# Quantum Information Notes - Part 1

Posted: 2024-01-27

Tags: Qiskit, Quantum, Notes

Notes to go along with the first lesson of IBM’s Basics of Quantum Information course.

## Classical Information

### Classical States

system - an abstraction of a physical device or a medium that stores information classical state - a configuration that can be recognized and described unambiguously

For a system to be useful, the set of classical states \(\Sigma\) should be of finite size and not empty.

### Probability Vectors

\(\Pr(\mathsf{X}=0) = \frac{3}{4}\) is read as “the probability that \(\mathsf{X}\) is in state \(0\) is \(3/4\).”

Any probabilistic state can be represented by a column vector satisfying two properties:

- All entries of the vector are nonnegative real numbers.
- The sum of the entries is equal to 1.

So, if \(\Pr(\mathsf{X}=0) = \frac{3}{4}\) and \(\Pr(\mathsf{X}=1) = \frac{1}{4}\), then this can be represented by: \(\begin{bmatrix} \frac{3}{4} \\ \frac{1}{4} \end{bmatrix}\)

Identifying probabilistic states as column vectors and operations as matrices allows the effect of an operation on probabilistic state to be represented as matrix–vector multiplication.

### Measuring Probabilistic States

By measuring a system, we mean that we look at the system and unambiguously recognize whatever classical state it is in.

Measurement changes our knowledge of the system, and therefore changes the probabilistic state that we associate with that system: if we recognize that \(\mathsf{X}\) is in the classical state \(a\in\Sigma\), then the new probability vector representing our knowledge of \(\mathsf{X}\) becomes a vector having a \(1\) in the entry corresponding to \(a\) and \(0\) for all other entries.

This vector indicates that \(\mathsf{X}\) is in the classical state \(a\) with certainty, which we know having just recognized it.