θout =Θin
in
No change if analyzerdoes nothing
analyzerβ
analyzerΘ = -30°
=2Θ
Θin=100°
Imagine final xy-sorter analyzes output beam into x and y-components.
x-Output is: 〈x|Θout〉= 〈x|x'〉〈x'|Θin〉+〈x|y'〉〈y'|Θin〉=cosΘcos(Θin-Θ) - sinΘsin(Θin-Θ)=cos Θin y-Output is: 〈y|Θout〉= 〈y|x'〉〈x'|Θin〉+〈y|y'〉〈y'|Θin〉=sinΘcos(Θin-Θ) - cosΘsin(Θin-Θ)=sin Θin. (Recall cos(a+b)=cosa cosb-sina sinb and sin(a+b)=sina cosb+cosa sinb )
Amplitude in x or y-channel is sum over x' and y'-amplitudes 〈x'|Θin〉=cos(Θin−Θ) 〈y'|Θin〉=sin(Θin−Θ) with relative angle Θin−Θ of Θin to Θ-analyzer axes-(x',y')in products with final xy-sorter: lab x-axis: 〈x|x'〉 = cosΘ = 〈y|y'〉 y-axis: 〈y|x'〉 = sinΘ = -〈x|y'〉.
Conclusion: 〈x|Θout〉 = cos Θout = cos Θin or: Θout= Θin so “Do-Nothing” Analyzer in fact does nothing.
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Review: Axioms 1-4 and“Do-Nothing”vs“ Do-Something” analyzers
Abstraction of Axiom-4 to define projection and unitary operators Projection operators and resolution of identity
Unitary operators and matrices that do something (or “nothing”) Diagonal unitary operators Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker ⊗-products Axiom-4 similarity transformation
Matrix representation of beam analyzers Non-unitary “killer” devices: Sorter-counter, filter Unitary “non-killer” devices: 1/2-wave plate, 1/4-wave plate
How analyzers “peek” and how that changes outcomes Peeking polarizers and coherence loss Classical Bayesian probability vs. Quantum probability
Feynman 〈j⏐k〉-axioms compared to Group axioms
6Tuesday, January 20, 2015
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Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉 may be “abstracted" three different ways
Abstraction of Axiom 4 to define projection and unitary operators
k=1
n
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Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉 may be “abstracted" three different ways
Left abstraction gives bra-transform:
〈j′′⏐=∑〈j′′⏐k〉〈k⏐
Abstraction of Axiom 4 to define projection and unitary operators
k=1
n
k=1
n
x x ' x y '
y x ' y y '
⎛
⎝⎜⎜
⎞
⎠⎟⎟= cosθ −sinθ
sinθ cosθ⎛
⎝⎜⎞
⎠⎟
Recall bra-ket Transformation Matrix
Tm,n′=〈m⏐ n′〉
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Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉 may be “abstracted" three different ways
Left abstraction gives bra-transform: Right abstraction gives ket-transform:
〈j′′⏐=∑〈j′′⏐k〉〈k⏐ ⏐m′〉=∑ ⏐k〉〈k⏐m′〉
Abstraction of Axiom 4 to define projection and unitary operators
k=1
n
k=1
n
k=1
n
x x ' x y '
y x ' y y '
⎛
⎝⎜⎜
⎞
⎠⎟⎟= cosθ −sinθ
sinθ cosθ⎛
⎝⎜⎞
⎠⎟
Recall bra-ket Transformation Matrix
Tm,n′=〈m⏐ n′〉
9Tuesday, January 20, 2015
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Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉 may be “abstracted" three different ways
Left abstraction gives bra-transform: Right abstraction gives ket-transform:
〈j′′⏐=∑〈j′′⏐k〉〈k⏐ ⏐m′〉=∑ ⏐k〉〈k⏐m′〉
Center abstraction gives ket-bra identity operator:
1=∑⏐k〉〈k⏐=∑⏐k′〉〈k′⏐=∑⏐k′′〉〈k′′⏐=...
Abstraction of Axiom 4 to define projection and unitary operators
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
x x ' x y '
y x ' y y '
⎛
⎝⎜⎜
⎞
⎠⎟⎟= cosθ −sinθ
sinθ cosθ⎛
⎝⎜⎞
⎠⎟
Recall bra-ket Transformation Matrix
Tm,n′=〈m⏐ n′〉
10Tuesday, January 20, 2015
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Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉 may be “abstracted" three different ways
Left abstraction gives bra-transform: Right abstraction gives ket-transform:
〈j′′⏐=∑〈j′′⏐k〉〈k⏐ ⏐m′〉=∑ ⏐k〉〈k⏐m′〉
Center abstraction gives ket-bra identity operator:
1=∑⏐k〉〈k⏐=∑⏐k′〉〈k′⏐=∑⏐k′′〉〈k′′⏐=...
Resolution of Identity into Projectors {⏐1〉〈1⏐, ⏐2〉〈2⏐..} or {⏐1′〉〈1′⏐, ⏐2′〉〈2′⏐..} or {⏐1′′〉〈1′′⏐, ⏐2′′〉〈2′′⏐..}
P1= ⏐1〉〈1⏐, P2= ⏐2〉〈2⏐,.. or P1′= ⏐1′〉〈1′⏐, P2′= ⏐2′〉〈2′⏐ etc.
Abstraction of Axiom 4 to define projection and unitary operators
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
x x ' x y '
y x ' y y '
⎛
⎝⎜⎜
⎞
⎠⎟⎟= cosθ −sinθ
sinθ cosθ⎛
⎝⎜⎞
⎠⎟
Recall bra-ket Transformation Matrix
Tm,n′=〈m⏐ n′〉
11Tuesday, January 20, 2015
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Review: Axioms 1-4 and“Do-Nothing”vs“ Do-Something” analyzers
Abstraction of Axiom-4 to define projection and unitary operators Projection operators and resolution of identity
Unitary operators and matrices that do something (or “nothing”) Diagonal unitary operators Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker ⊗-products Axiom-4 similarity transformation
Matrix representation of beam analyzers Non-unitary “killer” devices: Sorter-counter, filter Unitary “non-killer” devices: 1/2-wave plate, 1/4-wave plate
How analyzers “peek” and how that changes outcomes Peeking polarizers and coherence loss Classical Bayesian probability vs. Quantum probability
Feynman 〈j⏐k〉-axioms compared to Group axioms
12Tuesday, January 20, 2015
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Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉 may be “abstracted" three different ways
Left abstraction gives bra-transform: Right abstraction gives ket-transform:
〈j′′⏐=∑〈j′′⏐k〉〈k⏐ ⏐m′〉=∑ ⏐k〉〈k⏐m′〉
Center abstraction gives ket-bra identity operator:
1=∑⏐k〉〈k⏐=∑⏐k′〉〈k′⏐=∑⏐k′′〉〈k′′⏐=...
Resolution of Identity into Projectors {⏐1〉〈1⏐, ⏐2〉〈2⏐..} or {⏐1′〉〈1′⏐, ⏐2′〉〈2′⏐..} or {⏐1′′〉〈1′′⏐, ⏐2′′〉〈2′′⏐..}
P1= ⏐1〉〈1⏐, P2= ⏐2〉〈2⏐,.. or P1′= ⏐1′〉〈1′⏐, P2′= ⏐2′〉〈2′⏐ etc.
Abstraction of Axiom 4 to define projection and unitary operators
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
x x ' x y '
y x ' y y '
⎛
⎝⎜⎜
⎞
⎠⎟⎟= cosθ −sinθ
sinθ cosθ⎛
⎝⎜⎞
⎠⎟
Recall bra-ket Transformation Matrix
Tm,n′=〈m⏐ n′〉
13Tuesday, January 20, 2015
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Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉 may be “abstracted" three different ways
Left abstraction gives bra-transform: Right abstraction gives ket-transform:
〈j′′⏐=∑〈j′′⏐k〉〈k⏐ ⏐m′〉=∑ ⏐k〉〈k⏐m′〉
Center abstraction gives ket-bra identity operator:
1=∑⏐k〉〈k⏐=∑⏐k′〉〈k′⏐=∑⏐k′′〉〈k′′⏐=...
Resolution of Identity into Projectors {⏐1〉〈1⏐, ⏐2〉〈2⏐..} or {⏐1′〉〈1′⏐, ⏐2′〉〈2′⏐..} or {⏐1′′〉〈1′′⏐, ⏐2′′〉〈2′′⏐..}
P1= ⏐1〉〈1⏐, P2= ⏐2〉〈2⏐,.. or P1′= ⏐1′〉〈1′⏐, P2′= ⏐2′〉〈2′⏐ etc.
Abstraction of Axiom 4 to define projection and unitary operators
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
x Py x x Py y
y Py x y Py y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 0 0
0 1⎛⎝⎜
⎞⎠⎟
Projections of unit vector ⏐x′〉 ontounit kets ⏐x〉 and ⏐y〉
Py!x"#=!y#$y!x"# =!y#sin !
Px!x"#=!x#$x!x"# =!x#cos !
!y#
!x#$y!x"#
$x!x"# !x"#
!
x x ' x y '
y x ' y y '
⎛
⎝⎜⎜
⎞
⎠⎟⎟= cosθ −sinθ
sinθ cosθ⎛
⎝⎜⎞
⎠⎟
Recall bra-ket Transformation Matrix
Tm,n′=〈m⏐ n′〉
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Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉 may be “abstracted" three different ways
Left abstraction gives bra-transform: Right abstraction gives ket-transform:
〈j′′⏐=∑〈j′′⏐k〉〈k⏐ ⏐m′〉=∑ ⏐k〉〈k⏐m′〉
Center abstraction gives ket-bra identity operator:
1=∑⏐k〉〈k⏐=∑⏐k′〉〈k′⏐=∑⏐k′′〉〈k′′⏐=...
Resolution of Identity into Projectors {⏐1〉〈1⏐, ⏐2〉〈2⏐..} or {⏐1′〉〈1′⏐, ⏐2′〉〈2′⏐..} or {⏐1′′〉〈1′′⏐, ⏐2′′〉〈2′′⏐..}
P1= ⏐1〉〈1⏐, P2= ⏐2〉〈2⏐,.. or P1′= ⏐1′〉〈1′⏐, P2′= ⏐2′〉〈2′⏐ etc.
Abstraction of Axiom 4 to define projection and unitary operators
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
Py!x"#=!y#$y!x"# =!y#sin !
Px!x"#=!x#$x!x"# =!x#cos !
!y#
!x#$y!x"#
$x!x"# !x"#
!x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
Py!"#=!y#$y!"# Px!"#=!x#$x!"#
!y#
!x#$y!"#
$x!"#
!"#
x Py x x Py y
y Py x y Py y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 0 0
0 1⎛⎝⎜
⎞⎠⎟
Projections of unit vector ⏐x′〉 ontounit kets ⏐x〉 and ⏐y〉
Projections of general state ⏐Ψ〉 ...
x x ' x y '
y x ' y y '
⎛
⎝⎜⎜
⎞
⎠⎟⎟= cosθ −sinθ
sinθ cosθ⎛
⎝⎜⎞
⎠⎟
Recall bra-ket Transformation Matrix
Tm,n′=〈m⏐ n′〉
15Tuesday, January 20, 2015
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Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉 may be “abstracted" three different ways
Left abstraction gives bra-transform: Right abstraction gives ket-transform:
〈j′′⏐=∑〈j′′⏐k〉〈k⏐ ⏐m′〉=∑ ⏐k〉〈k⏐m′〉
Center abstraction gives ket-bra identity operator:
1=∑⏐k〉〈k⏐=∑⏐k′〉〈k′⏐=∑⏐k′′〉〈k′′⏐=...
Resolution of Identity into Projectors {⏐1〉〈1⏐, ⏐2〉〈2⏐..} or {⏐1′〉〈1′⏐, ⏐2′〉〈2′⏐..} or {⏐1′′〉〈1′′⏐, ⏐2′′〉〈2′′⏐..}
P1= ⏐1〉〈1⏐, P2= ⏐2〉〈2⏐,.. or P1′= ⏐1′〉〈1′⏐, P2′= ⏐2′〉〈2′⏐ etc.
Abstraction of Axiom 4 to define projection and unitary operators
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
Projections of unit vector ⏐x′〉 ontounit kets ⏐x〉 and ⏐y〉
Py!"#=!y#$y!"# Px!"#=!x#$x!"#
!y#
!x#$y!"#
$x!"#
!"#Projections of general state ⏐Ψ〉 ...
x Py x x Py y
y Py x y Py y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 0 0
0 1⎛⎝⎜
⎞⎠⎟
...must add up to⏐Ψ〉Px⏐Ψ〉 + Py⏐Ψ〉 =⏐Ψ〉(Px + Py)⏐Ψ〉 =⏐Ψ〉
Py!x"#=!y#$y!x"# =!y#sin !
Px!x"#=!x#$x!x"# =!x#cos !
!y#
!x#$y!x"#
$x!x"# !x"#
!
x x ' x y '
y x ' y y '
⎛
⎝⎜⎜
⎞
⎠⎟⎟= cosθ −sinθ
sinθ cosθ⎛
⎝⎜⎞
⎠⎟
Recall bra-ket Transformation Matrix
Tm,n′=〈m⏐ n′〉
16Tuesday, January 20, 2015
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Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉 may be “abstracted" three different ways
Left abstraction gives bra-transform: Right abstraction gives ket-transform:
〈j′′⏐=∑〈j′′⏐k〉〈k⏐ ⏐m′〉=∑ ⏐k〉〈k⏐m′〉
Center abstraction gives ket-bra identity operator:
1=∑⏐k〉〈k⏐=∑⏐k′〉〈k′⏐=∑⏐k′′〉〈k′′⏐=...
Resolution of Identity into Projectors {⏐1〉〈1⏐, ⏐2〉〈2⏐..} or {⏐1′〉〈1′⏐, ⏐2′〉〈2′⏐..} or {⏐1′′〉〈1′′⏐, ⏐2′′〉〈2′′⏐..}
P1= ⏐1〉〈1⏐, P2= ⏐2〉〈2⏐,.. or P1′= ⏐1′〉〈1′⏐, P2′= ⏐2′〉〈2′⏐ etc.
Abstraction of Axiom 4 to define projection and unitary operators
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
Projections of unit vector ⏐x′〉 ontounit kets ⏐x〉 and ⏐y〉
Py!"#=!y#$y!"# Px!"#=!x#$x!"#
!y#
!x#$y!"#
$x!"#
!"#Projections of general state ⏐Ψ〉 ...
x Py x x Py y
y Py x y Py y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 0 0
0 1⎛⎝⎜
⎞⎠⎟
...must add up to⏐Ψ〉Px⏐Ψ〉 + Py⏐Ψ〉 =⏐Ψ〉(Px + Py)⏐Ψ〉 =⏐Ψ〉
Py!x"#=!y#$y!x"# =!y#sin !
Px!x"#=!x#$x!x"# =!x#cos !
!y#
!x#$y!x"#
$x!x"# !x"#
!
...and so Pm projectors must add up to identity operator... 1 = Px + Py
x x ' x y '
y x ' y y '
⎛
⎝⎜⎜
⎞
⎠⎟⎟= cosθ −sinθ
sinθ cosθ⎛
⎝⎜⎞
⎠⎟
Recall bra-ket Transformation Matrix
Tm,n′=〈m⏐ n′〉
17Tuesday, January 20, 2015
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Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉 may be “abstracted" three different ways
Left abstraction gives bra-transform: Right abstraction gives ket-transform:
〈j′′⏐=∑〈j′′⏐k〉〈k⏐ ⏐m′〉=∑ ⏐k〉〈k⏐m′〉
Center abstraction gives ket-bra identity operator:
1=∑⏐k〉〈k⏐=∑⏐k′〉〈k′⏐=∑⏐k′′〉〈k′′⏐=...
Resolution of Identity into Projectors {⏐1〉〈1⏐, ⏐2〉〈2⏐..} or {⏐1′〉〈1′⏐, ⏐2′〉〈2′⏐..} or {⏐1′′〉〈1′′⏐, ⏐2′′〉〈2′′⏐..}
P1= ⏐1〉〈1⏐, P2= ⏐2〉〈2⏐,.. or P1′= ⏐1′〉〈1′⏐, P2′= ⏐2′〉〈2′⏐ etc.
Abstraction of Axiom 4 to define projection and unitary operators
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
Projections of unit vector ⏐x′〉 ontounit kets ⏐x〉 and ⏐y〉
Py!"#=!y#$y!"# Px!"#=!x#$x!"#
!y#
!x#$y!"#
$x!"#
!"#Projections of general state ⏐Ψ〉 ...
x Py x x Py y
y Py x y Py y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 0 0
0 1⎛⎝⎜
⎞⎠⎟
...must add up to⏐Ψ〉Px⏐Ψ〉 + Py⏐Ψ〉 =⏐Ψ〉(Px + Py)⏐Ψ〉 =⏐Ψ〉
Py!x"#=!y#$y!x"# =!y#sin !
Px!x"#=!x#$x!x"# =!x#cos !
!y#
!x#$y!x"#
$x!x"# !x"#
!
...and so Pm projectors must add up to identity operator... 1 = Px + Py
and identity matrix... 1 00 1
⎛⎝⎜
⎞⎠⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟+ 0 0
0 1⎛⎝⎜
⎞⎠⎟
x x ' x y '
y x ' y y '
⎛
⎝⎜⎜
⎞
⎠⎟⎟= cosθ −sinθ
sinθ cosθ⎛
⎝⎜⎞
⎠⎟
Recall bra-ket Transformation Matrix
Tm,n′=〈m⏐ n′〉
18Tuesday, January 20, 2015
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Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉 may be “abstracted" three different ways
Left abstraction gives bra-transform: Right abstraction gives ket-transform:
〈j′′⏐=∑〈j′′⏐k〉〈k⏐ ⏐m′〉=∑ ⏐k〉〈k⏐m′〉
Center abstraction gives ket-bra identity operator:
1=∑⏐k〉〈k⏐=∑⏐k′〉〈k′⏐=∑⏐k′′〉〈k′′⏐=...
Resolution of Identity into Projectors {⏐1〉〈1⏐, ⏐2〉〈2⏐..} or {⏐1′〉〈1′⏐, ⏐2′〉〈2′⏐..} or {⏐1′′〉〈1′′⏐, ⏐2′′〉〈2′′⏐..}
P1= ⏐1〉〈1⏐, P2= ⏐2〉〈2⏐,.. or P1′= ⏐1′〉〈1′⏐, P2′= ⏐2′〉〈2′⏐ etc.
Abstraction of Axiom 4 to define projection and unitary operators
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
Projections of unit vector ⏐x′〉 ontounit kets ⏐x〉 and ⏐y〉
Py!"#=!y#$y!"# Px!"#=!x#$x!"#
!y#
!x#$y!"#
$x!"#
!"#Projections of general state ⏐Ψ〉 ...
x Py x x Py y
y Py x y Py y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 0 0
0 1⎛⎝⎜
⎞⎠⎟
...must add up to⏐Ψ〉Px⏐Ψ〉 + Py⏐Ψ〉 =⏐Ψ〉(Px + Py)⏐Ψ〉 =⏐Ψ〉
Py!x"#=!y#$y!x"# =!y#sin !
Px!x"#=!x#$x!x"# =!x#cos !
!y#
!x#$y!x"#
$x!x"# !x"#
!
...and so Pm projectors must add up to identity operator... 1 = Px + Py
and identity matrix... 1 00 1
⎛⎝⎜
⎞⎠⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟+ 0 0
0 1⎛⎝⎜
⎞⎠⎟
..as required by Axiom 4:
x x ' x y '
y x ' y y '
⎛
⎝⎜⎜
⎞
⎠⎟⎟= cosθ −sinθ
sinθ cosθ⎛
⎝⎜⎞
⎠⎟
Recall bra-ket Transformation Matrix
Tm,n′=〈m⏐ n′〉
19Tuesday, January 20, 2015
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Review: Axioms 1-4 and“Do-Nothing”vs“ Do-Something” analyzers
Abstraction of Axiom-4 to define projection and unitary operators Projection operators and resolution of identity
Unitary operators and matrices that do something (or “nothing”) Diagonal unitary operators Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker ⊗-products Axiom-4 similarity transformation
Matrix representation of beam analyzers Non-unitary “killer” devices: Sorter-counter, filter Unitary “non-killer” devices: 1/2-wave plate, 1/4-wave plate
How analyzers “peek” and how that changes outcomes Peeking polarizers and coherence loss Classical Bayesian probability vs. Quantum probability
Feynman 〈j⏐k〉-axioms compared to Group axioms
20Tuesday, January 20, 2015
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|Ψ〉T|Ψ〉
|Ψ〉
analyzer
Tanalyzer
T|Ψ〉T|Ψ〉 input stateoutput state
TTUnitary operators and matrices that do something (or “nothing”)
Fig. 3.1.1 Effect of analyzer
represented by ket vector transformation of ⏐Ψ〉
to new ket vector T⏐Ψ〉 .
21Tuesday, January 20, 2015
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|Ψ〉T|Ψ〉
|Ψ〉
analyzer
Tanalyzer
T|Ψ〉T|Ψ〉 input stateoutput state
TTUnitary operators and matrices that do something (or “nothing”)
First is the “do-nothing” identity operator 1... 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐ = Px + Py
k=1
2
Fig. 3.1.1 Effect of analyzer
represented by ket vector transformation of ⏐Ψ〉
to new ket vector T⏐Ψ〉 .
22Tuesday, January 20, 2015
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|Ψ〉T|Ψ〉
|Ψ〉
analyzer
Tanalyzer
T|Ψ〉T|Ψ〉 input stateoutput state
TTUnitary operators and matrices that do something (or “nothing”)
First is the “do-nothing” identity operator 1... 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐ = Px + Py and matrix representation: 1 0
0 1⎛⎝⎜
⎞⎠⎟
= 1 00 0
⎛⎝⎜
⎞⎠⎟
+ 0 00 1
⎛⎝⎜
⎞⎠⎟
k=1
2
Fig. 3.1.1 Effect of analyzer
represented by ket vector transformation of ⏐Ψ〉
to new ket vector T⏐Ψ〉 .
23Tuesday, January 20, 2015
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|Ψ〉T|Ψ〉
|Ψ〉
analyzer
Tanalyzer
T|Ψ〉T|Ψ〉 input stateoutput state
TT Fig. 3.1.1 Effect of analyzer
represented by ket vector transformation of ⏐Ψ〉
to new ket vector T⏐Ψ〉 .
