# Lecture 010 - Linear Algebra

Some Pondering:

Quantum Aliasing Conjecture: The world is continuous. The reason why we observe discrete is due to sampling a continuous signal in a discrete way. It explains how rotational symmetry is compatible with Plank length.

Ryan's Game of Life Conjecture: There isn't such thing called a photon in nature: a photon is indeed traveling oscillating electromagnetic wave, much like a glider in Conway's Game of Life where each pixel update according to some update rule. A photon is just a pattern occur in phenomenon that we human identify such pattern and called it light.

In quantum, we have 4 representation

• box representation: for at most 3 qubits, each vertex represent a label

• cartesian representation: for 1 qubit, axis represent $|0\rangle$ or $|1\rangle$ (x-coordinate is the amplitude on $|0\rangle$, y-coordinate is the amplitude on $|1\rangle$), unit vector represent current cubit state. Each dimension is a possible string, so there are $2^n$ dimensions.

• grover's: // TODO

• 2 overlapping cartesian basis: two maximally entangled states, each basis represent one qubit, if one gets measured to $x$ basis vector, the other has to be $x$ basis vector.

## Some Physics

Bits: two state $\{0, 1\}$

Trits: three state $\{0, 1, 2\}$

Dits: four states $\{0, 1, 2, 3\}$

Law of Physics: if any object can be in $d$ basic states $\{0, 1, ..., d-1\}$, it can also be in a superposition state $\alpha_0 |\square\rangle + \alpha_1 |\square\rangle + ... + \alpha_{d-1} |\square\rangle$ where $|\alpha_0|^2 + |\alpha_1|^2 + ... + |\alpha_k|^2 = 1$

### Property of Photon

Polarity: any quantum state can be represented as polarity

• "Horizontal" ($|0\rangle$ or $|H\rangle$)

• "Vertical" ($|1\rangle$ or $|V\rangle$)

• any state correspond to a polarity

• you can build up a joined states of 2 qubits: $\alpha_0|HL\rangle + \alpha_1|HR\rangle + \alpha_2|VL\rangle + \alpha_3|VR\rangle$

Horizontal Polarizing Filter:

def HPF(A):
b = measure(A)  # print A
if b == "Vertical":
generate_heat()
return None
else: # "Horizontal"
make_into_deterministic_zero_state(A)
// QUESTION: instead of function, it should guarantee
return A
def VPF(A):
b = measure(A)  # print A
if b == "Horizontal":
generate_heat()
return None
else: # "Vertical"
make_into_deterministic_zero_state(A)
// QUESTION: instead of function, it should guarantee
return A


You can make a 3D glasses with a horizontal polarizing filter and a vertical polarizing filter, where you shine a screen displaying two different images, one with completely horizontal polarized image, the other with completely vertical polarized image.

Importantly, there exists no strict basis for our 3D world, meaning you can rotate a horizontal polarizing filter to a vertical polarizing filter.

// QUESTION: what happen when you stack 3 polarizing filters? // QUESTION: why does it generate heat? cuz a portion of photon disappeared? you are just filtering, not transforming right?

Position: position $(x, y, z)$ distributed in amplitude space

Reflection on objects is due to non-deterministic superposition of lights passing through glass.

Spin:

• "Down"

• "Up"

## Some Linear Algebra

We view quantum state as unit vector because their magnitude is always $1$.

Discussion on Notations: introducing Dirac Bra-Ket Notation for Linear Algebra (ket)

• $\textbf{v}$: bolded $v$, nearly invisible

• $\vec{v}$: arrowed $v$, good, but hard to get the meaning

• $\vec{\text{foo}}$: good, but you don't know whether its column or row

• $|v\rangle$: "ket" tells you exactly it is column vector named $v$

Notation on basis vector:

• 3 dimensional: $\vec{i}, \vec{j}, \vec{k}, \vec{l}$

• $n$ dimensional: $\vec{e_0}, \vec{e_1}, ..., \vec{e_{d-1}}$

• $n$ dimensional: $|0\rangle, |1\rangle, ..., |d-1\rangle$

|0\rangle = \begin{bmatrix} 1\\ 0\\ 0\\ \dots\\ 0\\ \end{bmatrix}, |1\rangle = \begin{bmatrix} 0\\ 1\\ 0\\ \dots\\ 0\\ \end{bmatrix}, |2\rangle = \begin{bmatrix} 0\\ 0\\ 1\\ \dots\\ 0\\ \end{bmatrix}, \dots, |d-1\rangle = \begin{bmatrix} 0\\ 0\\ 0\\ \dots\\ 1\\ \end{bmatrix}
|00\rangle = \begin{bmatrix} 1\\ 0\\ 0\\ 0\\ \end{bmatrix}, |01\rangle = \begin{bmatrix} 0\\ 1\\ 0\\ 0\\ \end{bmatrix}, |10\rangle = \begin{bmatrix} 0\\ 0\\ 1\\ 0\\ \end{bmatrix}, \dots, |11\rangle = \begin{bmatrix} 0\\ 0\\ 0\\ 1\\ \end{bmatrix}

Notice all above vector describe a "deterministic" state with respect to standard basis. So you can think $|v\rangle$ as deterministic when $v$ are $0$ and $1$s.

Dealing with sparse data structure is also easy:

\alpha_0 |\square\rangle + \alpha_1 |\square\rangle + ... + \alpha_{d-1} |\square\rangle

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