Introduction to Quantum Mechanics
QCSYS 2012
Outline
1. Polarization
2. Double-slit experiment
3. Photoelectric effect
4. No-cloning theorem
Polarization
Superposition
A basic feature of quantum mechanics is the principle of superposition:
If a quantum system can be in the state or in the state , then it can also be in state for any complex numbers (subject to normalization).
| i |�i↵| i+ �|�i ↵,�
Superposition
A basic feature of quantum mechanics is the principle of superposition:
Example:
|+i = 1p2|0i+ 1p
2|1i 1p
2
✓11
◆=
1p2
✓10
◆+
1p2
✓01
◆
If a quantum system can be in the state or in the state , then it can also be in state for any complex numbers (subject to normalization).
| i |�i↵| i+ �|�i ↵,�
Superposition
A basic feature of quantum mechanics is the principle of superposition:
The superposition principle is also shared by classical waves.
Example:
|+i = 1p2|0i+ 1p
2|1i 1p
2
✓11
◆=
1p2
✓10
◆+
1p2
✓01
◆
If a quantum system can be in the state or in the state , then it can also be in state for any complex numbers (subject to normalization).
| i |�i↵| i+ �|�i ↵,�
Superposition
A basic feature of quantum mechanics is the principle of superposition:
The superposition principle is also shared by classical waves.
We’ll explore superposition in the context of polarization of light.
Example:
|+i = 1p2|0i+ 1p
2|1i 1p
2
✓11
◆=
1p2
✓10
◆+
1p2
✓01
◆
If a quantum system can be in the state or in the state , then it can also be in state for any complex numbers (subject to normalization).
| i |�i↵| i+ �|�i ↵,�
The electromagnetic field
In classical electromagnetism, there is an
at every spacetime point .(x, y, z, t)
electric field ~
E(x, y, z, t)~
B(x, y, z, t)magnetic field
The electromagnetic field
In classical electromagnetism, there is an
at every spacetime point .(x, y, z, t)
electric field ~
E(x, y, z, t)~
B(x, y, z, t)magnetic field
These fields obey the Maxwell equations:
~r · ~E = ⇢ ~r⇥ ~E = �@ ~B
@t
~r · ~B = 0 ~r⇥ ~B = ~J +1
c2@ ~E
@t
Traveling wavesThe Maxwell equations have solutions that correspond to waves propagating through space.
Traveling wavesThe Maxwell equations have solutions that correspond to waves propagating through space.
Example: Plane wave propagating in the directionz
~
B(x, y, z, t) = z ⇥ ~
E(x, y, z, t)
~
E(x, y, z, t) = Re
0
@
0
@↵
x
↵
y
0
1
Ae
2⇡i(z�ct)/�
1
A
Traveling wavesThe Maxwell equations have solutions that correspond to waves propagating through space.
The vector indicates the polarization of the wave.
✓↵x
↵y
◆
Example: Plane wave propagating in the directionz
~
B(x, y, z, t) = z ⇥ ~
E(x, y, z, t)
~
E(x, y, z, t) = Re
0
@
0
@↵
x
↵
y
0
1
Ae
2⇡i(z�ct)/�
1
A
Superposition of polarization states
Horizontal polarization: |!i =✓10
◆Vertical polarization: |"i =
✓01
◆
Superposition of polarization states
Horizontal polarization: |!i =✓10
◆Vertical polarization: |"i =
✓01
◆
45° diagonal polarization (normalized states):
|%i = 1p2|!i+ 1p
2|"i = 1p
2
✓11
◆
|&i = 1p2|!i � 1p
2|"i = 1p
2
✓1�1
◆
Superposition of polarization states
Horizontal polarization: |!i =✓10
◆Vertical polarization: |"i =
✓01
◆
45° diagonal polarization (normalized states):
|%i = 1p2|!i+ 1p
2|"i = 1p
2
✓11
◆
|&i = 1p2|!i � 1p
2|"i = 1p
2
✓1�1
◆
The 45° states also form an orthonormal basis:
|!i = 1p2|%i+ 1p
2|&i
|"i = 1p2|%i � 1p
2|&i
Polarizing filters
Most sources of light are unpolarized.
Polarizing filters
Most sources of light are unpolarized.
We can create polarized light using a filter that only allows one of two orthogonal polarizations to pass.
Polarizing filters
Most sources of light are unpolarized.
We can create polarized light using a filter that only allows one of two orthogonal polarizations to pass.
Mathematically, this implements a projection onto a the polarization direction of the filter.
• The component along the filter direction passes through.
• The component orthogonal to the filter direction is blocked.
Polarizing filters
Most sources of light are unpolarized.
We can create polarized light using a filter that only allows one of two orthogonal polarizations to pass.
Mathematically, this implements a projection onto a the polarization direction of the filter.
• The component along the filter direction passes through.
• The component orthogonal to the filter direction is blocked.
Analogous to quantum measurement:
• The fraction of light that passes through the filter is given by the inner product squared with the filter direction.
• The outgoing light is entirely in the same direction as the filter.
Polarizing filter examples
Incident polarization: |%i
How much light passes?
Polarizing filter orientation: !
Polarizing filter examples
Incident polarization: |%i
How much light passes? 50%
Polarizing filter orientation: !
Polarizing filter examples
Incident polarization: |%i
How much light passes? 50%
Outgoing polarization?
Polarizing filter orientation: !
Polarizing filter examples
Incident polarization: |%i
How much light passes? 50%
Outgoing polarization? |!i
Polarizing filter orientation: !
Polarizing filter examples
Incident polarization: |%i
How much light passes? 50%
Outgoing polarization? |!i
Polarizing filter orientation: !
Incident polarization:
µPolarizing filter orientation: !
cos ✓|!i+ sin ✓|"i =✓cos ✓sin ✓
◆
Polarizing filter examples
Incident polarization: |%i
How much light passes? 50%
Outgoing polarization? |!i
Polarizing filter orientation: !