Unitary operators and matrices that do something (or “nothing”)
First is the “do-nothing” identity operator 1... 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐ = Px + Py and matrix representation: 1 0
0 1⎛⎝⎜
⎞⎠⎟
= 1 00 0
⎛⎝⎜
⎞⎠⎟
+ 0 00 1
⎛⎝⎜
⎞⎠⎟
k=1
2
Next is the diagonal “do-something” unitary* operator T... T=∑⏐k〉e-iΩkt〈k⏐= ⏐x〉e-iΩxt〈x⏐ + ⏐y〉e-iΩyt〈y⏐ = e-iΩxt Px + e-iΩyt Py and its matrix representation: e− iΩxt 0
0 e− iΩxt
⎛
⎝⎜
⎞
⎠⎟ =
e− iΩxt 00 0
⎛
⎝⎜⎞
⎠⎟+ 0 0
0 e− iΩxt
⎛
⎝⎜⎞
⎠⎟
24Tuesday, January 20, 2015
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|Ψ〉T|Ψ〉
|Ψ〉
analyzer
Tanalyzer
T|Ψ〉T|Ψ〉 input stateoutput state
TT Fig. 3.1.1 Effect of analyzer
represented by ket vector transformation of ⏐Ψ〉
to new ket vector T⏐Ψ〉 .
Unitary operators and matrices that do something (or “nothing”)
First is the “do-nothing” identity operator 1... 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐ = Px + Py and matrix representation: 1 0
0 1⎛⎝⎜
⎞⎠⎟
= 1 00 0
⎛⎝⎜
⎞⎠⎟
+ 0 00 1
⎛⎝⎜
⎞⎠⎟
k=1
2
Next is the diagonal “do-something” unitary* operator T... T=∑⏐k〉e-iΩkt〈k⏐= ⏐x〉e-iΩxt〈x⏐ + ⏐y〉e-iΩyt〈y⏐ = e-iΩxt Px + e-iΩyt Py and its matrix representation: e− iΩxt 0
0 e− iΩxt
⎛
⎝⎜
⎞
⎠⎟ =
e− iΩxt 00 0
⎛
⎝⎜⎞
⎠⎟+ 0 0
0 e− iΩxt
⎛
⎝⎜⎞
⎠⎟
*Unitary here meansinverse-T-1= T†= TT*=transpose-conjugate-T
(Time-Reversal-Symmetry)
25Tuesday, January 20, 2015
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|Ψ〉T|Ψ〉
|Ψ〉
analyzer
Tanalyzer
T|Ψ〉T|Ψ〉 input stateoutput state
TT Fig. 3.1.1 Effect of analyzer
represented by ket vector transformation of ⏐Ψ〉
to new ket vector T⏐Ψ〉 .
Unitary operators and matrices that do something (or “nothing”)
First is the “do-nothing” identity operator 1... 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐ = Px + Py and matrix representation: 1 0
0 1⎛⎝⎜
⎞⎠⎟
= 1 00 0
⎛⎝⎜
⎞⎠⎟
+ 0 00 1
⎛⎝⎜
⎞⎠⎟
k=1
2
Next is the diagonal “do-something” unitary* operator T... T=∑⏐k〉e-iΩkt〈k⏐= ⏐x〉e-iΩxt〈x⏐ + ⏐y〉e-iΩyt〈y⏐ = e-iΩxt Px + e-iΩyt Py and its matrix representation:
Most “do-something” operators T′ are not diagonal, that is, not just ⏐x〉〈x⏐ and ⏐y〉〈y⏐ combinations. T′=∑⏐k′〉e-iΩk′t〈k′⏐= ⏐x′〉e-iΩx′t〈x′⏐ + ⏐y′〉e-iΩy′t〈y′⏐ = e-iΩx′t Px′ + e-iΩy′t Py′
e− iΩxt 00 e− iΩxt
⎛
⎝⎜
⎞
⎠⎟ =
e− iΩxt 00 0
⎛
⎝⎜⎞
⎠⎟+ 0 0
0 e− iΩxt
⎛
⎝⎜⎞
⎠⎟
*Unitary here meansinverse-T-1= T†= TT*=transpose-conjugate-T
(Time-Reversal-Symmetry)
26Tuesday, January 20, 2015
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(Matrix representation of T′ is a little more complicated. See following pages.)
|Ψ〉T|Ψ〉
|Ψ〉
analyzer
Tanalyzer
T|Ψ〉T|Ψ〉 input stateoutput state
TT Fig. 3.1.1 Effect of analyzer
represented by ket vector transformation of ⏐Ψ〉
to new ket vector T⏐Ψ〉 .
Unitary operators and matrices that do something (or “nothing”)
First is the “do-nothing” identity operator 1... 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐ = Px + Py and matrix representation: 1 0
0 1⎛⎝⎜
⎞⎠⎟
= 1 00 0
⎛⎝⎜
⎞⎠⎟
+ 0 00 1
⎛⎝⎜
⎞⎠⎟
k=1
2
Next is the diagonal “do-something” unitary* operator T... T=∑⏐k〉e-iΩkt〈k⏐= ⏐x〉e-iΩxt〈x⏐ + ⏐y〉e-iΩyt〈y⏐ = e-iΩxt Px + e-iΩyt Py and its matrix representation:
Most “do-something” operators T′ are not diagonal, that is, not just ⏐x〉〈x⏐ and ⏐y〉〈y⏐ combinations. T′=∑⏐k′〉e-iΩk′t〈k′⏐= ⏐x′〉e-iΩx′t〈x′⏐ + ⏐y′〉e-iΩy′t〈y′⏐ = e-iΩx′t Px′ + e-iΩy′t Py′
e− iΩxt 00 e− iΩxt
⎛
⎝⎜
⎞
⎠⎟ =
e− iΩxt 00 0
⎛
⎝⎜⎞
⎠⎟+ 0 0
0 e− iΩxt
⎛
⎝⎜⎞
⎠⎟
*Unitary here meansinverse-T-1= T†= TT*=transpose-conjugate-T
(Time-Reversal-Symmetry)
27Tuesday, January 20, 2015
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Review: Axioms 1-4 and“Do-Nothing”vs“ Do-Something” analyzers
Abstraction of Axiom-4 to define projection and unitary operators Projection operators and resolution of identity
Unitary operators and matrices that do something (or “nothing”) Diagonal unitary operators Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker ⊗-products Axiom-4 similarity transformation
Matrix representation of beam analyzers Non-unitary “killer” devices: Sorter-counter, filter Unitary “non-killer” devices: 1/2-wave plate, 1/4-wave plate
How analyzers “peek” and how that changes outcomes Peeking polarizers and coherence loss Classical Bayesian probability vs. Quantum probability
Feynman 〈j⏐k〉-axioms compared to Group axioms
28Tuesday, January 20, 2015
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Unitary operators U satisfy “easy inversion” relations: U-1= U†= UT* They are “designed” to conserve probability and overlap so each transformed ket ⏐Ψ′〉=U⏐Ψ〉 has the same probability 〈Ψ|Ψ〉=〈Ψ′|Ψ′〉=〈Ψ|U†U|Ψ〉and all transformed kets ⏐Φ′〉=U⏐Φ〉 have the same overlap 〈Ψ|Φ〉=〈Ψ′|Φ′〉=〈Ψ|U†U|Φ〉where transformed bras are defined by 〈Ψ′⏐=〈Ψ⏐U† or 〈Φ′⏐=〈Φ⏐U† implying 1=U†U=UU†
29Tuesday, January 20, 2015
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Unitary operators U satisfy “easy inversion” relations: U-1= U†= UT* They are “designed” to conserve probability and overlap so each transformed ket ⏐Ψ′〉=U⏐Ψ〉 has the same probability 〈Ψ|Ψ〉=〈Ψ′|Ψ′〉=〈Ψ|U†U|Ψ〉and all transformed kets ⏐Φ′〉=U⏐Φ〉 have the same overlap 〈Ψ|Φ〉=〈Ψ′|Φ′〉=〈Ψ|U†U|Φ〉where transformed bras are defined by 〈Ψ′⏐=〈Ψ⏐U† or 〈Φ′⏐=〈Φ⏐U† implying 1=U†U=UU†
Example U transfomation:
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
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Unitary operators U satisfy “easy inversion” relations: U-1= U†= UT* They are “designed” to conserve probability and overlap so each transformed ket ⏐Ψ′〉=U⏐Ψ〉 has the same probability 〈Ψ|Ψ〉=〈Ψ′|Ψ′〉=〈Ψ|U†U|Ψ〉and all transformed kets ⏐Φ′〉=U⏐Φ〉 have the same overlap 〈Ψ|Φ〉=〈Ψ′|Φ′〉=〈Ψ|U†U|Φ〉where transformed bras are defined by 〈Ψ′⏐=〈Ψ⏐U† or 〈Φ′⏐=〈Φ⏐U† implying 1=U†U=UU†
Example U transfomation:
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
Ket definition: ⏐x′〉=U⏐x〉 implies: U†⏐x′〉=⏐x〉 implies: 〈x⏐=〈x′⏐U implies: 〈x⏐U† =〈x′⏐
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Unitary operators U satisfy “easy inversion” relations: U-1= U†= UT* They are “designed” to conserve probability and overlap so each transformed ket ⏐Ψ′〉=U⏐Ψ〉 has the same probability 〈Ψ|Ψ〉=〈Ψ′|Ψ′〉=〈Ψ|U†U|Ψ〉and all transformed kets ⏐Φ′〉=U⏐Φ〉 have the same overlap 〈Ψ|Φ〉=〈Ψ′|Φ′〉=〈Ψ|U†U|Φ〉where transformed bras are defined by 〈Ψ′⏐=〈Ψ⏐U† or 〈Φ′⏐=〈Φ⏐U† implying 1=U†U=UU†
Example U transfomation:
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
Ket definition: ⏐x′〉=U⏐x〉 implies: U†⏐x′〉=⏐x〉 implies: 〈x⏐=〈x′⏐U implies: 〈x⏐U† =〈x′⏐ Ket definition: ⏐y′〉=U⏐y〉 implies: U†⏐y′〉=⏐y〉 implies: 〈y⏐=〈y′⏐U implies: 〈y⏐U† =〈y′⏐
32Tuesday, January 20, 2015
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Unitary operators U satisfy “easy inversion” relations: U-1= U†= UT* They are “designed” to conserve probability and overlap so each transformed ket ⏐Ψ′〉=U⏐Ψ〉 has the same probability 〈Ψ|Ψ〉=〈Ψ′|Ψ′〉=〈Ψ|U†U|Ψ〉and all transformed kets ⏐Φ′〉=U⏐Φ〉 have the same overlap 〈Ψ|Φ〉=〈Ψ′|Φ′〉=〈Ψ|U†U|Φ〉where transformed bras are defined by 〈Ψ′⏐=〈Ψ⏐U† or 〈Φ′⏐=〈Φ⏐U† implying 1=U†U=UU†
Example U transfomation:
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
Ket definition: ⏐x′〉=U⏐x〉 implies: U†⏐x′〉=⏐x〉 implies: 〈x⏐=〈x′⏐U implies: 〈x⏐U† =〈x′⏐ Ket definition: ⏐y′〉=U⏐y〉 implies: U†⏐y′〉=⏐y〉 implies: 〈y⏐=〈y′⏐U implies: 〈y⏐U† =〈y′⏐
x U x x U y
y U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...implies matrix representation of operator U
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Unitary operators U satisfy “easy inversion” relations: U-1= U†= UT* They are “designed” to conserve probability and overlap so each transformed ket ⏐Ψ′〉=U⏐Ψ〉 has the same probability 〈Ψ|Ψ〉=〈Ψ′|Ψ′〉=〈Ψ|U†U|Ψ〉and all transformed kets ⏐Φ′〉=U⏐Φ〉 have the same overlap 〈Ψ|Φ〉=〈Ψ′|Φ′〉=〈Ψ|U†U|Φ〉where transformed bras are defined by 〈Ψ′⏐=〈Ψ⏐U† or 〈Φ′⏐=〈Φ⏐U† implying 1=U†U=UU†
Example U transfomation: (Rotation by φ=30°)
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
Ket definition: ⏐x′〉=U⏐x〉 implies: U†⏐x′〉=⏐x〉 implies: 〈x⏐=〈x′⏐U implies: 〈x⏐U† =〈x′⏐ Ket definition: ⏐y′〉=U⏐y〉 implies: U†⏐y′〉=⏐y〉 implies: 〈y⏐=〈y′⏐U implies: 〈y⏐U† =〈y′⏐
x U x x U y
y U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x U ′x ′x U ′y
′y U ′x ′y U ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...implies matrix representation of operator U in either of the bases it connects is exactly the same.
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Unitary operators U satisfy “easy inversion” relations: U-1= U†= UT* They are “designed” to conserve probability and overlap so each transformed ket ⏐Ψ′〉=U⏐Ψ〉 has the same probability 〈Ψ|Ψ〉=〈Ψ′|Ψ′〉=〈Ψ|U†U|Ψ〉and all transformed kets ⏐Φ′〉=U⏐Φ〉 have the same overlap 〈Ψ|Φ〉=〈Ψ′|Φ′〉=〈Ψ|U†U|Φ〉where transformed bras are defined by 〈Ψ′⏐=〈Ψ⏐U† or 〈Φ′⏐=〈Φ⏐U† implying 1=U†U=UU†
Example U transfomation: (Rotation by φ=30°)
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
Ket definition: ⏐x′〉=U⏐x〉 implies: U†⏐x′〉=⏐x〉 implies: 〈x⏐=〈x′⏐U implies: 〈x⏐U† =〈x′⏐ Ket definition: ⏐y′〉=U⏐y〉 implies: U†⏐y′〉=⏐y〉 implies: 〈y⏐=〈y′⏐U implies: 〈y⏐U† =〈y′⏐
x U x x U y
y U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x U ′x ′x U ′y
′y U ′x ′y U ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...implies matrix representation of operator U in either of the bases it connects is exactly the same.
x U† x x U† y
y U† x y U† y
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ =
′x x ′x y
′y x ′y y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ sinφ−sinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=′x U† ′x ′x U† ′y
′y U† ′x ′y U† ′y
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
So also is the inverse U†
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Unitary operators U satisfy “easy inversion” relations: U-1= U†= UT* They are “designed” to conserve probability and overlap so each transformed ket ⏐Ψ′〉=U⏐Ψ〉 has the same probability 〈Ψ|Ψ〉=〈Ψ′|Ψ′〉=〈Ψ|U†U|Ψ〉and all transformed kets ⏐Φ′〉=U⏐Φ〉 have the same overlap 〈Ψ|Φ〉=〈Ψ′|Φ′〉=〈Ψ|U†U|Φ〉where transformed bras are defined by 〈Ψ′⏐=〈Ψ⏐U† or 〈Φ′⏐=〈Φ⏐U† implying 1=U†U=UU†
Example U transfomation: (Rotation by φ=30°)
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
Ket definition: ⏐x′〉=U⏐x〉 implies: U†⏐x′〉=⏐x〉 implies: 〈x⏐=〈x′⏐U implies: 〈x⏐U† =〈x′⏐ Ket definition: ⏐y′〉=U⏐y〉 implies: U†⏐y′〉=⏐y〉 implies: 〈y⏐=〈y′⏐U implies: 〈y⏐U† =〈y′⏐
x U x x U y
y U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x U ′x ′x U ′y
′y U ′x ′y U ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...implies matrix representation of operator U in either of the bases it connects is exactly the same.
x U† x x U† y
y U† x y U† y
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ =
′x x ′x y
′y x ′y y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ sinφ−sinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=′x U† ′x ′x U† ′y
′y U† ′x ′y U† ′y
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
=x ′x * y ′x *
x ′y * y ′y *
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
Axiom-3 consistent with inverse U =tranpose-conjugate U† = UT*
So also is the inverse U†
36Tuesday, January 20, 2015
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Unitary operators U satisfy “easy inversion” relations: U-1= U†= UT* They are “designed” to conserve probability and overlap so each transformed ket ⏐Ψ′〉=U⏐Ψ〉 has the same probability 〈Ψ|Ψ〉=〈Ψ′|Ψ′〉=〈Ψ|U†U|Ψ〉and all transformed kets ⏐Φ′〉=U⏐Φ〉 have the same overlap 〈Ψ|Φ〉=〈Ψ′|Φ′〉=〈Ψ|U†U|Φ〉where transformed bras are defined by 〈Ψ′⏐=〈Ψ⏐U† or 〈Φ′⏐=〈Φ⏐U† implying 1=U†U=UU†
Example U transfomation: (Rotation by φ=30°)
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
Ket definition: ⏐x′〉=U⏐x〉 implies: U†⏐x′〉=⏐x〉 implies: 〈x⏐=〈x′⏐U implies: 〈x⏐U† =〈x′⏐ Ket definition: ⏐y′〉=U⏐y〉 implies: U†⏐y′〉=⏐y〉 implies: 〈y⏐=〈y′⏐U implies: 〈y⏐U† =〈y′⏐
x U x x U y
y U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x U ′x ′x U ′y
′y U ′x ′y U ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
...implies matrix representation of operator U in either of the bases it connects is exactly the same.
x U† x x U† y
y U† x y U† y
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ =
′x x ′x y
′y x ′y y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ sinφ−sinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=′x U† ′x ′x U† ′y
′y U† ′x ′y U† ′y
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
= cosφ sinφ−sinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x ′x * y ′x *
x ′y * y ′y *
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
Axiom-3 consistent with inverse U =tranpose-conjugate U† = UT*
So also is the inverse U†
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
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Review: Axioms 1-4 and“Do-Nothing”vs“ Do-Something” analyzers
Abstraction of Axiom-4 to define projection and unitary operators Projection operators and resolution of identity
Unitary operators and matrices that do something (or “nothing”) Diagonal unitary operators Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker ⊗-products Axiom-4 similarity transformation
Matrix representation of beam analyzers Non-unitary “killer” devices: Sorter-counter, filter Unitary “non-killer” devices: 1/2-wave plate, 1/4-wave plate
How analyzers “peek” and how that changes outcomes Peeking polarizers and coherence loss Classical Bayesian probability vs. Quantum probability
Feynman 〈j⏐k〉-axioms compared to Group axioms
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′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x x ′x ′x x x ′y
′y x x ′x ′y x x ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Projector Px=⏐x〉〈x⏐ in φ-tilted polarization bases {⏐x′〉, ⏐y′〉} is not diagonal.
x U x x U y
y U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=′x U ′x ′x U ′y
′y U ′x ′y U ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
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′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x x ′x ′x x x ′y
′y x x ′x ′y x x ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Projector Px=⏐x〉〈x⏐ in φ-tilted polarization bases {⏐x′〉, ⏐y′〉} is not diagonal.
x U x x U y
y U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=′x U ′x ′x U ′y
′y U ′x ′y U ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
Projector Px=⏐x〉〈x⏐ is what is called an outer or Kronecker tensor (⊗) product of ket ⏐x〉 and bra 〈x⏐.
40Tuesday, January 20, 2015
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′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x x ′x ′x x x ′y
′y x x ′x ′y x x ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Projector Px=⏐x〉〈x⏐ in φ-tilted polarization bases {⏐x′〉, ⏐y′〉} is not diagonal.
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x x ′x ′x x x ′y
′y x x ′x ′y x x ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=′x x
′y x
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗ x ′x x ′y( )
x U x x U y
y U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=′x U ′x ′x U ′y
′y U ′x ′y U ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
Projector Px=⏐x〉〈x⏐ is what is called an outer or Kronecker tensor (⊗) product of ket ⏐x〉 and bra 〈x⏐.
41Tuesday, January 20, 2015
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′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x x ′x ′x x x ′y
′y x x ′x ′y x x ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Projector Px=⏐x〉〈x⏐ is what is called an outer or Kronecker tensor (⊗) product of ket ⏐x〉 and bra 〈x⏐.
Projector Px=⏐x〉〈x⏐ in φ-tilted polarization bases {⏐x′〉, ⏐y′〉} is not diagonal.
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x x ′x ′x x x ′y
′y x x ′x ′y x x ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=′x x
′y x
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗ x ′x x ′y( )
x U x x U y
y U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=′x U ′x ′x U ′y
′y U ′x ′y U ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
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′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x x ′x ′x x x ′y
′y x x ′x ′y x x ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Projector Px=⏐x〉〈x⏐ is what is called an outer or Kronecker tensor (⊗) product of ket ⏐x〉 and bra 〈x⏐.
Projector Px=⏐x〉〈x⏐ in φ-tilted polarization bases {⏐x′〉, ⏐y′〉} is not diagonal.
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x x ′x ′x x x ′y
′y x x ′x ′y x x ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=′x x
′y x
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗ x ′x x ′y( )
x U x x U y
y U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=′x U ′x ′x U ′y
′y U ′x ′y U ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
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′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x x ′x ′x x x ′y
′y x x ′x ′y x x ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Projector Px=⏐x〉〈x⏐ in φ-tilted polarization bases {⏐x′〉, ⏐y′〉} is not diagonal.
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x x ′x ′x x x ′y
′y x x ′x ′y x x ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=′x x
′y x
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗ x ′x x ′y( )
Px = x x →cosφ−sinφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗ cosφ −sinφ( )
=cos2φ −sinφ cosφ
−sinφ cosφ sin2φ
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛
⎝⎜⎞
⎠⎟ for φ=0( )
The x'y'-representation of Px:
x U x x U y
y U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=′x U ′x ′x U ′y
′y U ′x ′y U ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
Projector Px=⏐x〉〈x⏐ is what is called an outer or Kronecker tensor (⊗) product of ket ⏐x〉 and bra 〈x⏐.
44Tuesday, January 20, 2015
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′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x x ′x ′x x x ′y
′y x x ′x ′y x x ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Projector Px=⏐x〉〈x⏐ in φ-tilted polarization bases {⏐x′〉, ⏐y′〉} is not diagonal.
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x x ′x ′x x x ′y
′y x x ′x ′y x x ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=′x x
′y x
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗ x ′x x ′y( )
Px = x x →cosφ−sinφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗ cosφ −sinφ( )
=cos2φ −sinφ cosφ
−sinφ cosφ sin2φ
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛
⎝⎜⎞
⎠⎟ for φ=0( )
The x'y'-representation of Px:
x U x x U y
y U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Py = y y →sinφcosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗ sinφ cosφ( )
==sin2φ sinφ cosφ
sinφ cosφ cos2φ
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 0 0
0 1⎛
⎝⎜⎞
⎠⎟ for φ=0( )
The x'y'-representation of Py:
=′x U ′x ′x U ′y
′y U ′x ′y U ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
UU|y 〉=U|y〉=-sinφ |x〉 + cosφ |y〉|x 〉=U|x〉= cosφ |x〉 + sinφ |y〉
sinφ
-sinφ
cosφcosφ
|x〉
|y〉|x 〉|y 〉
```
`
Projector Px=⏐x〉〈x⏐ is what is called an outer or Kronecker tensor (⊗) product of ket ⏐x〉 and bra 〈x⏐.
45Tuesday, January 20, 2015
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Review: Axioms 1-4 and“Do-Nothing”vs“ Do-Something” analyzers
Abstraction of Axiom-4 to define projection and unitary operators Projection operators and resolution of identity
Unitary operators and matrices that do something (or “nothing”) Diagonal unitary operators Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker ⊗-products Axiom-4 similarity transformation
Matrix representation of beam analyzers Non-unitary “killer” devices: Sorter-counter, filter Unitary “non-killer” devices: 1/2-wave plate, 1/4-wave plate
How analyzers “peek” and how that changes outcomes Peeking polarizers and coherence loss Classical Bayesian probability vs. Quantum probability
Feynman 〈j⏐k〉-axioms compared to Group axioms
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Axiom-4 is basically a matrix product as seen by comparing the following.
Axiom-4 similarity transformations (Using: 1=∑⏐k〉〈k⏐ )
j" m ' = j" 1 m ' =
k=1
n∑ j" k k m '
1" 1 ' 1" 2 ' 1" n '
2" 1' 2" 2 ' 2" n '
n" 1' n" 2 ' n" n '
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
=
1" 1 1" 2 1" n
2" 1 2" 2 2" n
n" 1 n" 2 n" n
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
•
1 1 ' 1 2 ' 1 n '
2 1' 2 2 ' 2 n '
n 1' n 2 ' n n '
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
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Axiom-4 is basically a matrix product as seen by comparing the following.
Axiom-4 similarity transformations (Using: 1=∑⏐k〉〈k⏐ )
j" m ' = j" 1 m ' =
k=1
n∑ j" k k m '
1" 1 ' 1" 2 ' 1" n '
2" 1' 2" 2 ' 2" n '
n" 1' n" 2 ' n" n '
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
=
1" 1 1" 2 1" n
2" 1 2" 2 2" n
n" 1 n" 2 n" n
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
•
1 1 ' 1 2 ' 1 n '
2 1' 2 2 ' 2 n '
n 1' n 2 ' n n '
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
Tj " m '
primeto
double − prime
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟=
k=1
n∑ Tj " k
unprimedto
double − prime
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
Tk m '
primeto
unprimed
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
T(b"← b ') = T(b"← b ) •T(b← b ')
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Axiom-4 is basically a matrix product as seen by comparing the following.
Axiom-4 similarity transformations (Using: 1=∑⏐k〉〈k⏐ )
j" m ' = j" 1 m ' =
k=1
n∑ j" k k m '
1" 1 ' 1" 2 ' 1" n '
2" 1' 2" 2 ' 2" n '
n" 1' n" 2 ' n" n '
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
=
1" 1 1" 2 1" n
2" 1 2" 2 2" n
n" 1 n" 2 n" n
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
•
1 1 ' 1 2 ' 1 n '
2 1' 2 2 ' 2 n '
n 1' n 2 ' n n '
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
Tj " m '
primeto
double − prime
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟=
k=1
n∑ Tj " k
unprimedto
double − prime
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
Tk m '
primeto
unprimed
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
T(b"← b ') = T(b"← b ) •T(b← b ')(1) The closure axiom Products ab = c are defined between any two group elements a and b, and the result c is contained in the group.
(2) The associativity axiom Products (ab)c and a(bc) are equal for all elements a, b, and c in the group .
Transformation Group axioms(3) The identity axiom There is a unique element 1 (the identity) such that 1.a = a = a.1 for all elements a in the group ..
4) The inverse axiom For all elements a in the group there is an inverse element a-1 such that a-1a = 1 = a.a-1.
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Axiom-4 is applied twice to transform operator matrix representation.Example: Find: given: and T-matrix:
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
The old “P=1·P·1-trick” where: 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐;
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Axiom-4 is applied twice to transform operator matrix representation.Example: Find: given: and T-matrix:
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
′x Px ′y = ′x 1·Px ·1 ′y = ′x x x + y y( )·Px · x x + y y( ) ′y
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
The old “P=1·P·1-trick” where: 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐;
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Axiom-4 is applied twice to transform operator matrix representation.Example: Find: given: and T-matrix:
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
′x Px ′y = ′x 1·Px ·1 ′y = ′x x x + y y( )·Px · x x + y y( ) ′y = ′x x x + ′x y y( )·Px · x x ′y + y y ′y( )
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
The old “P=1·P·1-trick” where: 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐;
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Axiom-4 is applied twice to transform operator matrix representation.Example: Find: given: and T-matrix:
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
′x Px ′y = ′x 1·Px ·1 ′y = ′x x x + y y( )·Px · x x + y y( ) ′y = ′x x x + ′x y y( )·Px · x x ′y + y y ′y( ) = ′x x x Px x x ′y + ...
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
The old “P=1·P·1-trick” where: 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐;
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Axiom-4 is applied twice to transform operator matrix representation.Example: Find: given: and T-matrix:
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
′x Px ′y = ′x 1·Px ·1 ′y = ′x x x + y y( )·Px · x x + y y( ) ′y = ′x x x + ′x y y( )·Px · x x ′y + y y ′y( ) = ′x x x Px x x ′y + ′x y y Px x x ′y + ...
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
The old “P=1·P·1-trick” where: 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐;
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Axiom-4 is applied twice to transform operator matrix representation.Example: Find: given: and T-matrix:
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
′x Px ′y = ′x 1·Px ·1 ′y = ′x x x + y y( )·Px · x x + y y( ) ′y = ′x x x + ′x y y( )·Px · x x ′y + y y ′y( ) = ′x x x Px x x ′y + ′x y y Px x x ′y + ′x x x Px y y ′y + ...
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
The old “P=1·P·1-trick” where: 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐;
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Axiom-4 is applied twice to transform operator matrix representation.Example: Find: given: and T-matrix:
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
′x Px ′y = ′x 1·Px ·1 ′y = ′x x x + y y( )·Px · x x + y y( ) ′y = ′x x x + ′x y y( )·Px · x x ′y + y y ′y( ) = ′x x x Px x x ′y + ′x y y Px x x ′y + ′x x x Px y y ′y + ′x y y Px y y ′y
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
The old “P=1·P·1-trick” where: 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐;
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Axiom-4 is applied twice to transform operator matrix representation.Example: Find: given: and T-matrix:
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
′x Px ′y = ′x 1·Px ·1 ′y = ′x x x + y y( )·Px · x x + y y( ) ′y = ′x x x + ′x y y( )·Px · x x ′y + y y ′y( ) = ′x x x Px x x ′y + ′x y y Px x x ′y + ′x x x Px y y ′y + ′x y y Px y y ′y
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x ′x y
′y x ′y y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px y
y Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
More elegant matrix product:
The old “P=1·P·1-trick” where: 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐;
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Axiom-4 is applied twice to transform operator matrix representation.Example: Find: given: and T-matrix:
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
′x Px ′y = ′x 1·Px ·1 ′y = ′x x x + y y( )·Px · x x + y y( ) ′y = ′x x x + ′x y y( )·Px · x x ′y + y y ′y( ) = ′x x x Px x x ′y + ′x y y Px x x ′y + ′x x x Px y y ′y + ′x y y Px y y ′y
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x ′x y
′y x ′y y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px y
y Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ sinφ−sinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px y
y Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ sinφ−sinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
1 00 0
⎛
⎝⎜⎞
⎠⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
More elegant matrix product:
The old “P=1·P·1-trick” where: 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐;
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Axiom-4 is applied twice to transform operator matrix representation.Example: Find: given: and T-matrix:
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
′x Px ′y = ′x 1·Px ·1 ′y = ′x x x + y y( )·Px · x x + y y( ) ′y = ′x x x + ′x y y( )·Px · x x ′y + y y ′y( ) = ′x x x Px x x ′y + ′x y y Px x x ′y + ′x x x Px y y ′y + ′x y y Px y y ′y
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x ′x y
′y x ′y y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px y
y Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ sinφ−sinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px y
y Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ sinφ−sinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
1 00 0
⎛
⎝⎜⎞
⎠⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
= cosφ 0−sinφ 0
⎛
⎝⎜⎜
⎞
⎠⎟⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cos2φ −cosφ sinφ
−sinφ cosφ sin2φ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
More elegant matrix product:
The old “P=1·P·1-trick” where: 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐;
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Axiom-4 is applied twice to transform operator matrix representation.Example: Find: given: and T-matrix:
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px yy Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛⎝⎜
⎞⎠⎟
′x Px ′y = ′x 1·Px ·1 ′y = ′x x x + y y( )·Px · x x + y y( ) ′y = ′x x x + ′x y y( )·Px · x x ′y + y y ′y( ) = ′x x x Px x x ′y + ′x y y Px x x ′y + ′x x x Px y y ′y + ′x y y Px y y ′y
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
′x Px ′x ′x Px ′y
′y Px ′x ′y Px ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
′x x ′x y
′y x ′y y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px y
y Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ sinφ−sinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x Px x x Px y
y Px x y Px y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
=cosφ sinφ−sinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
1 00 0
⎛
⎝⎜⎞
⎠⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
= cosφ 0−sinφ 0
⎛
⎝⎜⎜
⎞
⎠⎟⎟
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cos2φ −cosφ sinφ
−sinφ cosφ sin2φ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Px = x x →cosφ−sinφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗ cosφ −sinφ( ) = cos2φ −sinφ cosφ
−sinφ cosφ sin2φ
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 0⎛
⎝⎜⎞
⎠⎟ for φ=0( )This checks with the previous result 4-pages back:
More elegant matrix product:
The old “P=1·P·1-trick” where: 1=∑⏐k〉〈k⏐= ⏐x〉〈x⏐ + ⏐y〉〈y⏐;
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Review: Axioms 1-4 and“Do-Nothing”vs“ Do-Something” analyzers
Abstraction of Axiom-4 to define projection and unitary operators Projection operators and resolution of identity
Unitary operators and matrices that do something (or “nothing”) Diagonal unitary operators Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker ⊗-products Axiom-4 similarity transformation
Matrix representation of beam analyzers Non-unitary “killer” devices: Sorter-counter, filter Unitary “non-killer” devices: 1/2-wave plate, 1/4-wave plate
How analyzers “peek” and how that changes outcomes Peeking polarizers and coherence loss Classical Bayesian probability vs. Quantum probability
Feynman 〈j⏐k〉-axioms compared to Group axioms
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(1) Optical analyzer in sorter-counter configuration
Initial polarization angleθ=β/2 = 30°
θ
x-counts~| 〈x|x'〉|2= cos2 θ =
y-counts~| 〈y|x'〉|2= sin2 θ=
xy-analyzer( βanalyzer =0°)
Analyzer reduced to a simple sorter-counter by blocking output of x-high-road and y-low-road with counters
Fig. 1.3.3 Simulated polarization analyzer set up as a sorter-counter
x T x x T yy T x y T y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
= 0 00 0
⎛⎝⎜
⎞⎠⎟
Analyzer matrix:
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(1) Optical analyzer in sorter-counter configuration
Initial polarization angleθ=β/2 = 30°
θ
x-counts~| 〈x|x'〉|2= cos2 θ =
y-counts~| 〈y|x'〉|2= sin2 θ=
xy-analyzer( βanalyzer =0°)
Analyzer reduced to a simple sorter-counter by blocking output of x-high-road and y-low-road with counters
Fig. 1.3.3 Simulated polarization analyzer set up as a sorter-counter
y-output~| 〈y|x'〉|2= sin2 θ=
x-counts~| 〈y|x'〉|2=(Blocked and filtered out)
Initial polarization angleθ=β/2 = 30°
θxy-analyzer( βanalyzer =0°)
Analyzer blocks one path which may have photon counter without affecting function.
