Qubits vs Probabilistic Bits
A qubit is not a random classical bit: complex amplitudes, interference, and entanglement set it fundamentally apart.
Superficial similarity
A probabilistic bit (pbit) is a classical bit that is 0 with probability p and 1 with probability 1-p. A qubit measured in the computational basis also gives 0 or 1 with some probabilities. This resemblance is a trap. The qubit's state before measurement is a vector of complex amplitudes, not a probability distribution, and that difference produces phenomena no pbit can imitate.
Amplitudes vs probabilities
A pbit's state is one real number p in [0,1]. A pure qubit's state is two complex amplitudes constrained to unit norm, equivalently a point on the Bloch sphere with two real parameters. Amplitudes can be negative or complex and can cancel through interference; probabilities are nonnegative and only ever add. This is why a sequence of two Hadamards returns |0> deterministically, while flipping a coin twice never restores certainty.
- pbit: one real p; state space is a line segment
- qubit: two complex amplitudes; state space is the Bloch ball
- pbit evolution: stochastic matrices (probabilities add)
- qubit evolution: unitary matrices (amplitudes interfere)
Correlations
Two pbits can be correlated, but their correlations always satisfy Bell inequalities and can be explained by shared randomness. Two qubits can be entangled, violating the CHSH inequality up to the Tsirelson bound and exhibiting monogamy. No probabilistic model over local hidden variables reproduces these correlations, which is the experimentally confirmed dividing line.
When a qubit looks classical
A qubit whose coherences have been destroyed by dephasing behaves like a pbit: its density matrix is diagonal and interference is gone. This is precisely decoherence, the process by which quantum systems lose their distinctive features and appear classical. Quantum computing is the discipline of protecting amplitudes and entanglement long enough to exploit exactly the properties that separate a qubit from a mere random bit.