A few things quantum

01: Spin

  • Magnet spinning around an axis
  • Rotation of the entire system:
    Clockwise to counter-clockwise

01: Spin

01: Spin

Electromagnetism: spinning magnets in an inhomogeneous magnetic field deflect,

depending on (counter)clockwise rotation

Stern-Gerlach apparatus

Quantum case:

No spectrum anymore! Either you are spin up or spin down, but nothing in between.

01: Spin

I'm cutting some corners here, but this is our first meeting with a unique quantum phenomenon called collapse

01: Spin

M

M

01: Spin

02: Cbits and Qbits

02: Cbits and Qbits

Now, we need to focus and distinguish 3 things:

1. Bits
2. Cbits
3. Qbits

1. Bits
2. Cbits
3. Qbits

Information

Logic

Mathematics

Encoding a message

Shannon

02: Cbits and Qbits

1. Bits
2. Cbits
3. Qbits

02: Cbits and Qbits

1

0

1. Bits
2. Cbits
3. Qbits

Light on/off

Transistor

Electronics

Physical System

Turing

02: Cbits and Qbits

1. Bits
2. Cbits
3. Qbits

02: Cbits and Qbits

1. Bits
2. Cbits
3. Qbits

Spin up/down

Quantum-physical System

Feynmann/Deutsch/Schor

02: Cbits and Qbits

1. Bits
2. Cbits
3. Qbits

02: Cbits and Qbits

02: Cbits and Qbits

Cbit

Qbit

| 0 \rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}
| 1 \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}
| \hat{\psi} \rangle = \begin{pmatrix} \psi_0 \\ \psi_1 \end{pmatrix}
\psi_0, \psi_1 \in \mathbb{C}, |\psi_0|^2 + |\psi_1|^2 = 1
| \kappa \rangle = \begin{pmatrix} \kappa_0 \\ \kappa_1 \end{pmatrix}
\kappa_0, \kappa_1 \in \{0,1\}

02: Cbits and Qbits

Qbit

| \hat{\psi} \rangle = \begin{pmatrix} \psi_0 \\ \psi_1 \end{pmatrix}
\psi_0, \psi_1 \in \mathbb{C}, |\psi_0|^2 + |\psi_1|^2 = 1

Again cutting some corners, but this is what people call superposition.

03: Quantum entanglement

03: Quantum entanglement

``God does not play dice''

- Albert E.

``A moving clock is slower'' (SR)

- Albert E.

``oh also, spacetime is a 4D manifold'' (GR)

- Albert E.

03: Quantum entanglement

``God does not play dice''

- Albert E.

Things, are real

Things are, in spacetime

Realism

Localism

03: Quantum entanglement

``God does not play dice''

- Albert E.

Realism

Localism

Turns out in QM you have to give up one!

(More than dice throwing, this was Einstein's actual problem with QM)

03: Quantum entanglement

| \hat{\psi_1} \rangle = \alpha |0 \rangle + \beta |1 \rangle

Remember, a single qubit looks like:

= \begin{pmatrix} \alpha \\ \beta \end{pmatrix}

03: Quantum entanglement

| \hat{\psi_1} \rangle = \alpha |0 \rangle + \beta |1 \rangle \\ = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}
| \hat{\psi_2} \rangle = \gamma |0 \rangle + \delta |1 \rangle \\ = \begin{pmatrix} \gamma \\ \delta \end{pmatrix}

03: Quantum entanglement

=
= \alpha \gamma | 00 \rangle + \alpha \delta | 01 \rangle + \beta \gamma | 10 \rangle + \beta \delta | 11 \rangle
(\alpha |0\rangle + \beta | 1 \rangle)
| \hat{\psi} \rangle =
(\gamma |0\rangle + \delta | 1 \rangle)
\otimes
= \begin{pmatrix} \alpha \gamma \\ \alpha \delta \\ \beta \gamma \\ \beta \delta\\ \end{pmatrix}

Composing systems goes via the tensor product

03: Quantum entanglement

\neq
| \hat{\psi} \rangle =
\begin{pmatrix} \psi_{00} \\ 0 \\ 0 \\ \psi_{11} \\ \end{pmatrix} = \begin{pmatrix} \alpha \gamma \\ \alpha \delta \\ \beta \gamma \\ \beta \delta\\ \end{pmatrix} \Rightarrow

Composing systems goes via the tensor product

?

\alpha \delta \beta \gamma = 0
\alpha \gamma \beta \delta = 0

03: Quantum entanglement

= \begin{pmatrix} \psi_{00} \\ 0 \\ 0\\ \psi_{11} \\ \end{pmatrix}
\neq
=

``Bell State''

= \psi_{00} | 00 \rangle + \psi_{11} | 11 \rangle

entanglement

03: Quantum entanglement

| \hat{\psi} \rangle = \begin{pmatrix} \psi_{00} \\ \psi_{01} \\ \psi_{10} \\ \psi_{11} \\ \end{pmatrix} = \begin{pmatrix} \psi_{00} \\ 0 \\ 0\\ \psi_{11} \\ \end{pmatrix}

04: Quantum communication

04: Quantum communication

Now, we're experts in superposition, collapse and entanglement

...but so what?

04: Quantum communication

\psi

Z-measurement

Q-channel

Q-state

C-channel

04: Quantum communication

\psi

collapse

superposition

04: Quantum communication

\psi
\psi

Z-measurement

X-measurement

04: Quantum communication

04: Quantum communication

=

Since you collapse on Z,

your state loses ALL information regarding X.

 

This is an example of complementarity

04: Quantum communication

04: Quantum communication

04: Quantum communication

BB84

Alice wants to send random bits to Bob,

over a Q-channel.

04: Quantum communication

BB84

Alice wants to send random bits to Bob,

over a Q-channel.

04: Quantum communication

BB84

Alice wants to send random bits to Bob,

over a Q-channel.

 

 

Evil Donald wants to eavesdrop!

 

 

04: Quantum communication

BB84

Alice wants to send random bits to Bob,

over a Q-channel.

 

 

Evil Donald wants to eavesdrop!

 

 

05: Quantum computation

05: Quantum Computation

Can all this quantum magic give us computing magic?

Yes! Do 2 things:

 

1. Exploit superposition

2. Fight collapse

 

05: Quantum Computation

05: Quantum Computation

05: Quantum Computation

05: Quantum Computation

05: Quantum Computation

Uniform superposition

05: Quantum Computation