(2) Optical analyzer in a filter configuration (Polaroid© sunglasses)
Fig. 1.3.4 Simulated polarization analyzer set up to filter out the x-polarized photons
x Py x x Py y
y Py x y Py y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
= 0 00 1
⎛⎝⎜
⎞⎠⎟
x T x x T yy T x y T y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
= 0 00 0
⎛⎝⎜
⎞⎠⎟
Analyzer matrix:
Analyzer matrix:
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Review: Axioms 1-4 and“Do-Nothing”vs“ Do-Something” analyzers
Abstraction of Axiom-4 to define projection and unitary operators Projection operators and resolution of identity
Unitary operators and matrices that do something (or “nothing”) Diagonal unitary operators Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker ⊗-products Axiom-4 similarity transformation
Matrix representation of beam analyzers Non-unitary “killer” devices: Sorter-counter, filter Unitary “non-killer” devices: 1/2-wave plate, 1/4-wave plate
How analyzers “peek” and how that changes outcomes Peeking polarizers and coherence loss Classical Bayesian probability vs. Quantum probability
64Tuesday, January 20, 2015
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(3) Optical analyzers in the "control" configuration: Half or Quarter wave plates
θ
Initial polarization angle
θ=β/2 = 30°(a)
Half-wave plate
(Ω=π)Final polarization angle
θ=β/2 = 150°(or -30°)
(b) Quarter-wave
plate
(Ω=π/2)Final polarization is
untilted elliptical
θ
Initial polarization angle
θ=β/2 = 30°
Ω
Ω
Analyzer phase lag
(activity angle)
Analyzer phase lag
(activity angle)
xy-analyzer
(βanalyzer
=0°)
xy-analyzer
(βanalyzer
=0°)
x U x x U yy U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 −1⎛⎝⎜
⎞⎠⎟
Analyzer matrix:
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(3) Optical analyzers in the "control" configuration: Half or Quarter wave plates
θ
Initial polarization angle
θ=β/2 = 30°(a)
Half-wave plate
(Ω=π)Final polarization angle
θ=β/2 = 150°(or -30°)
(b) Quarter-wave
plate
(Ω=π/2)Final polarization is
untilted elliptical
θ
Initial polarization angle
θ=β/2 = 30°
Ω
Ω
Analyzer phase lag
(activity angle)
Analyzer phase lag
(activity angle)
xy-analyzer
(βanalyzer
=0°)
xy-analyzer
(βanalyzer
=0°)
Fig. 1.3.5 Polarization control set to shift phase by (a) Half-wave (Ω = π) , (b) Quarter wave (Ω= π/2)
x U x x U yy U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 −1⎛⎝⎜
⎞⎠⎟
x U x x U yy U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 1 0
0 −i⎛⎝⎜
⎞⎠⎟
x U x x U yy U x y U y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= e− iΩxt 0
0 e− iΩxt
⎛
⎝⎜
⎞
⎠⎟
Analyzer matrix:
Analyzer matrix:
Analyzer matrix:
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Similar to "do-nothing" analyzer but has extra phase factor e-iΩx′ = 0.94-i 0.34 on the x'-path (top).x-output: y-output:
(b)Simulation
x'-polarized light
y'-polarized light
β/2=Θ= -30°
Θin =βin/2=100°plane-polarized light
x'
y
x
y'
Elliptically
polarized light ω=20°phase shift
(a)Analyzer Experiment
Phase shift→ Ω
Output polarization
changed by analyzer
phase shift
setting of
input
polarization
2Θin =βin=200°Θin=100°
Θ=-30°
=2Θanalyzerβ
x Ψout = x ′x e−iΩ ′x ′x Ψin + x ′y ′y Ψin = e−iΩ ′x cosΘcos Θin −Θ( )− sinΘsin Θin −Θ( )
y Ψout = y ′x e−iΩ ′x ′x Ψin + x ′y ′y Ψin = e−iΩ ′x sinΘcos Θin −Θ( ) + cosΘsin Θin −Θ( )
′x U ′x ′x U ′y
′y U ′x ′y U ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟
= e− iΩ ′x t 00 e− iΩ ′x t
⎛
⎝⎜
⎞
⎠⎟
Analyzer matrix:
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ReΨy
ReΨx
ReΨy
ReΨx-√3/2
1/2
-30°
1/2√3/2
θ=
ω t = 0ω t = 0
ω t = πω t = π
0 12
3
4
5
6
78-7-6
-5
-4
-3
-2-1
01
2
3 4 5
6
78
-7-6
-5
-4-3
-2-1
(2,4)(3,5)(4,6)
(5,7)
(6,8)
(7,-7)
x-position
x-velocity vx/ω(c) 2-D Oscillator
Phasor Plot
(8,-6)(9,-5)
y-velocity
vy/ω
(x-Phase
45° behind the
y-Phase)
y-position
φ counter-clockwise
if y is behind x
clockwise
orbit
if x is behind y
(1,3) Left-
handed
Right-
handed
(0,2)
Fig. 1.3.6 Polarization states for (a) Half-wave (Ω = π) , (b) Quarter wave (Ω= π/2) (c) (Ω=−π/4)
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Review: Axioms 1-4 and“Do-Nothing”vs“ Do-Something” analyzers
Abstraction of Axiom-4 to define projection and unitary operators Projection operators and resolution of identity
Unitary operators and matrices that do something (or “nothing”) Diagonal unitary operators Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker ⊗-products Axiom-4 similarity transformation
Matrix representation of beam analyzers Non-unitary “killer” devices: Sorter-counter, filter Unitary “non-killer” devices: 1/2-wave plate, 1/4-wave plate
How analyzers “peek” and how that changes outcomes Peeking polarizers and coherence loss Classical Bayesian probability vs. Quantum probability
Feynman 〈j⏐k〉-axioms compared to Group axioms
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A "peeking" eye
If eye sees an x-photonthen the output particleis 100% x-polarized.(75% probability for that.)
If eye sees no x-photonthen the output particleis 100% y-polarized(25% probability.)
θ
θ=β/2 = 30°
θ
θ=β/2 = 30°
(Looks for x-photons)
xy-analyzer(βanalyzer=0°)
xy-analyzer(βanalyzer=0°)
Initial polarization angle
Initial polarization angle
Fig. 1.3.7 Simulated polarization analyzer set up to "peek" if the photon is x-or y-polarized
How analyzers may “peek” and how that changes outcomes
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Initial
Without
“Peeking Eye” angle
θ=β/2 = 30°
Initial
polarization
angle
θ=β/2 = 30°
With
“Peeking Eye”
polarization(a)
(b)
xy-analyzer
(βanalyzer
=0°)
xy-analyzer
(βanalyzer
=0°)
βanalyzer
=60°
Θanalyzer
=30°
βanalyzer
=60°
Θanalyzer
=30°
Reconstructs
x (30°) beam
Cancels
y (30°) beamNo
y (30°)
appear
Only
x (30°)
appears
=2θ
5 to 8 odds
for x (30°)
to appear
3 to 8 odds
for y (30°)
to appear
30°
30°
(empty path)
Fig. 1.3.8 Output with β/2=30° input to: (a) Coherent xy-"Do nothing" or (b) Incoherent xy-"Peeking" devices
How analyzers “peek” and how that changes outcomesSimulations
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Fig. 1.3.9 Beams-amplitudes of (a) xy-"Do nothing" and (b) xy-"Peeking" analyzer each with input
〈y|x'〉=1/2
〈x|x'〉=√3/2
〈y'|x〉= -1/2
〈x'|x〉=√3/2
〈x'|y〉=1/2
〈y'|y〉=√3/2
〈x'|x〉〈x|x'〉=√3/2 √3/2
〈x'|y〉〈y|x'〉=1/2 1/2
〈y'|y〉〈y|x'〉=√3/2 1/2
〈y'|x〉〈x|x'〉= -1/2 √3/2
〈x'|x〉〈x|x'〉+〈x'|y〉〈y|x'〉=√3/2 √3/2 +1/2 1/2 = 1
〈y'|x〉〈x|x' 〉+〈y'|y〉〈y|x'〉=-1/2 √3/2+√3/2 1/2 =0
|x〉-beam
|y〉-beam
|x〉-beam
|y〉-beam
|x〉
|x〉
|x'〉-beam|x'〉-beam
|y'〉-beam (zero)
|x'〉|y〉
Withouta
“Peeking Eye”
Witha
“Peeking Eye”
(a)
(b)
|x'〉-beam|y〉
|x'〉
|y'〉
|y'〉
|x'〉
x '
How analyzers “peek” and how that changes outcomes
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
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A(x') = x' x x x' + x' y y x'
= 43 + 4
1 = 1
Amplitude A(n′) and Probability P(n′) at counter n′ WITHOUT “peeking”
Without“Peeking Eye”
(a)
xy-analyzer(βanalyzer=0°)
βanalyzer=60°Θanalyzer=30°
Reconstructsx (30°) beam
Cancelsy (30°) beam
30°
(empty path)θ=β/2= 30°
A( y') = y' x x x' + y' y y x'
= 4− 3 + 4
3 = 0
=P(x′)
=P(y′)
Do-Nothing-analyzer
!x"#$x"!x#=%3/2!x"#
!y"#$y"!x#=1/2!y"#
!x"=!x#"$x#!x"+!y#"$y#!x"
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
cosφ −sinφsinφ cosφ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
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Suppose "x-eye" puts phase eiφ on each x-photon with random φ distributed over unit circle (-π< φ <π). So eiφ averages to zero!
A(x') = x' x 1( ) x x' + x' y y x'
= 43 1( ) + 4
1 = 1
A( y') = y' x 1( ) x x' + y' y y x'
= 4− 3 1( ) + 4
3 = 0
A(x') = x' x eiφ( ) x x' + x' y y x'
= 43 eiφ( ) + 4
1
Amplitude A(n′) and Probability P(n′) at counter n′ WITHOUT “peeking”
Amplitude A(n′) and Probability P(n′) at counter n′ WITH “peeking”
Without“Peeking Eye”
(a)
xy-analyzer(βanalyzer=0°)
βanalyzer=60°Θanalyzer=30°
Reconstructsx (30°) beam
Cancelsy (30°) beam
30°
(empty path)θ=β/2= 30°
(b)
xy-analyzer(βanalyzer=0°)
βanalyzer=60°Θanalyzer=30°
5 to 8 oddsfor x (30°)to appear
3 to 8 oddsfor y (30°)to appear
30°
With“Peeking Eye”
With“Peeking Eye”
θ=β/2= 30°
Do-Nothing-analyzer
=P(x′)
=P(y′)
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
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Suppose "x-eye" puts phase eiφ on each x-photon with random φ distributed over unit circle (-π< φ <π). So eiφ averages to zero!
A(x') = x' x 1( ) x x' + x' y y x'
= 43 1( ) + 4
1 = 1
A( y') = y' x 1( ) x x' + y' y y x'
= 4− 3 1( ) + 4
3 = 0
A(x') = x' x eiφ( ) x x' + x' y y x'
= 43 eiφ( ) + 4
1
Amplitude A(n′) and Probability P(n′) at counter n′ WITHOUT “peeking”
Amplitude A(n′) and Probability P(n′) at counter n′ WITH “peeking”
Without“Peeking Eye”
(a)
xy-analyzer(βanalyzer=0°)
βanalyzer=60°Θanalyzer=30°
Reconstructsx (30°) beam
Cancelsy (30°) beam
30°
(empty path)θ=β/2= 30°
(b)
xy-analyzer(βanalyzer=0°)
βanalyzer=60°Θanalyzer=30°
5 to 8 oddsfor x (30°)to appear
3 to 8 oddsfor y (30°)to appear
30°
With“Peeking Eye”
With“Peeking Eye”
θ=β/2= 30°
Do-Nothing-analyzer
=P(x′)
=P(y′)
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
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Suppose "x-eye" puts phase eiφ on each x-photon with random φ distributed over unit circle (-π< φ <π). So eiφ averages to zero!
A(x') = x' x 1( ) x x' + x' y y x'
= 43 1( ) + 4
1 = 1
A( y') = y' x 1( ) x x' + y' y y x'
= 4− 3 1( ) + 4
3 = 0
A(x') = x' x eiφ( ) x x' + x' y y x'
= 43 eiφ( ) + 4
1
P(x')= 43 eiφ( ) +4
1( )* 43 eiφ( ) +4
1( ) = 5
8+ 3
16e−iφ + eiφ( ) = 5+ 3cosφ
8
Amplitude A(n′) and Probability P(n′) at counter n′ WITHOUT “peeking”
Amplitude A(n′) and Probability P(n′) at counter n′ WITH “peeking”
Without“Peeking Eye”
(a)
xy-analyzer(βanalyzer=0°)
βanalyzer=60°Θanalyzer=30°
Reconstructsx (30°) beam
Cancelsy (30°) beam
30°
(empty path)θ=β/2= 30°
(b)
xy-analyzer(βanalyzer=0°)
βanalyzer=60°Θanalyzer=30°
5 to 8 oddsfor x (30°)to appear
3 to 8 oddsfor y (30°)to appear
30°
With“Peeking Eye”
With“Peeking Eye”
θ=β/2= 30°
Do-Nothing-analyzer
=P(x′)
=P(y′)
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
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Suppose "x-eye" puts phase eiφ on each x-photon with random φ distributed over unit circle (-π< φ <π). So eiφ averages to zero!
A(x') = x' x 1( ) x x' + x' y y x'
= 43 1( ) + 4
1 = 1
A( y') = y' x 1( ) x x' + y' y y x'
= 4− 3 1( ) + 4
3 = 0
A(x') = x' x eiφ( ) x x' + x' y y x'
= 43 eiφ( ) + 4
1
A( y') = y' x eiφ( ) x x' + y' y y x'
= 4− 3 eiφ( ) + 4
3
P(x')= 43 eiφ( ) +4
1( )* 43 eiφ( ) +4
1( ) = 5
8+ 3
16e−iφ + eiφ( ) = 5+ 3cosφ
8
Amplitude A(n′) and Probability P(n′) at counter n′ WITHOUT “peeking”
Amplitude A(n′) and Probability P(n′) at counter n′ WITH “peeking”
Without“Peeking Eye”
(a)
xy-analyzer(βanalyzer=0°)
βanalyzer=60°Θanalyzer=30°
Reconstructsx (30°) beam
Cancelsy (30°) beam
30°
(empty path)θ=β/2= 30°
(b)
xy-analyzer(βanalyzer=0°)
βanalyzer=60°Θanalyzer=30°
5 to 8 oddsfor x (30°)to appear
3 to 8 oddsfor y (30°)to appear
30°
With“Peeking Eye”
With“Peeking Eye”
θ=β/2= 30°
Do-Nothing-analyzer
=P(x′)
=P(y′)
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
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Suppose "x-eye" puts phase eiφ on each x-photon with random φ distributed over unit circle (-π< φ <π). So eiφ averages to zero!
A(x') = x' x 1( ) x x' + x' y y x'
= 43 1( ) + 4
1 = 1
A( y') = y' x 1( ) x x' + y' y y x'
= 4− 3 1( ) + 4
3 = 0
A(x') = x' x eiφ( ) x x' + x' y y x'
= 43 eiφ( ) + 4
1
A( y') = y' x eiφ( ) x x' + y' y y x'
= 4− 3 eiφ( ) + 4
3
P(x')= 43 eiφ( ) +4
1( )* 43 eiφ( ) +4
1( ) = 5
8+ 3
16e−iφ + eiφ( ) = 5+ 3cosφ
8
P( y')= 4− 3 eiφ( ) + 4
3( )* 4− 3 eiφ( ) + 4
3( ) = 3
8− 3
16e−iφ + eiφ( ) = 3− 3cosφ
8
Amplitude A(n′) and Probability P(n′) at counter n′ WITHOUT “peeking”
Amplitude A(n′) and Probability P(n′) at counter n′ WITH “peeking”
Without“Peeking Eye”
(a)
xy-analyzer(βanalyzer=0°)
βanalyzer=60°Θanalyzer=30°
Reconstructsx (30°) beam
Cancelsy (30°) beam
30°
(empty path)θ=β/2= 30°
(b)
xy-analyzer(βanalyzer=0°)
βanalyzer=60°Θanalyzer=30°
5 to 8 oddsfor x (30°)to appear
3 to 8 oddsfor y (30°)to appear
30°
With“Peeking Eye”
With“Peeking Eye”
θ=β/2= 30°
Do-Nothing-analyzer
=P(x′)
=P(y′)
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
78Tuesday, January 20, 2015
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Review: Axioms 1-4 and“Do-Nothing”vs“ Do-Something” analyzers
Abstraction of Axiom-4 to define projection and unitary operators Projection operators and resolution of identity
Unitary operators and matrices that do something (or “nothing”) Diagonal unitary operators Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker ⊗-products Axiom-4 similarity transformation
Matrix representation of beam analyzers Non-unitary “killer” devices: Sorter-counter, filter Unitary “non-killer” devices: 1/2-wave plate, 1/4-wave plate
How analyzers “peek” and how that changes outcomes Peeking polarizers and coherence loss Classical Bayesian probability vs. Quantum probability
Feynman 〈j⏐k〉-axioms compared to Group axioms
79Tuesday, January 20, 2015
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Probability thatphoton in x'-input
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
classical
=
probability thatphoton in x-beam
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
⋅
probability thatphoton in x'-input
becomesphoton in x-beam
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
+
probability thatphoton in y-beam
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
⋅
probability thatphoton in x'-input
becomesphoton in y-beam
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
Probability thatphoton in x'-input
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
classical
= x' x2⎛
⎝⎜⎞⎠⎟ ⋅ x x'
2⎛⎝⎜
⎞⎠⎟ + x' y
2⎛⎝⎜
⎞⎠⎟ ⋅ y x'
2⎛⎝⎜
⎞⎠⎟
Classical Bayesian probability vs. Quantum probability
80Tuesday, January 20, 2015
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Probability thatphoton in x'-input
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
classical
= x' x2⎛
⎝⎜⎞⎠⎟ · x x'
2⎛⎝⎜
⎞⎠⎟ + x' y
2⎛⎝⎜
⎞⎠⎟ · y x'
2⎛⎝⎜
⎞⎠⎟ =
32
2⎛
⎝
⎜⎜
⎞
⎠
⎟⎟· 3
2
2⎛
⎝
⎜⎜
⎞
⎠
⎟⎟+ −1
2
2⎛
⎝⎜⎜
⎞
⎠⎟⎟· 1
2
2⎛
⎝⎜⎜
⎞
⎠⎟⎟= 5
8
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Classical Bayesian probability vs. Quantum probability
Probability thatphoton in x'-input
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
classical
=
probability thatphoton in x-beam
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
⋅
probability thatphoton in x'-input
becomesphoton in x-beam
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
+
probability thatphoton in y-beam
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
⋅
probability thatphoton in x'-input
becomesphoton in y-beam
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
81Tuesday, January 20, 2015
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Quantum probabilityat x'-counter
⎛
⎝⎜⎜
⎞
⎠⎟⎟= x' x eiφ( ) x x' + x' y y x'
2
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Classical Bayesian probability vs. Quantum probability
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Classical Bayesian probability vs. Quantum probability
Probability thatphoton in x'-input
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
classical
= x' x2⎛
⎝⎜⎞⎠⎟ · x x'
2⎛⎝⎜
⎞⎠⎟ + x' y
2⎛⎝⎜
⎞⎠⎟ · y x'
2⎛⎝⎜
⎞⎠⎟ =
32
2⎛
⎝
⎜⎜
⎞
⎠
⎟⎟· 3
2
2⎛
⎝
⎜⎜
⎞
⎠
⎟⎟+ −1
2
2⎛
⎝⎜⎜
⎞
⎠⎟⎟· 1
2
2⎛
⎝⎜⎜
⎞
⎠⎟⎟= 5
8
Probability thatphoton in x'-input
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
classical
=
probability thatphoton in x-beam
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
⋅
probability thatphoton in x'-input
becomesphoton in x-beam
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
+
probability thatphoton in y-beam
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
⋅
probability thatphoton in x'-input
becomesphoton in y-beam
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
82Tuesday, January 20, 2015
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Quantum probabilityat x'-counter
⎛
⎝⎜⎜
⎞
⎠⎟⎟= x' x eiφ( ) x x' + x' y y x'
2
= x' x x x'2+ x' y y x'
2+ e−iφ x' x * x x' * x' y y x' + eiφ x' x x x' x' y * y x' *
=1
=( classical probability ) + ( Phase-sensitive or quantum interference terms )
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Classical Bayesian probability vs. Quantum probability
Probability thatphoton in x'-input
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
classical
= x' x2⎛
⎝⎜⎞⎠⎟ · x x'
2⎛⎝⎜
⎞⎠⎟ + x' y
2⎛⎝⎜
⎞⎠⎟ · y x'
2⎛⎝⎜
⎞⎠⎟ =
32
2⎛
⎝
⎜⎜
⎞
⎠
⎟⎟· 3
2
2⎛
⎝
⎜⎜
⎞
⎠
⎟⎟+ −1
2
2⎛
⎝⎜⎜
⎞
⎠⎟⎟· 1
2
2⎛
⎝⎜⎜
⎞
⎠⎟⎟= 5
8
Probability thatphoton in x'-input
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
classical
=
probability thatphoton in x-beam
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
⋅
probability thatphoton in x'-input
becomesphoton in x-beam
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
+
probability thatphoton in y-beam
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
⋅
probability thatphoton in x'-input
becomesphoton in y-beam
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
83Tuesday, January 20, 2015
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Review: Axioms 1-4 and“Do-Nothing”vs“ Do-Something” analyzers
Abstraction of Axiom-4 to define projection and unitary operators Projection operators and resolution of identity
Unitary operators and matrices that do something (or “nothing”) Diagonal unitary operators Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker ⊗-products Axiom-4 similarity transformation
Matrix representation of beam analyzers Non-unitary “killer” devices: Sorter-counter, filter Unitary “non-killer” devices: 1/2-wave plate, 1/4-wave plate
How analyzers “peek” and how that changes outcomes Peeking polarizers and coherence loss Classical Bayesian probability vs. Quantum probability
Feynman 〈j⏐k〉-axioms compared to Group axioms
84Tuesday, January 20, 2015
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Probability thatphoton in x'-input
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
classical
=
probability thatphoton in x-beam
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
*
probability thatphoton in x'-input
becomesphoton in x-beam
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
+
probability thatphoton in y-beam
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
*
probability thatphoton in x'-input
becomesphoton in y-beam
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
Probability thatphoton in x'-input
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
classical
= x' x2⎛
⎝⎜⎞⎠⎟ * x x'
2⎛⎝⎜
⎞⎠⎟ + x' y
2⎛⎝⎜
⎞⎠⎟ * y x'
2⎛⎝⎜
⎞⎠⎟ =
32
2⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
* 32
2⎛
⎝
⎜⎜
⎞
⎠
⎟⎟+ −1
2
2⎛
⎝⎜⎜
⎞
⎠⎟⎟
* 12
2⎛
⎝⎜⎜
⎞
⎠⎟⎟= 5
8
Quantum probabilityat x'-counter
⎛
⎝⎜⎜
⎞
⎠⎟⎟= x' x eiφ( ) x x' + x' y y x'
2
= x' x x x'2+ x' y y x'
2+ e−iφ x' x * x x' * x' y y x' + eiφ x' x x x' x' y * y x' *
=1