How much light passes?
Incident polarization:
µPolarizing filter orientation: !
cos ✓|!i+ sin ✓|"i =✓cos ✓sin ✓
◆
Polarizing filter examples
Incident polarization: |%i
How much light passes? 50%
Outgoing polarization? |!i
Polarizing filter orientation: !
How much light passes? cos
2 ✓
Incident polarization:
µPolarizing filter orientation: !
cos ✓|!i+ sin ✓|"i =✓cos ✓sin ✓
◆
Polarizing filter examples
Incident polarization: |%i
How much light passes? 50%
Outgoing polarization? |!i
Polarizing filter orientation: !
How much light passes?
Outgoing polarization?
cos
2 ✓
Incident polarization:
µPolarizing filter orientation: !
cos ✓|!i+ sin ✓|"i =✓cos ✓sin ✓
◆
Polarizing filter examples
Incident polarization: |%i
How much light passes? 50%
Outgoing polarization? |!i
Polarizing filter orientation: !
How much light passes?
Outgoing polarization?
cos
2 ✓
Incident polarization:
µPolarizing filter orientation: !
cos ✓|!i+ sin ✓|"i =✓cos ✓sin ✓
◆
|!i
Polarizing filter demo
What happens to the incident light if we orient polarizers as follows?
• crossed polarizers (0 and 90 degrees)
• diagonal polarizers (0 and 45 degrees)
• polarizers at 0, 45, 90 degrees
Another polarizing filter example
Incident polarization:
Polarizing filter orientation:
|!i ✓cos ✓sin ✓
◆
Another polarizing filter example
Incident polarization:
Polarizing filter orientation:
|!i ✓cos ✓sin ✓
◆
|!i =✓1
0
◆
= cos ✓
✓cos ✓sin ✓
◆� sin ✓
✓� sin ✓cos ✓
◆+
Another polarizing filter example
Incident polarization:
Polarizing filter orientation:
|!i ✓cos ✓sin ✓
◆
|!i =✓1
0
◆
= cos ✓
✓cos ✓sin ✓
◆� sin ✓
✓� sin ✓cos ✓
◆
Another polarizing filter example
How much light passes?
Incident polarization:
Polarizing filter orientation:
|!i ✓cos ✓sin ✓
◆
|!i =✓1
0
◆
= cos ✓
✓cos ✓sin ✓
◆� sin ✓
✓� sin ✓cos ✓
◆
Another polarizing filter example
How much light passes? cos
2 ✓
Incident polarization:
Polarizing filter orientation:
|!i ✓cos ✓sin ✓
◆
|!i =✓1
0
◆
= cos ✓
✓cos ✓sin ✓
◆� sin ✓
✓� sin ✓cos ✓
◆
Another polarizing filter example
How much light passes?
Outgoing polarization?
cos
2 ✓
Incident polarization:
Polarizing filter orientation:
|!i ✓cos ✓sin ✓
◆
|!i =✓1
0
◆
= cos ✓
✓cos ✓sin ✓
◆� sin ✓
✓� sin ✓cos ✓
◆
Another polarizing filter example
How much light passes?
Outgoing polarization?
cos
2 ✓
Incident polarization:
Polarizing filter orientation:
|!i ✓cos ✓sin ✓
◆
|!i =✓1
0
◆
= cos ✓
✓cos ✓sin ✓
◆� sin ✓
✓� sin ✓cos ✓
◆
✓cos ✓sin ✓
◆= cos ✓|!i+ sin ✓|"i
Circular polarization
We can also consider superpositions involving complex numbers.
Circular polarization
We can also consider superpositions involving complex numbers.
| i = 1p2(|!i � i|"i)|�i = 1p
2(|!i+ i|"i)
Examples:
Circular polarization
We can also consider superpositions involving complex numbers.
The direction of the electric field moves in a circle as the wave propagates, so this is called circular polarization.
| i = 1p2(|!i � i|"i)|�i = 1p
2(|!i+ i|"i)
Examples:
QubitsSo far, we have used polarization vectors to describe classical light.
Single photons also have polarization.
Then the polarization vector describes the state of a quantum bit, or qubit.
↵|!i+ �|"i
QubitsSo far, we have used polarization vectors to describe classical light.
Single photons also have polarization.
Then the polarization vector describes the state of a quantum bit, or qubit.
↵|!i+ �|"i
Many other systems can be used to store qubits, and the components of the state vector need not represent directions in space:
• Atom:• Electron spin:• Photon number:• Superconducting flux:• ...
↵|groundi+ �|excitedi↵|upi+ �|downi
↵|0i+ �|1i↵|cwi+ �|ccwi
Exercise: Stacked polarizers
Suppose we stack polarizers so that the angle between the polarization direction of each filter and the next is . What fraction of the light passing the first polarizer passes the last polarizer?
n⇡/n
a.Compute exact values for .b.Give a symbolic expression for general .c.Using a computer, plot the values for through .d.What happens in the limit as ?
n = 2, 3, 4
n
n = 2 50
n ! 1
Double-slit experiment
Firing bullets at a slit
Firing bullets at a slit
Firing bullets at a slit
Firing bullets at a slit
Firing bullets at a slit
Firing bullets at a double slit
Firing bullets at a double slit
Firing bullets at a double slit
Firing bullets at a double slit
Firing bullets at a double slit
Double slit with waves
Double slit with waves
Double slit with waves
Double slit with waves
Double slit with waves
Double slit with waves
Double slit with waves
Demo
Interference
Interference
wavelength ¸
Interference
wavelength ¸
amplitude
Amplitude can be positive (water is above sea level)or negative (water is below sea level)
Interference
wavelength ¸
amplitude
Amplitude can be positive (water is above sea level)or negative (water is below sea level)
+
Interference
wavelength ¸
amplitude
Amplitude can be positive (water is above sea level)or negative (water is below sea level)
+ =
Interference
wavelength ¸
amplitude
Amplitude can be positive (water is above sea level)or negative (water is below sea level)
+
+
=
Interference
wavelength ¸
amplitude
Amplitude can be positive (water is above sea level)or negative (water is below sea level)
+
+
=
=
Form of the interference pattern
Form of the interference pattern
¢
Form of the interference pattern
...
