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Page 1: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Introduction to Quantum Mechanics

QCSYS 2012

Page 2: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Outline

1. Polarization

2. Double-slit experiment

3. Photoelectric effect

4. No-cloning theorem

Page 3: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Polarization

Page 4: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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 ↵,�

Page 5: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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 ↵,�

Page 6: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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 ↵,�

Page 7: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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 ↵,�

Page 8: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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

Page 9: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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

Page 10: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Traveling wavesThe Maxwell equations have solutions that correspond to waves propagating through space.

Page 11: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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

Page 12: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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

Page 13: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Superposition of polarization states

Horizontal polarization: |!i =✓10

◆Vertical polarization: |"i =

✓01

Page 14: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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

Page 15: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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

Page 16: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Polarizing filters

Most sources of light are unpolarized.

Page 17: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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.

Page 18: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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.

Page 19: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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.

Page 20: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Polarizing filter examples

Incident polarization: |%i

How much light passes?

Polarizing filter orientation: !

Page 21: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Polarizing filter examples

Incident polarization: |%i

How much light passes? 50%

Polarizing filter orientation: !

Page 22: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Polarizing filter examples

Incident polarization: |%i

How much light passes? 50%

Outgoing polarization?

Polarizing filter orientation: !

Page 23: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Polarizing filter examples

Incident polarization: |%i

How much light passes? 50%

Outgoing polarization? |!i

Polarizing filter orientation: !

Page 24: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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 ✓

Page 25: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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 ✓

Page 26: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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 ✓

Page 27: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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 ✓

Page 28: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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

Page 29: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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

Page 30: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Another polarizing filter example

Incident polarization:

Polarizing filter orientation:

|!i ✓cos ✓sin ✓

Page 31: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Another polarizing filter example

Incident polarization:

Polarizing filter orientation:

|!i ✓cos ✓sin ✓

|!i =✓1

0

= cos ✓

✓cos ✓sin ✓

◆� sin ✓

✓� sin ✓cos ✓

◆+

Page 32: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Another polarizing filter example

Incident polarization:

Polarizing filter orientation:

|!i ✓cos ✓sin ✓

|!i =✓1

0

= cos ✓

✓cos ✓sin ✓

◆� sin ✓

✓� sin ✓cos ✓

Page 33: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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 ✓

Page 34: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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 ✓

Page 35: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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 ✓

Page 36: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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

Page 37: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Circular polarization

We can also consider superpositions involving complex numbers.

Page 38: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Circular polarization

We can also consider superpositions involving complex numbers.

| i = 1p2(|!i � i|"i)|�i = 1p

2(|!i+ i|"i)

Examples:

Page 39: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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:

Page 40: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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

Page 41: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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

Page 42: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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

Page 43: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Double-slit experiment

Page 44: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Firing bullets at a slit

Page 45: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Firing bullets at a slit

Page 46: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Firing bullets at a slit

Page 47: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Firing bullets at a slit

Page 48: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Firing bullets at a slit

Page 49: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Firing bullets at a double slit

Page 50: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Firing bullets at a double slit

Page 51: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Firing bullets at a double slit

Page 52: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Firing bullets at a double slit

Page 53: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Firing bullets at a double slit

Page 54: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Double slit with waves

Page 55: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Double slit with waves

Page 56: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Double slit with waves

Page 57: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Double slit with waves

Page 58: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Double slit with waves

Page 59: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Double slit with waves

Page 60: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Double slit with waves

Page 61: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Demo

Page 62: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Interference

Page 63: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Interference

wavelength ¸

Page 64: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Interference

wavelength ¸

amplitude

Amplitude can be positive (water is above sea level)or negative (water is below sea level)

Page 65: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Interference

wavelength ¸

amplitude

Amplitude can be positive (water is above sea level)or negative (water is below sea level)

+

Page 66: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Interference

wavelength ¸

amplitude

Amplitude can be positive (water is above sea level)or negative (water is below sea level)

+ =

Page 67: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Interference

wavelength ¸

amplitude

Amplitude can be positive (water is above sea level)or negative (water is below sea level)

+

+

=

Page 68: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Interference

wavelength ¸

amplitude

Amplitude can be positive (water is above sea level)or negative (water is below sea level)

+

+

=

=

Page 69: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Form of the interference pattern

Page 70: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Form of the interference pattern

¢

Page 71: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Form of the interference pattern

...