=( classical probability ) + ( Phase-sensitive or quantum interference terms )
Quantum probabilityat x'-counter
⎛
⎝⎜⎜
⎞
⎠⎟⎟= x' x x x' + x' y y x'
2
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Classical Bayesian probability vs. Quantum probability
Square of sum 85Tuesday, January 20, 2015
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Probability thatphoton in x'-input
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
classical
=
probability thatphoton in x-beam
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
*
probability thatphoton in x'-input
becomesphoton in x-beam
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
+
probability thatphoton in y-beam
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
*
probability thatphoton in x'-input
becomesphoton in y-beam
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
Probability thatphoton in x'-input
becomesphoton in x'-counter
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
classical
= x' x2⎛
⎝⎜⎞⎠⎟ * x x'
2⎛⎝⎜
⎞⎠⎟ + x' y
2⎛⎝⎜
⎞⎠⎟ * y x'
2⎛⎝⎜
⎞⎠⎟ =
32
2⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
* 32
2⎛
⎝
⎜⎜
⎞
⎠
⎟⎟+ −1
2
2⎛
⎝⎜⎜
⎞
⎠⎟⎟
* 12
2⎛
⎝⎜⎜
⎞
⎠⎟⎟= 5
8
Quantum probabilityat x'-counter
⎛
⎝⎜⎜
⎞
⎠⎟⎟= x' x eiφ( ) x x' + x' y y x'
2
= x' x x x'2+ x' y y x'
2+ e−iφ x' x * x x' * x' y y x' + eiφ x' x x x' x' y * y x' *
=( classical probability ) + ( Phase-sensitive or quantum interference terms )
Quantum probabilityat x'-counter
⎛
⎝⎜⎜
⎞
⎠⎟⎟= x' x x x' + x' y y x'
2
Classical probabilityat x'-counter
⎛
⎝⎜⎜
⎞
⎠⎟⎟= x' x x x'
2+ x' y y x'
2
x ′x x ′y
y ′x y ′y
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟= 3 / 2 −1/ 2
1/ 2 3 / 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Classical Bayesian probability vs. Quantum probability
Square of sum Sum of squares 86Tuesday, January 20, 2015
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Review: Axioms 1-4 and“Do-Nothing”vs“ Do-Something” analyzers
Abstraction of Axiom-4 to define projection and unitary operators Projection operators and resolution of identity
Unitary operators and matrices that do something (or “nothing”) Diagonal unitary operators Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker ⊗-products Axiom-4 similarity transformation
Matrix representation of beam analyzers Non-unitary “killer” devices: Sorter-counter, filter Unitary “non-killer” devices: 1/2-wave plate, 1/4-wave plate
How analyzers “peek” and how that changes outcomes Peeking polarizers and coherence loss Classical Bayesian probability vs. Quantum probability
Feynman 〈j⏐k〉-axioms compared to Group axioms
87Tuesday, January 20, 2015
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(1) The closure axiom Products ab = c are defined between any two group elements a and b, and the result c is contained in the group.
(2) The associativity axiom Products (ab)c and a(bc) are equal for all elements a, b, and c in the group .
Group axioms
(3) The identity axiom There is a unique element 1 (the identity) such that 1.a = a = a.1 for all elements a in the group ..
4) The inverse axiom For all elements a in the group there is an inverse element a-1 such that a-1a = 1 = a.a-1.
Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉
k=1
nAxiom 3: 〈j⏐k〉=δjk=〈j′⏐k′〉=〈j′′⏐ k′′〉
Axiom 2: 〈j′′⏐m′〉*=〈m′⏐ j′′〉
Axiom 1: j′′⇔m′ probability equals |〈j′′⏐m′〉|2 = |〈m′⏐ j′′〉|2
Feynman 〈j⏐k〉-axioms compared to Group axioms
(Probability) (T-reversal Conjugation) (Orthonormality) (Completeness)
88Tuesday, January 20, 2015
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(1) The closure axiom Products ab = c are defined between any two group elements a and b, and the result c is contained in the group.
(2) The associativity axiom Products (ab)c and a(bc) are equal for all elements a, b, and c in the group .
Group axioms
(3) The identity axiom There is a unique element 1 (the identity) such that 1.a = a = a.1 for all elements a in the group ..
4) The inverse axiom For all elements a in the group there is an inverse element a-1 such that a-1a = 1 = a.a-1.
Feynman Axiom-4 consistent with group axiom 1 since analyzer-A following analyzer-B is analyzer-AB =C and analyzer-B following analyzer-A is analyzer-BA =D
Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉
k=1
nAxiom 3: 〈j⏐k〉=δjk=〈j′⏐k′〉=〈j′′⏐ k′′〉
Axiom 2: 〈j′′⏐m′〉*=〈m′⏐ j′′〉
Axiom 1: j′′⇔m′ probability equals |〈j′′⏐m′〉|2 = |〈m′⏐ j′′〉|2
Feynman 〈j⏐k〉-axioms compared to Group axioms
(Probability) (T-reversal Conjugation) (Orthonormality) (Completeness)
89Tuesday, January 20, 2015
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(1) The closure axiom Products ab = c are defined between any two group elements a and b, and the result c is contained in the group.
(2) The associativity axiom Products (ab)c and a(bc) are equal for all elements a, b, and c in the group .
Group axioms
(3) The identity axiom There is a unique element 1 (the identity) such that 1.a = a = a.1 for all elements a in the group ..
4) The inverse axiom For all elements a in the group there is an inverse element a-1 such that a-1a = 1 = a.a-1.
Feynman Axiom-4 consistent with group axiom 1 since analyzer-A following analyzer-B is analyzer-AB =C and analyzer-B following analyzer-A is analyzer-BA =D
Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉
k=1
nAxiom 3: 〈j⏐k〉=δjk=〈j′⏐k′〉=〈j′′⏐ k′′〉
Axiom 2: 〈j′′⏐m′〉*=〈m′⏐ j′′〉
Axiom 1: j′′⇔m′ probability equals |〈j′′⏐m′〉|2 = |〈m′⏐ j′′〉|2
Feynman 〈j⏐k〉-axioms compared to Group axioms
Feynman Axiom-4 consistent with group axiom 2 since analyzer matrix multiplication is associative
(Probability) (T-reversal Conjugation) (Orthonormality) (Completeness)
90Tuesday, January 20, 2015
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(1) The closure axiom Products ab = c are defined between any two group elements a and b, and the result c is contained in the group.
(2) The associativity axiom Products (ab)c and a(bc) are equal for all elements a, b, and c in the group .
Group axioms
(3) The identity axiom There is a unique element 1 (the identity) such that 1.a = a = a.1 for all elements a in the group ..
4) The inverse axiom For all elements a in the group there is an inverse element a-1 such that a-1a = 1 = a.a-1.
Feynman Axiom-2 consistent with group axiom 3 since “Do Nothing” analyzer =identity operator=1
Feynman Axiom-4 consistent with group axiom 1 since analyzer-A following analyzer-B is analyzer-AB =C and analyzer-B following analyzer-A is analyzer-BA =D
Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉
k=1
nAxiom 3: 〈j⏐k〉=δjk=〈j′⏐k′〉=〈j′′⏐ k′′〉
Axiom 2: 〈j′′⏐m′〉*=〈m′⏐ j′′〉
Axiom 1: j′′⇔m′ probability equals |〈j′′⏐m′〉|2 = |〈m′⏐ j′′〉|2
Feynman 〈j⏐k〉-axioms compared to Group axioms
Feynman Axiom-4 consistent with group axiom 2 since analyzer matrix multiplication is associative
(Probability) (T-reversal Conjugation) (Orthonormality) (Completeness)
91Tuesday, January 20, 2015
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(1) The closure axiom Products ab = c are defined between any two group elements a and b, and the result c is contained in the group.
(2) The associativity axiom Products (ab)c and a(bc) are equal for all elements a, b, and c in the group .
Group axioms
(3) The identity axiom There is a unique element 1 (the identity) such that 1.a = a = a.1 for all elements a in the group ..
4) The inverse axiom For all elements a in the group there is an inverse element a-1 such that a-1a = 1 = a.a-1.
Feynman Axiom-3 consistent with group axiom 4 since inverse U =tranpose-conjugate U† = UT*
Feynman Axiom-2 consistent with group axiom 3 since “Do Nothing” analyzer =identity operator=1
Feynman Axiom-4 consistent with group axiom 1 since analyzer-A following analyzer-B is analyzer-AB =C and analyzer-B following analyzer-A is analyzer-BA =D
Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉
k=1
nAxiom 3: 〈j⏐k〉=δjk=〈j′⏐k′〉=〈j′′⏐ k′′〉
Axiom 2: 〈j′′⏐m′〉*=〈m′⏐ j′′〉
Axiom 1: j′′⇔m′ probability equals |〈j′′⏐m′〉|2 = |〈m′⏐ j′′〉|2
Feynman 〈j⏐k〉-axioms compared to Group axioms
Feynman Axiom-4 consistent with group axiom 2 since analyzer matrix multiplication is associative
(Probability) (T-reversal Conjugation) (Orthonormality) (Completeness)
92Tuesday, January 20, 2015
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(1) The closure axiom Products ab = c are defined between any two group elements a and b, and the result c is contained in the group.
(2) The associativity axiom Products (ab)c and a(bc) are equal for all elements a, b, and c in the group .
Group axioms
(3) The identity axiom There is a unique element 1 (the identity) such that 1.a = a = a.1 for all elements a in the group ..
4) The inverse axiom For all elements a in the group there is an inverse element a-1 such that a-1a = 1 = a.a-1.
Feynman Axiom-3 consistent with group axiom 4 since inverse U =tranpose-conjugate U† = UT*
Feynman Axiom-2 consistent with group axiom 3 since “Do Nothing” analyzer =identity operator=1
Feynman Axiom-4 consistent with group axiom 1 since analyzer-A following analyzer-B is analyzer-AB =C and analyzer-B following analyzer-A is analyzer-BA =D
Axiom 4: 〈j′′⏐m′〉=∑〈j′′⏐k〉〈k⏐m′〉
k=1
nAxiom 3: 〈j⏐k〉=δjk=〈j′⏐k′〉=〈j′′⏐ k′′〉
Axiom 2: 〈j′′⏐m′〉*=〈m′⏐ j′′〉
Axiom 1: j′′⇔m′ probability equals |〈j′′⏐m′〉|2 = |〈m′⏐ j′′〉|2
Feynman 〈j⏐k〉-axioms compared to Group axioms
Feynman Axiom-4 consistent with group axiom 2 since analyzer matrix multiplication is associative
(5) The commutative axiom (Abelian groups only) All elements a in an Abelian group are mutually commuting: a.b = b.a.
Most analyzer sets (and most groups)are not Abelian (commutative)
(Probability) (T-reversal Conjugation) (Orthonormality) (Completeness)
93Tuesday, January 20, 2015