Quantum Bits: From Classical Bits to Qubits • Astro Theme OpenBlog

A qubit is the basic information unit in quantum computing.

Classical Bit vs Qubit

Classical bit: exactly one of 1.
Qubit: a normalized linear combination of basis states $|0\rangle$ and $|1\rangle$ .

|\psi\rangle = \alpha|0\rangle + \beta|1\rangle, \quad |\alpha|^2 + |\beta|^2 = 1

The coefficients $\alpha$ and $\beta$ are complex amplitudes.

When measured in the computational basis:

After measurement, the state collapses to the observed basis state.

Any single qubit pure state can be represented on a sphere:

|\psi\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\phi}\sin\frac{\theta}{2}|1\rangle

The angles $(\theta, \phi)$ provide geometric intuition for single qubit operations.

For

|\psi\rangle = \frac{\sqrt{3}}{2}|0\rangle + \frac{1}{2}|1\rangle

measurement gives:

Qubits are probabilistic at readout but deterministic under unitary evolution.

Two states that differ only by a global phase are physically equivalent:

|\psi'\rangle = e^{i\gamma}|\psi\rangle

But relative phase changes interference behavior. Compare:

\frac{|0\rangle + |1\rangle}{\sqrt{2}} \quad \text{vs} \quad \frac{|0\rangle - |1\rangle}{\sqrt{2}}

They produce different outcomes after a Hadamard, so relative phase carries computational meaning.

Not every qubit state is perfectly known. A statistical ensemble is represented by a density matrix:

\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|

For a pure state, $\rho = |\psi\rangle\langle\psi|$ and $\mathrm{Tr}(\rho^2)=1$ . For a mixed state, $\mathrm{Tr}(\rho^2)<1$ .

This formalism is essential once noise and partial information enter the picture.

Measurement can be defined in any orthonormal basis $\{|u_k\rangle\}$ . Outcome probability is:

P(k)=|\langle u_k|\psi\rangle|^2

So measurement is fundamentally “projection onto a basis,” not tied only to $|0\rangle,|1\rangle$ .