Quantum States & State Vectors

Overview

To talk about quantum computing correctly, you need a clean definition of a quantum state. For a single qubit, the state is a state vector—a mathematical object that predicts the probabilities of different measurement outcomes. This page introduces the general single-qubit state, explains the normalization rule, and clarifies what amplitudes mean.

Intuition

A classical bit is fully described by a value: 0 or 1. A qubit is described by something closer to a direction with extra structure. You cannot “peek” at that direction directly; instead, you can only ask questions (measurements) and get probabilistic answers.

The purpose of the state vector is therefore practical: it is a compact way to store everything needed to predict measurement statistics.

Formal Description

Fix the computational basis $\{|0\rangle,|1\rangle\}$. A general single-qubit pure state is written

$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle,$$

where $\alpha$ and $\beta$ are complex numbers.

Normalization. The state must be normalized so that the total probability of all outcomes is 1:

$$|\alpha|^2 + |\beta|^2 = 1.$$

In words: if you measure in the computational basis, you must get either 0 or 1, so the probabilities of those two outcomes must sum to 1.

Meaning of amplitudes. The amplitudes $\alpha$ and $\beta$ are not directly observable numbers you can read out. Their meaning is operational:

$$P(0)=|\alpha|^2,\quad P(1)=|\beta|^2.$$

The fact that $\alpha$ and $\beta$ can be complex is what allows phase effects and interference later.

Worked Example

Consider

$$|\psi\rangle = \left(\tfrac{1}{2} + \tfrac{i}{2}\right)|0\rangle + \tfrac{1}{\sqrt{2}}|1\rangle.$$

Compute probabilities in the computational basis.

First, $\alpha = \tfrac{1+i}{2}$. Its magnitude squared is

$$|\alpha|^2 = \left|\tfrac{1+i}{2}\right|^2 = \tfrac{1}{4}|1+i|^2 = \tfrac{1}{4}(1^2+1^2) = \tfrac{1}{2}.$$

Second, $\beta = \tfrac{1}{\sqrt{2}}$, so $|\beta|^2 = \tfrac{1}{2}$. The state is normalized and measurement yields 0 and 1 with equal probability.

This example shows a key point: complex amplitudes can still lead to simple probabilities, but the complex structure matters for how states combine and evolve.

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What’s Next

We now have the formal language to describe a qubit. The next page focuses on one of the most misunderstood words in quantum computing—superposition—and explains what it does and does not mean.