...
µ¢
Form of the interference pattern
...
...
µ¢
Form of the interference pattern
...
...
µ
µ
¢
Form of the interference pattern
...
...
µ
µ
¢
� sin ✓
Form of the interference pattern
...
...
µ
µ
¢
� sin ✓
One fringe: � sin ✓ = �
Form of the interference pattern
...
...
µ
µ
¢
� sin ✓
One fringe: � sin ✓ = � ✓ ⇡ �
�
Form of the interference pattern
...
...
µ
µ
¢
� sin ✓
One fringe: � sin ✓ = � ✓ ⇡ �
�
Screen at a distance d away: fringe spacing is approximately d · ✓ ⇡ d · ��
Double slit with electrons
How will the experiment behave if we use electrons instead of bullets or water waves?
Double slit with electrons
How will the experiment behave if we use electrons instead of bullets or water waves?
• Electrons come in discrete chunks, like bullets.
Double slit with electrons
How will the experiment behave if we use electrons instead of bullets or water waves?
• Electrons come in discrete chunks, like bullets.
• Nevertheless, the experiment shows an interference pattern!
Double slit with electrons
How will the experiment behave if we use electrons instead of bullets or water waves?
• Electrons come in discrete chunks, like bullets.
• Nevertheless, the experiment shows an interference pattern!
What does this mean? Is an electron a particle or a wave?
Double slit with electrons
How will the experiment behave if we use electrons instead of bullets or water waves?
• Electrons come in discrete chunks, like bullets.
• Nevertheless, the experiment shows an interference pattern!
What does this mean? Is an electron a particle or a wave?
• Yes. (“wave-particle duality”)
• De Broglie: ,� = h/p h = 6.626⇥ 10�34 J · s
Double slit with electrons
What if we observe which slit the electrons go through?
How will the experiment behave if we use electrons instead of bullets or water waves?
• Electrons come in discrete chunks, like bullets.
• Nevertheless, the experiment shows an interference pattern!
What does this mean? Is an electron a particle or a wave?
• Yes. (“wave-particle duality”)
• De Broglie: ,� = h/p h = 6.626⇥ 10�34 J · s
Double slit with electrons
What if we observe which slit the electrons go through?
How will the experiment behave if we use electrons instead of bullets or water waves?
• Electrons come in discrete chunks, like bullets.
• Nevertheless, the experiment shows an interference pattern!
Same behavior with light, which is composed of individual photons.
What does this mean? Is an electron a particle or a wave?
• Yes. (“wave-particle duality”)
• De Broglie: ,� = h/p h = 6.626⇥ 10�34 J · s
Uncertainty principle
In classical mechanics, nothing prevents us from measuring the state of a particle (its position and momentum) with arbitrary precision.
Uncertainty principle
In classical mechanics, nothing prevents us from measuring the state of a particle (its position and momentum) with arbitrary precision.
Quantum mechanics forbids this: it places fundamental limitations on the kinds of measurements that can be carried out.
Uncertainty principle
In classical mechanics, nothing prevents us from measuring the state of a particle (its position and momentum) with arbitrary precision.
Quantum mechanics forbids this: it places fundamental limitations on the kinds of measurements that can be carried out.
Heisenberg uncertainty principle: �x�p � ~2
(~ = h2⇡ )
Uncertainty principle and diffraction
Exercise: Double slit with laser light
Suppose you perform the double slit experiment using a green laser with a wavelength of 523 nm and slits spaced by 1 mm.
a. What is the angular spacing between two adjacent fringes of the interference pattern?
b. If the pattern is projected onto a screen at a distance of 5 m, what is the distance between adjacent fringes?
Photoelectric effect
The photoelectric effect
⊝⊝⊝
⊝⊝
⊝⊝⊝
⊝⊝
⊝⊝⊝
⊝⊝
⊝⊝⊝
⊝⊝
⊝⊝⊝
⊝⊝
⊝⊝⊝
⊝⊝
⊝⊝⊝
⊝⊝
⊝⊝⊝
The photoelectric effect
⊝⊝⊝
⊝⊝
⊝⊝⊝
⊝⊝
⊝⊝⊝
⊝⊝
⊝⊝⊝
⊝
⊝⊝⊝
⊝⊝
⊝⊝⊝
⊝⊝
⊝⊝⊝
⊝⊝
⊝⊝⊝
⊝
Photons and electrons
Photons and electrons
Light is made up of massless particles called photons.
Photons and electrons
Light is made up of massless particles called photons.
Photons are characterized by their wavelength �or equivalently, their frequency .⌫
Photons and electrons
Light is made up of massless particles called photons.
Photons are characterized by their wavelength �or equivalently, their frequency .⌫
E = h⌫ =hc
�
h = 6.62⇥ 10�34 J · sc = 3⇥ 108 m/s
Photons and electrons
Light is made up of massless particles called photons.
Photons are characterized by their wavelength �or equivalently, their frequency .⌫
E = h⌫ =hc
�
h = 6.62⇥ 10�34 J · sc = 3⇥ 108 m/s
Electrons are massive particles with negative electric charge.
Photons and electrons
Light is made up of massless particles called photons.
Photons are characterized by their wavelength �or equivalently, their frequency .⌫
E = h⌫ =hc
�
h = 6.62⇥ 10�34 J · sc = 3⇥ 108 m/s
Electrons are massive particles with negative electric charge.
E =1
2mv2 + �
m = 9.11⇥ 10�31 kg
e = 1.60⇥ 10�19 C
The work function
Removing an electron from a material costs energy.
The work function
Removing an electron from a material costs energy.
In the context of the photoelectric effect, this is called the work function, denoted . Its value depends on the material. Typical values are a few electron volts ( ).
�1 eV = 1.602⇥ 10�19 J
The work function
Removing an electron from a material costs energy.
By conservation of energy, for a photon of frequency to eject an electron with velocity , we havev
⌫
h⌫ =1
2mv2 + �
In the context of the photoelectric effect, this is called the work function, denoted . Its value depends on the material. Typical values are a few electron volts ( ).
�1 eV = 1.602⇥ 10�19 J
The work function
Whether emission occurs depends on the frequency of the light, not on its intensity (the number of photons arriving per unit time).
Removing an electron from a material costs energy.
By conservation of energy, for a photon of frequency to eject an electron with velocity , we havev
⌫
h⌫ =1
2mv2 + �
In the context of the photoelectric effect, this is called the work function, denoted . Its value depends on the material. Typical values are a few electron volts ( ).
�1 eV = 1.602⇥ 10�19 J
Photoelectric effect experiment
Photoelectric effect experiment
Photoelectric effect experiment
⊝
Photoelectric effect experiment
⊝
Photoelectric effect experiment
⊝
h⌫ =1
2mv2 + �
Photoelectric effect experiment
⊝
h⌫ =1
2mv2 + �
Photoelectric effect experiment
⊝
h⌫ =1
2mv2 + �
1
2mv2 = eV0 where is the stopping potential, the voltage that
must be applied to stop the current from flowingV0
Photoelectric effect experiment: results
V0
⌫
Photoelectric effect experiment: results
V0
⌫
Photoelectric effect experiment: results
V0
⌫
h⌫ =1
2mv2 + �
1
2mv2 = eV0
Photoelectric effect experiment: results
V0
⌫
h⌫ =1
2mv2 + �
1
2mv2 = eV0
V0 =h⌫ � �
e
Photoelectric effect experiment: results
V0
⌫
h⌫ =1
2mv2 + �
1
2mv2 = eV0
V0 =h⌫ � �
e
�/h
Photoelectric effect experiment: results
V0
⌫
h⌫ =1
2mv2 + �
1
2mv2 = eV0
V0 =h⌫ � �
e
�/h
h/e
slope
Exercise: Photoelectric effect in platinum
For this problem, the following values may be useful:
h = 4.14⇥ 10�15 eV · sc = 3⇥ 108 m/s
a. When a platinum electrode is illuminated with light of wavelength 150 nm, the stopping potential is 2 V. What is the work function of platinum in eV?
b. What is the maximum wavelength of light that will eject electrons from platinum?
Experiment(end of the week)
No-cloning theorem
Classical cloning
cloning machine
Classical cloning
cloning machine
Classical cloning
cloning machine
Classical cloning
cloning machine
Classical cloning
In principle, such a device is possible.
cloning machine
Classical cloning (digital)
cloning machine
Classical cloning (digital)
cloning machine
file1.txt
Lorem ipsum dolor sit amet, consectetur adipisicing elit...
file2.txt
Classical cloning (digital)
cloning machine
file1.txt
Lorem ipsum dolor sit amet, consectetur adipisicing elit...
file2.txt
file1.txt
Lorem ipsum dolor sit amet, consectetur adipisicing elit...
file2.txt
Lorem ipsum dolor sit amet, consectetur adipisicing elit...
Quantum cloning?
cloning machine
Quantum cloning?
cloning machine
Quantum cloning?
cloning machine
Quantum cloning?
cloning machine
The uncertainty principle prevents us from learning an unknown quantum state.
Quantum cloning?
cloning machine
The uncertainty principle prevents us from learning an unknown quantum state.
Such a device is impossible!
Quantum cloning?
cloning machine
Quantum cloning?
cloning machine
|0i
↵|0i+ �|1i
Quantum cloning?
cloning machine
|0i
↵|0i+ �|1i ↵|0i+ �|1i
↵|0i+ �|1i
Quantum cloning?
cloning machine
|0i
↵|0i+ �|1i ↵|0i+ �|1i
↵|0i+ �|1i
This is also impossible.
Quantum cloning?
Even digital quantum information (qubits) cannot be cloned.
cloning machine
|0i
↵|0i+ �|1i ↵|0i+ �|1i
↵|0i+ �|1i
This is also impossible.
No-cloning theorem
Theorem [Wootters, Zurek, Dieks 1982]: There is no valid quantum process that takes as input an unknown quantum state and an ancillary system in a known state, and outputs two copies of .
| i| i
|0i
| i | i
| iU
Proof of the no-cloning theorem
Proof of the no-cloning theorem
U(| i ⌦ |0i) = | i ⌦ | iU(|�i ⌦ |0i) = |�i ⌦ |�i
Consider two orthogonal states . By the definition of cloning,| i, |�i
Proof of the no-cloning theorem
By linearity,
U [(↵| i+ �|�i)⌦ |0i]
U(| i ⌦ |0i) = | i ⌦ | iU(|�i ⌦ |0i) = |�i ⌦ |�i
Consider two orthogonal states . By the definition of cloning,| i, |�i
Proof of the no-cloning theorem
By linearity,
U [(↵| i+ �|�i)⌦ |0i] = ↵U(| i ⌦ |0i) + �U(|�i ⌦ |0i)
U(| i ⌦ |0i) = | i ⌦ | iU(|�i ⌦ |0i) = |�i ⌦ |�i
Consider two orthogonal states . By the definition of cloning,| i, |�i
Proof of the no-cloning theorem
By linearity,
U [(↵| i+ �|�i)⌦ |0i] = ↵U(| i ⌦ |0i) + �U(|�i ⌦ |0i)= ↵| i ⌦ | i+ �|�i ⌦ |�i
U(| i ⌦ |0i) = | i ⌦ | iU(|�i ⌦ |0i) = |�i ⌦ |�i
Consider two orthogonal states . By the definition of cloning,| i, |�i
Proof of the no-cloning theorem
By linearity,
U [(↵| i+ �|�i)⌦ |0i] = ↵U(| i ⌦ |0i) + �U(|�i ⌦ |0i)= ↵| i ⌦ | i+ �|�i ⌦ |�i
U(| i ⌦ |0i) = | i ⌦ | iU(|�i ⌦ |0i) = |�i ⌦ |�i
Consider two orthogonal states . By the definition of cloning,| i, |�i
But again by the definition of cloning, we should have
U [(↵| i+ �|�i)⌦ |0i]
Proof of the no-cloning theorem
By linearity,
U [(↵| i+ �|�i)⌦ |0i] = ↵U(| i ⌦ |0i) + �U(|�i ⌦ |0i)= ↵| i ⌦ | i+ �|�i ⌦ |�i
U(| i ⌦ |0i) = | i ⌦ | iU(|�i ⌦ |0i) = |�i ⌦ |�i
Consider two orthogonal states . By the definition of cloning,| i, |�i
But again by the definition of cloning, we should have
U [(↵| i+ �|�i)⌦ |0i]= (↵| i+ �|�i)⌦ (↵| i+ �|�i)
Proof of the no-cloning theorem
By linearity,
U [(↵| i+ �|�i)⌦ |0i] = ↵U(| i ⌦ |0i) + �U(|�i ⌦ |0i)= ↵| i ⌦ | i+ �|�i ⌦ |�i
U(| i ⌦ |0i) = | i ⌦ | iU(|�i ⌦ |0i) = |�i ⌦ |�i
Consider two orthogonal states . By the definition of cloning,| i, |�i
But again by the definition of cloning, we should have
U [(↵| i+ �|�i)⌦ |0i]= (↵| i+ �|�i)⌦ (↵| i+ �|�i)= ↵2| i ⌦ | i+ ↵�| i ⌦ |�i+ ↵�|�i ⌦ | i+ �2|�i ⌦ |�i
Proof of the no-cloning theorem
By linearity,
U [(↵| i+ �|�i)⌦ |0i] = ↵U(| i ⌦ |0i) + �U(|�i ⌦ |0i)= ↵| i ⌦ | i+ �|�i ⌦ |�i
U(| i ⌦ |0i) = | i ⌦ | iU(|�i ⌦ |0i) = |�i ⌦ |�i
Consider two orthogonal states . By the definition of cloning,| i, |�i
But again by the definition of cloning, we should have
U [(↵| i+ �|�i)⌦ |0i]= (↵| i+ �|�i)⌦ (↵| i+ �|�i)= ↵2| i ⌦ | i+ ↵�| i ⌦ |�i+ ↵�|�i ⌦ | i+ �2|�i ⌦ |�i
Therefore ↵2 = ↵, ↵� = 0, �2 = �
Proof of the no-cloning theorem
By linearity,
U [(↵| i+ �|�i)⌦ |0i] = ↵U(| i ⌦ |0i) + �U(|�i ⌦ |0i)= ↵| i ⌦ | i+ �|�i ⌦ |�i
U(| i ⌦ |0i) = | i ⌦ | iU(|�i ⌦ |0i) = |�i ⌦ |�i
Consider two orthogonal states . By the definition of cloning,| i, |�i
But again by the definition of cloning, we should have
U [(↵| i+ �|�i)⌦ |0i]= (↵| i+ �|�i)⌦ (↵| i+ �|�i)= ↵2| i ⌦ | i+ ↵�| i ⌦ |�i+ ↵�|�i ⌦ | i+ �2|�i ⌦ |�i
Therefore ↵2 = ↵, ↵� = 0, �2 = �
So either or .↵ = 0 � = 0
Cloning in a fixed basis
While we cannot copy quantum information, we can copy classical information. In particular, we can copy quantum states in a fixed basis.
Cloning in a fixed basis
While we cannot copy quantum information, we can copy classical information. In particular, we can copy quantum states in a fixed basis.
Example: Controlled-not gate
|00i 7! |00i|01i 7! |01i|10i 7! |11i|11i 7! |10i
ASSIGNMENT 1: Solutions CO 481/CS 467/PHYS 467 (Winter 2010)
1. Universality of reversible logic gates.
(a) [3 points] The cccnot (triple-controlled not) gate is a four-bit reversible gate that flipsits fourth bit if and only if the first three bits are all in the state 1. Show how to implementa cccnot gate using To�oli gates. You may use additional workspace as needed. You mayassume that bits in the workspace start with a particular value, either 0 or 1, provided youreturn them to that value. For a bonus point, give a circuit that works regardless of thevalues of any bits of workspace.
Solution: The following circuit shows a simple construction using one bit of workspace inthe 0 state:
•••���⌧⇠⇡⇢�
0
=
• •• •
•���⌧⇠⇡⇢�0 ���⌧⇠⇡⇢� • ���⌧⇠⇡⇢�
The first gate computes the and of the first two bits in the fifth (workspace) bit. Thesecond gate computes the and of the third and fifth bits (i.e., the and of the first threebits) in the fourth (target) bit. The final gate uncomputes the value in the workspace.This circuit can be modified to work for a workspace bit with any value as follows:
•••���⌧⇠⇡⇢� =
• •• •
• •���⌧⇠⇡⇢� ���⌧⇠⇡⇢����⌧⇠⇡⇢� • ���⌧⇠⇡⇢� •(b) [5 points] Show that a To�oli gate cannot be implemented using any number of cnot gates,
with any amount of workspace. Hence the cnot gate alone is not universal. (Hint: It maybe helpful to think of the gates as performing arithmetic operations on integers mod 2.)
