OE-4.2 Qubits & Quantum States

What this is and why it exists

There is one sentence about qubits that almost everyone hears first and that is wrong. A qubit is not a bit that is secretly both values at once.

It is a unit vector in a two-dimensional complex space. Superposition is that vector not lying along either axis. That is all superposition means, and stating it that way removes most of the mystique without removing any of the content.

The notation looks strange for about a day and then stops. It is column and row vectors in disguise.

The vocabulary

• Qubit — a unit vector in a two-dimensional complex space.
• Basis state — one of the two chosen reference directions.
• Amplitude — the complex number multiplying a basis state.
• Superposition — a state that is a combination of basis states.
• Normalised — the squared magnitudes of the amplitudes add to one.
• Bra-ket notation — the standard shorthand: a ket is a column vector, a bra a row vector.
• Bloch sphere — a picture of every single-qubit state as a point on a sphere.
• Tensor product — the operation that combines two systems into one.

The mental model

A qubit is a unit vector in a two-dimensional complex space. Write it as an amplitude times the first basis state plus another amplitude times the second. Here is the normalisation condition, in words. The squared magnitude of the first amplitude plus the squared magnitude of the second equals one. Those squared magnitudes are the measurement probabilities, which is why they must add to one.

Superposition is then not a mystery. It is the vector not lying along either axis. Nothing is "both values". The vector has components in both directions.

Bra-ket notation is compact once you accept what it is. A ket is a column vector. A bra is the corresponding row vector, conjugated. A bra next to a ket is an inner product, which returns a number — the overlap from the previous topic. A ket next to a bra is an outer product, which returns an operator. The notation makes the distinction visible in the order of the symbols, which is genuinely helpful once noticed.

The Bloch sphere puts every single-qubit state on the surface of a sphere. It is a very good picture with one limit worth stating. It works for one qubit and does not extend to two. Do not go looking for the two-qubit version.

Which brings us to the tensor product, the step where things get large. Combining two qubits gives a four-dimensional space. Three gives eight. The dimension doubles with each qubit added. That is why simulating fifty qubits on ordinary computers is out of reach, and why the field is interesting at all. That exponential growth is the resource being exploited.

The physics background — wave behaviour, the photoelectric effect, matter waves, the uncertainty principle — is context for why any of this describes reality. Read it as motivation rather than as material to be examined on.

What you should now be able to explain or do

Say what a qubit is without saying "both at once". State the normalisation condition in words and say why it must hold. Read and write bra-ket notation, and tell an inner product from an outer one by the order of symbols. Place a state on the Bloch sphere and say why it does not extend to two qubits. