My notes about the basics of quantum information - part 1

Last updated: 2/2/2024

This lesson on the basics of quantum information explains the concept of applying deterministic operations to probabilistic state of a system. The  key idea is that when you apply a deterministic operation on a probabilistic state, it transforms into different probabilistic state within the same set. This interaction can be described using matrix-vector multiplication. To solidify my understanding and refresh my memory on matrices, I decided to explore all possible functions acting on the set {0, 1}.  Calculated the matrices, ensuring I understand every nuance of the lesson and could confidently continue with the course. Bellow are my notes. 

There is a set  Σ with all possible states for a system: 0 and 1. Let's say the system is a lamp that can be turned OFF, representing state 0, and turned ON, representing state 1.

Σ = {0,1} 

A deterministic operation can be applied to this system. This means that depending on the lamp's initial state and the specific operation performed, we will always get the same outcome. For example, if the lamp is on we flip the switch (let's call that function f), the lamp will turn off. If the lamp is off flipping the switch will turn it on.  f changes the lamp from one possible state (a ∈ Σ) to another (b ∈ Σ ), written as  f: Σ -> Σ.

The lesson's example explores four basic functions for the set {0,1} : two constant functions, an identity function and the so called bit-flipping function.

Each function can be represented by a unique matrix calculated based on specific rules:

• Matrix elements: 0 or 1 (false or true).
• Element value: Determined by comparing the column index to the function's result applied to the row index.

Or written as: 

How those four functions are applied  on a : 

Those are the generated matrices based on the rule: 

Here are the calculations demonstrating that each function's matrix, when multiplied by vector representing the initial state, results in a column probabilistic vector representing the outcome of the function. 

Part 2: 

Some extra calculation with the bra and ket forms of the vectors. 

Bra and ket are special vectors.

Bra is a row vector, while ket is the column vector. Both have single element equal to 1 ant 0 on the rest of the positions.

Multiplying ket and bra gives a matrix that has single element 1 and the rest are zero. 

The next section dives into the more formal mathematical approach to calculate the matrix, as demonstrated in the lesson. Essentially, a unique matrix maps the transformation from one classical sate of a system to another. While the first three screenshots above follow a logical way to construct the matrix, that was used in the beginning of the course, a more efficient method involves summing the outer products of the ket vector representing the final state and the bra vector representing the initial state. 

Marked with * is the matrix constructed following the rule from the beginning of the post:

Marked with # is the matrix of the sums of the outer product:

Probabilistic operations - introduce randomness:

Stochastic matrices : 

All elements are real nonnegative numbers. Every column is a probability vector - the sum of all elements in a given column is 1.

Composing operations: