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: a photon is like a glider in game of life. There isn't such thing called a photon in nature. A photon is just a pattern occur in phenomenon.

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 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.

Property of Electron

Spin:

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)

Notation on basis vector:

|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 1s.

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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