Solution: The cnot gate acts as
x • xy ���⌧⇠⇡⇢� x� y
(where � denotes addition modulo 2), so composing any number of cnot gates gives outputbits that can be expressed as a sum modulo 2 of a subset of input bits. However, the To�oligate acts as
x • xy • y
z ���⌧⇠⇡⇢� xy � z
and the product xy (representing logical and) cannot be expressed as a sum modulo 2 ofterms involving x and y (and the constants 0 and 1 if we allow work bits with known initialstates), the only possibilities being 0, x, y, x� y (and their negations if we allow work bitsknown to be in the 1 state).
1
|xi
|yi |x� yi
|xi0
BB@
1 0 0 00 1 0 00 0 0 10 0 1 0
1
CCA
Cloning in a fixed basis
While we cannot copy quantum information, we can copy classical information. In particular, we can copy quantum states in a fixed basis.
Example: Controlled-not gate
|00i 7! |00i|01i 7! |01i|10i 7! |11i|11i 7! |10i
ASSIGNMENT 1: Solutions CO 481/CS 467/PHYS 467 (Winter 2010)
1. Universality of reversible logic gates.
(a) [3 points] The cccnot (triple-controlled not) gate is a four-bit reversible gate that flipsits fourth bit if and only if the first three bits are all in the state 1. Show how to implementa cccnot gate using To�oli gates. You may use additional workspace as needed. You mayassume that bits in the workspace start with a particular value, either 0 or 1, provided youreturn them to that value. For a bonus point, give a circuit that works regardless of thevalues of any bits of workspace.
Solution: The following circuit shows a simple construction using one bit of workspace inthe 0 state:
•••���⌧⇠⇡⇢�
0
=
• •• •
•���⌧⇠⇡⇢�0 ���⌧⇠⇡⇢� • ���⌧⇠⇡⇢�
The first gate computes the and of the first two bits in the fifth (workspace) bit. Thesecond gate computes the and of the third and fifth bits (i.e., the and of the first threebits) in the fourth (target) bit. The final gate uncomputes the value in the workspace.This circuit can be modified to work for a workspace bit with any value as follows:
•••���⌧⇠⇡⇢� =
• •• •
• •���⌧⇠⇡⇢� ���⌧⇠⇡⇢����⌧⇠⇡⇢� • ���⌧⇠⇡⇢� •(b) [5 points] Show that a To�oli gate cannot be implemented using any number of cnot gates,
with any amount of workspace. Hence the cnot gate alone is not universal. (Hint: It maybe helpful to think of the gates as performing arithmetic operations on integers mod 2.)
Solution: The cnot gate acts as
x • xy ���⌧⇠⇡⇢� x� y
(where � denotes addition modulo 2), so composing any number of cnot gates gives outputbits that can be expressed as a sum modulo 2 of a subset of input bits. However, the To�oligate acts as
x • xy • y
z ���⌧⇠⇡⇢� xy � z
and the product xy (representing logical and) cannot be expressed as a sum modulo 2 ofterms involving x and y (and the constants 0 and 1 if we allow work bits with known initialstates), the only possibilities being 0, x, y, x� y (and their negations if we allow work bitsknown to be in the 1 state).
1
|xi
|yi |x� yi
|xi0
BB@
1 0 0 00 1 0 00 0 0 10 0 1 0
1
CCA
Inputting non-basis states produces an entangled state:
ASSIGNMENT 1: Solutions CO 481/CS 467/PHYS 467 (Winter 2010)
1. Universality of reversible logic gates.
(a) [3 points] The cccnot (triple-controlled not) gate is a four-bit reversible gate that flipsits fourth bit if and only if the first three bits are all in the state 1. Show how to implementa cccnot gate using To�oli gates. You may use additional workspace as needed. You mayassume that bits in the workspace start with a particular value, either 0 or 1, provided youreturn them to that value. For a bonus point, give a circuit that works regardless of thevalues of any bits of workspace.
Solution: The following circuit shows a simple construction using one bit of workspace inthe 0 state:
•••���⌧⇠⇡⇢�
0
=
• •• •
•���⌧⇠⇡⇢�0 ���⌧⇠⇡⇢� • ���⌧⇠⇡⇢�
The first gate computes the and of the first two bits in the fifth (workspace) bit. Thesecond gate computes the and of the third and fifth bits (i.e., the and of the first threebits) in the fourth (target) bit. The final gate uncomputes the value in the workspace.This circuit can be modified to work for a workspace bit with any value as follows:
•••���⌧⇠⇡⇢� =
• •• •
• •���⌧⇠⇡⇢� ���⌧⇠⇡⇢����⌧⇠⇡⇢� • ���⌧⇠⇡⇢� •(b) [5 points] Show that a To�oli gate cannot be implemented using any number of cnot gates,
with any amount of workspace. Hence the cnot gate alone is not universal. (Hint: It maybe helpful to think of the gates as performing arithmetic operations on integers mod 2.)
Solution: The cnot gate acts as
x • xy ���⌧⇠⇡⇢� x� y
(where � denotes addition modulo 2), so composing any number of cnot gates gives outputbits that can be expressed as a sum modulo 2 of a subset of input bits. However, the To�oligate acts as
x • xy • y
z ���⌧⇠⇡⇢� xy � z
and the product xy (representing logical and) cannot be expressed as a sum modulo 2 ofterms involving x and y (and the constants 0 and 1 if we allow work bits with known initialstates), the only possibilities being 0, x, y, x� y (and their negations if we allow work bitsknown to be in the 1 state).
1
|0i
1p2(|0i+ |1i)
Cloning in a fixed basis
While we cannot copy quantum information, we can copy classical information. In particular, we can copy quantum states in a fixed basis.
Example: Controlled-not gate
|00i 7! |00i|01i 7! |01i|10i 7! |11i|11i 7! |10i
ASSIGNMENT 1: Solutions CO 481/CS 467/PHYS 467 (Winter 2010)
1. Universality of reversible logic gates.
(a) [3 points] The cccnot (triple-controlled not) gate is a four-bit reversible gate that flipsits fourth bit if and only if the first three bits are all in the state 1. Show how to implementa cccnot gate using To�oli gates. You may use additional workspace as needed. You mayassume that bits in the workspace start with a particular value, either 0 or 1, provided youreturn them to that value. For a bonus point, give a circuit that works regardless of thevalues of any bits of workspace.
