1. Postulates: State Space

1. Postulates: State Space

We have finished the pre-requisites (Math and Logic). Now we enter Physics.

Quantum Mechanics is built on four "Postulates"—axioms that cannot be proven, only verified by experiment. If you accept them, everything else follows.

Postulate 1: The State Space

The Rule: Associated to any isolated physical system is a complex vector space with an inner product (a Hilbert Space) known as the state space of the system. The system is completely described by its state vector ($|\psi\rangle$), which is a unit vector in the system's state space.

1. The Requirement: Normalization

The most critical phrase above is "unit vector." Because the coefficients of the vector represent probability amplitudes, and total probability must equal 100% (1.0), every valid quantum state must satisfy the Normalization Condition:

$$\langle \psi | \psi \rangle = 1$$

If $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, then $|\alpha|^2 + |\beta|^2 = 1$.

If you have a vector where the squared magnitudes do not sum to 1, it is not a valid quantum state. It is just a mathematical abstraction until you normalize it.

2. The Implication: The Exponential Explosion

The state space for a single qubit is $\mathbb{C}^2$ (2 complex dimensions). The state space for $N$ qubits is the tensor product of the individual spaces: $\mathbb{C}^{2^N}$.

• 1 Qubit: 2 complex numbers.
• 10 Qubits: $2^{10} = 1,024$ complex numbers.
• 50 Qubits: $2^{50} \approx 1.1 \times 10^{15}$ (Petabytes of RAM to simulate).
• 300 Qubits: $2^{300}$ (More numbers than atoms in the visible universe).

This is why we build quantum computers. To simulate a 300-qubit system on a classical computer is physically impossible. Nature manages this information effortlessly.

Exponential growth: the state space dimension scales as $2^N$.

Your Task: The "Illegal" State

You are given the following vector representing a potential system state:

$$|\phi\rangle = 3|0\rangle + 4i|1\rangle$$

1. Check Validity: Calculate the squared magnitude of the vector (the sum of the squared moduli of the coefficients). Does it equal 1?
2. Fix It: Normalize the vector. Find a constant $N$ such that if you multiply $|\phi\rangle$ by $\frac{1}{N}$, the resulting vector is a valid unit vector. (Hint: $N = \sqrt{\langle \phi | \phi \rangle}$).

Warning: Don't forget that $|4i|^2$ is $16$, not $-16$. Modulus is always positive.