Quantum States and State Vectors

A pure quantum state is a unit vector in a complex Hilbert space, written as a ket, encoding all predictable information about the system.

States as vectors

A pure quantum state is a normalised vector in a complex vector space. For a single qubit that space is two-dimensional, and the state |psi> = a|0> + b|1> is the column vector (a, b) with |a|^2 + |b|^2 = 1. Everything predictable about the system is contained in this vector.

Dirac notation

The bra-ket notation writes column vectors as kets |psi> and their conjugate-transpose row vectors as bras is a complex number measuring overlap; the outer product |psi>

Normalisation and phase

Two conditions matter. First, states are normalised: = 1, so total probability is one. Second, an overall global phase is physically meaningless: |psi> and e^{i alpha}|psi> describe the same state. Strictly, physical states are rays, not vectors, but the vector representation is the working tool.

Superposition and basis choice

Because states are vectors, any state can be expanded in any orthonormal basis. The computational basis {|0>, |1>} is conventional, but the {|+>, |->} basis or any rotated basis is equally valid. The same physical state has different coordinate lists in different bases, exactly like a geometric vector.

Limits of the pure-state picture

Not every situation is a pure state. When a system is entangled with something you cannot access, or when there is genuine classical uncertainty about its preparation, the state vector is insufficient and you must use a density matrix instead. Pure states are the special case of perfectly known, isolated systems.

import numpy as np
psi = np.array([1, 1j])/np.sqrt(2)   # (|0> + i|1>)/sqrt2
print(np.vdot(psi, psi).real)         # 1.0  (normalised)
print(abs(psi)**2)                    # [0.5 0.5] outcome probs