Solution: The following circuit shows a simple construction using one bit of workspace inthe 0 state:
•••���⌧⇠⇡⇢�
0
=
• •• •
•���⌧⇠⇡⇢�0 ���⌧⇠⇡⇢� • ���⌧⇠⇡⇢�
The first gate computes the and of the first two bits in the fifth (workspace) bit. Thesecond gate computes the and of the third and fifth bits (i.e., the and of the first threebits) in the fourth (target) bit. The final gate uncomputes the value in the workspace.This circuit can be modified to work for a workspace bit with any value as follows:
•••���⌧⇠⇡⇢� =
• •• •
• •���⌧⇠⇡⇢� ���⌧⇠⇡⇢����⌧⇠⇡⇢� • ���⌧⇠⇡⇢� •(b) [5 points] Show that a To�oli gate cannot be implemented using any number of cnot gates,
with any amount of workspace. Hence the cnot gate alone is not universal. (Hint: It maybe helpful to think of the gates as performing arithmetic operations on integers mod 2.)
Solution: The cnot gate acts as
x • xy ���⌧⇠⇡⇢� x� y
(where � denotes addition modulo 2), so composing any number of cnot gates gives outputbits that can be expressed as a sum modulo 2 of a subset of input bits. However, the To�oligate acts as
x • xy • y
z ���⌧⇠⇡⇢� xy � z
and the product xy (representing logical and) cannot be expressed as a sum modulo 2 ofterms involving x and y (and the constants 0 and 1 if we allow work bits with known initialstates), the only possibilities being 0, x, y, x� y (and their negations if we allow work bitsknown to be in the 1 state).
1
|xi
|yi |x� yi
|xi0
BB@
1 0 0 00 1 0 00 0 0 10 0 1 0
1
CCA
Inputting non-basis states produces an entangled state:
ASSIGNMENT 1: Solutions CO 481/CS 467/PHYS 467 (Winter 2010)
1. Universality of reversible logic gates.
(a) [3 points] The cccnot (triple-controlled not) gate is a four-bit reversible gate that flipsits fourth bit if and only if the first three bits are all in the state 1. Show how to implementa cccnot gate using To�oli gates. You may use additional workspace as needed. You mayassume that bits in the workspace start with a particular value, either 0 or 1, provided youreturn them to that value. For a bonus point, give a circuit that works regardless of thevalues of any bits of workspace.
Solution: The following circuit shows a simple construction using one bit of workspace inthe 0 state:
•••���⌧⇠⇡⇢�
0
=
• •• •
•���⌧⇠⇡⇢�0 ���⌧⇠⇡⇢� • ���⌧⇠⇡⇢�
The first gate computes the and of the first two bits in the fifth (workspace) bit. Thesecond gate computes the and of the third and fifth bits (i.e., the and of the first threebits) in the fourth (target) bit. The final gate uncomputes the value in the workspace.This circuit can be modified to work for a workspace bit with any value as follows:
•••���⌧⇠⇡⇢� =
• •• •
• •���⌧⇠⇡⇢� ���⌧⇠⇡⇢����⌧⇠⇡⇢� • ���⌧⇠⇡⇢� •(b) [5 points] Show that a To�oli gate cannot be implemented using any number of cnot gates,
with any amount of workspace. Hence the cnot gate alone is not universal. (Hint: It maybe helpful to think of the gates as performing arithmetic operations on integers mod 2.)
Solution: The cnot gate acts as
x • xy ���⌧⇠⇡⇢� x� y
(where � denotes addition modulo 2), so composing any number of cnot gates gives outputbits that can be expressed as a sum modulo 2 of a subset of input bits. However, the To�oligate acts as
x • xy • y
z ���⌧⇠⇡⇢� xy � z
and the product xy (representing logical and) cannot be expressed as a sum modulo 2 ofterms involving x and y (and the constants 0 and 1 if we allow work bits with known initialstates), the only possibilities being 0, x, y, x� y (and their negations if we allow work bitsknown to be in the 1 state).
1
|0i
1p2(|0i+ |1i)
1p2(|00i+ |11i)
Exercise: Distinguishing non-orthogonal states
No device can clone two non-orthogonal states, and in particular, it is not possible to perfectly distinguish such states. But if we want to distinguish them, how well can we do?
Suppose Alice prepares the state or , each with probability .
|0i |+i = 1p2(|0i+ |1i)
12
a. If you measure in the basis , with what probability can you correctly guess which state Alice prepared?
b. What if you measure in the basis , where ?
c. Can you think of another measurement that distinguishes the states with higher probability? (Hint: Consider the given states as polarizations of light. How would you orient a polarizer to get the most information about which polarization was prepared?)
{|+i, |�i}
{|0i, |1i}
|�i = 1p2(|0i � |1i)
Mach-Zehnder interferometer
A simple experiment
laser
detectors
50/50 beamsplitter (half-silvered mirror)
A simple experiment
laser 50%
50%
detectors
50/50 beamsplitter (half-silvered mirror)
Interferometer
laser
Interferometer
laser
Interferometer
laser
Interferometer
laser
Interferometer
laser
Interferometer
laser
50%?
50%?
Interferometer
0%
100%
laser
Mathematical model
laser
Mathematical model
laser
|0i =✓10
◆
Mathematical model
laser
|1i =✓01
◆
|0i =✓10
◆
Mathematical model
laser
|1i =✓01
◆
|0i =✓10
◆
H =1p2
✓1 11 �1
◆
Mathematical model
laser
|1i =✓01
◆
|0i =✓10
◆
|0iproject onto
H =1p2
✓1 11 �1
◆
Mathematical model
laser
|1i =✓01
◆
|0i =✓10
◆
project onto |1i
|0iproject onto
H =1p2
✓1 11 �1
◆
Calculation
|0i
|1i
Calculation
|0i
|1i
Initial state:
Calculation
|0i
|1i
Initial state: |0i =✓10
◆
Calculation
|0i
|1i
Initial state: |0i =✓10
◆
After first beamsplitter:
Calculation
|0i
|1i
Initial state: |0i =✓10
◆
After first beamsplitter:
H|0i = 1p2
✓1 11 �1
◆✓10
◆
Calculation
|0i
|1i
Initial state: |0i =✓10
◆
After first beamsplitter:
H|0i = 1p2
✓1 11 �1
◆✓10
◆
=1p2
✓11
◆
Calculation
|0i
|1i
Initial state: |0i =✓10
◆
After first beamsplitter:
H|0i = 1p2
✓1 11 �1
◆✓10
◆
=1p2
✓11
◆
After second beamsplitter:
Calculation
|0i
|1i
Initial state: |0i =✓10
◆
After first beamsplitter:
H|0i = 1p2
✓1 11 �1
◆✓10
◆
=1p2
✓11
◆
After second beamsplitter:
H1p2
✓11
◆=
1
2
✓1 11 �1
◆✓11
◆
Calculation
|0i
|1i
Initial state: |0i =✓10
◆
After first beamsplitter:
H|0i = 1p2
✓1 11 �1
◆✓10
◆
=1p2
✓11
◆
After second beamsplitter:
H1p2
✓11
◆=
1
2
✓1 11 �1
◆✓11
◆
=
✓10
◆
Calculation
|0i
|1i
Initial state: |0i =✓10
◆
After first beamsplitter:
H|0i = 1p2
✓1 11 �1
◆✓10
◆
=1p2
✓11
◆
After second beamsplitter:
H1p2
✓11
◆=
1
2
✓1 11 �1
◆✓11
◆
=
✓10
◆
Probability of measuring :|0i
Calculation
|0i
|1i
Initial state: |0i =✓10
◆
After first beamsplitter:
H|0i = 1p2
✓1 11 �1
◆✓10
◆
=1p2
✓11
◆
After second beamsplitter:
H1p2
✓11
◆=
1
2
✓1 11 �1
◆✓11
◆
=
✓10
◆
Probability of measuring :|0i|h0|0i|2 = 1
Calculation
|0i
|1i
Initial state: |0i =✓10
◆
After first beamsplitter:
H|0i = 1p2
✓1 11 �1
◆✓10
◆
=1p2
✓11
◆
After second beamsplitter:
H1p2
✓11
◆=
1
2
✓1 11 �1
◆✓11
◆
=
✓10
◆
Probability of measuring :|0i|h0|0i|2 = 1
Probability of measuring :|1i
Calculation
|0i
|1i
Initial state: |0i =✓10
◆
After first beamsplitter:
H|0i = 1p2
✓1 11 �1
◆✓10
◆
=1p2
✓11
◆
After second beamsplitter:
H1p2
✓11
◆=
1
2
✓1 11 �1
◆✓11
◆
=
✓10
◆
Probability of measuring :|0i|h0|0i|2 = 1
Probability of measuring :|1i|h1|0i|2 = 0
Phase shifter
Another simple optical element is a phase shifter, which shifts the phase of the light passing through it by some amount.
Phase shifter
Another simple optical element is a phase shifter, which shifts the phase of the light passing through it by some amount.
φ
Phase shifter
Another simple optical element is a phase shifter, which shifts the phase of the light passing through it by some amount.
φ multiplies this portion of the state by ei'
Phase shifter
Another simple optical element is a phase shifter, which shifts the phase of the light passing through it by some amount.
φ multiplies this portion of the state by ei'
φ1
φ0 |0i
|1i
Phase shifter
Another simple optical element is a phase shifter, which shifts the phase of the light passing through it by some amount.
φ multiplies this portion of the state by ei'
φ1
φ0 |0i
|1i
✓ei'0 00 ei'1
◆implements the unitary transformation
Exercise: Interferometry with phase shifts
laser
φ1
φ0
?
?
Exercise: Interferometry with phase shifts
laser
φ1
φ0
sin2 '1�'0
2
cos
2 '1�'0
2
Deutsch’s problemGiven: A function f: {0,1} → {0,1}
Task: Determine whether f is constant.
(As a black box: You can call the function f, but you can’t read its source code.)
Deutsch’s problem
Four possible functions:
x f1(x)
0 0
1 0
x f2(x)
0 1
1 1
x f3(x)
0 0
1 1
x f4(x)
0 1
1 0
Given: A function f: {0,1} → {0,1}
Task: Determine whether f is constant.
(As a black box: You can call the function f, but you can’t read its source code.)
Deutsch’s problem
Four possible functions:
x f1(x)
0 0
1 0
x f2(x)
0 1
1 1
x f3(x)
0 0
1 1
x f4(x)
0 1
1 0constant not constant
Given: A function f: {0,1} → {0,1}
Task: Determine whether f is constant.
(As a black box: You can call the function f, but you can’t read its source code.)
Deutsch’s problem
Four possible functions:
x f1(x)
0 0
1 0
x f2(x)
0 1
1 1
x f3(x)
0 0
1 1
x f4(x)
0 1
1 0constant not constant
Classically, two function calls are required to solve this problem.
Given: A function f: {0,1} → {0,1}
Task: Determine whether f is constant.
(As a black box: You can call the function f, but you can’t read its source code.)
Deutsch’s algorithm as interferometry
laser
Deutsch’s algorithm as interferometry
laser
π f(1)
π f(0)
Deutsch’s algorithm as interferometry
laser
π f(1)
π f(0)
f is not constant
f is constant
Exercise: More linear optics
What unitary transformation is implemented by the following optical setup?
π/2
π/2
|0i
|1i