...

µ¢

Page 72: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Form of the interference pattern

...

...

µ¢

Page 73: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Form of the interference pattern

...

...

µ

µ

¢

Page 74: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Form of the interference pattern

...

...

µ

µ

¢

� sin ✓

Page 75: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Form of the interference pattern

...

...

µ

µ

¢

� sin ✓

One fringe: � sin ✓ = �

Page 76: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Form of the interference pattern

...

...

µ

µ

¢

� sin ✓

One fringe: � sin ✓ = � ✓ ⇡ �

Page 77: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Form of the interference pattern

...

...

µ

µ

¢

� sin ✓

One fringe: � sin ✓ = � ✓ ⇡ �

Screen at a distance d away: fringe spacing is approximately d · ✓ ⇡ d · ��

Page 78: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Double slit with electrons

How will the experiment behave if we use electrons instead of bullets or water waves?

Page 79: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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.

Page 80: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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!

Page 81: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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?

Page 82: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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

Page 83: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

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

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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

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Uncertainty principle

In classical mechanics, nothing prevents us from measuring the state of a particle (its position and momentum) with arbitrary precision.

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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.

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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⇡ )

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Uncertainty principle and diffraction

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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?

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Photoelectric effect

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The photoelectric effect

⊝⊝⊝

⊝⊝

⊝⊝⊝

⊝⊝

⊝⊝⊝

⊝⊝

⊝⊝⊝

⊝⊝

⊝⊝⊝

⊝⊝

⊝⊝⊝

⊝⊝

⊝⊝⊝

⊝⊝

⊝⊝⊝

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The photoelectric effect

⊝⊝⊝

⊝⊝

⊝⊝⊝

⊝⊝

⊝⊝⊝

⊝⊝

⊝⊝⊝

⊝⊝⊝

⊝⊝

⊝⊝⊝

⊝⊝

⊝⊝⊝

⊝⊝

⊝⊝⊝

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Photons and electrons

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Photons and electrons

Light is made up of massless particles called photons.

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Photons and electrons

Light is made up of massless particles called photons.

Photons are characterized by their wavelength �or equivalently, their frequency .⌫

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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

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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.

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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

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The work function

Removing an electron from a material costs energy.

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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

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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

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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

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Photoelectric effect experiment

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Photoelectric effect experiment

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Photoelectric effect experiment

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Photoelectric effect experiment

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Photoelectric effect experiment

h⌫ =1

2mv2 + �

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Photoelectric effect experiment

h⌫ =1

2mv2 + �

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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

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Photoelectric effect experiment: results

V0

Page 111: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Photoelectric effect experiment: results

V0

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Photoelectric effect experiment: results

V0

h⌫ =1

2mv2 + �

1

2mv2 = eV0

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Photoelectric effect experiment: results

V0

h⌫ =1

2mv2 + �

1

2mv2 = eV0

V0 =h⌫ � �

e

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Photoelectric effect experiment: results

V0

h⌫ =1

2mv2 + �

1

2mv2 = eV0

V0 =h⌫ � �

e

�/h

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Photoelectric effect experiment: results

V0

h⌫ =1

2mv2 + �

1

2mv2 = eV0

V0 =h⌫ � �

e

�/h

h/e

slope

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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?

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Experiment(end of the week)

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No-cloning theorem

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Classical cloning

cloning machine

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Classical cloning

cloning machine

Page 121: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Classical cloning

cloning machine

Page 122: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Classical cloning

cloning machine

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Classical cloning

In principle, such a device is possible.

cloning machine

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Classical cloning (digital)

cloning machine

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Classical cloning (digital)

cloning machine

file1.txt

Lorem ipsum dolor sit amet, consectetur adipisicing elit...

file2.txt

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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...

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Quantum cloning?

cloning machine

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Quantum cloning?

cloning machine

Page 129: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Quantum cloning?

cloning machine

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Quantum cloning?

cloning machine

The uncertainty principle prevents us from learning an unknown quantum state.

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Quantum cloning?

cloning machine

The uncertainty principle prevents us from learning an unknown quantum state.

Such a device is impossible!

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Quantum cloning?

cloning machine

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Quantum cloning?

cloning machine

|0i

↵|0i+ �|1i

Page 134: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Quantum cloning?

cloning machine

|0i

↵|0i+ �|1i ↵|0i+ �|1i

↵|0i+ �|1i

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Quantum cloning?

cloning machine

|0i

↵|0i+ �|1i ↵|0i+ �|1i

↵|0i+ �|1i

This is also impossible.

Page 136: Introduction to Quantum Mechanics - University Of …amchilds/talks/qcsys12.pdf · Superposition A basic feature of quantum mechanics is the principle of superposition: If a quantum

Quantum cloning?

Even digital quantum information (qubits) cannot be cloned.

cloning machine

|0i

↵|0i+ �|1i ↵|0i+ �|1i

↵|0i+ �|1i

This is also impossible.

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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

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Proof of the no-cloning theorem

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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

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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

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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

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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

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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]

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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)

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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

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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 = �

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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

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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.

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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

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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)

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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)

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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)

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Mach-Zehnder interferometer

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A simple experiment

laser

detectors

50/50 beamsplitter (half-silvered mirror)

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A simple experiment

laser 50%

50%

detectors

50/50 beamsplitter (half-silvered mirror)

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Interferometer

laser

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Interferometer

laser

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Interferometer

laser

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Interferometer

laser

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Interferometer

laser

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Interferometer

laser

50%?

50%?

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Interferometer

0%

100%

laser

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Mathematical model

laser

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Mathematical model

laser

|0i =✓10

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Mathematical model

laser

|1i =✓01

|0i =✓10

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Mathematical model

laser

|1i =✓01

|0i =✓10

H =1p2

✓1 11 �1

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Mathematical model

laser

|1i =✓01

|0i =✓10

|0iproject onto

H =1p2

✓1 11 �1

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Mathematical model

laser

|1i =✓01

|0i =✓10

project onto |1i

|0iproject onto

H =1p2

✓1 11 �1

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Calculation

|0i

|1i

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Calculation

|0i

|1i

Initial state:

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Calculation

|0i

|1i

Initial state: |0i =✓10

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Calculation

|0i

|1i

Initial state: |0i =✓10

After first beamsplitter:

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Calculation

|0i

|1i

Initial state: |0i =✓10

After first beamsplitter:

H|0i = 1p2

✓1 11 �1

◆✓10

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Calculation

|0i

|1i

Initial state: |0i =✓10

After first beamsplitter:

H|0i = 1p2

✓1 11 �1

◆✓10

=1p2

✓11

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Calculation

|0i

|1i

Initial state: |0i =✓10

After first beamsplitter:

H|0i = 1p2

✓1 11 �1

◆✓10

=1p2

✓11

After second beamsplitter:

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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

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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

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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

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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

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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

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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

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Phase shifter

Another simple optical element is a phase shifter, which shifts the phase of the light passing through it by some amount.

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Phase shifter

Another simple optical element is a phase shifter, which shifts the phase of the light passing through it by some amount.

φ

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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'

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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

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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

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Exercise: Interferometry with phase shifts

laser

φ1

φ0

?

?

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Exercise: Interferometry with phase shifts

laser

φ1

φ0

sin2 '1�'0

2

cos

2 '1�'0

2

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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.)

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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.)

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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.)

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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.)

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Deutsch’s algorithm as interferometry

laser

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Deutsch’s algorithm as interferometry

laser

π f(1)

π f(0)

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Deutsch’s algorithm as interferometry

laser

π f(1)

π f(0)

f is not constant

f is constant

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Exercise: More linear optics

What unitary transformation is implemented by the following optical setup?

π/2

π/2

|0i

|1i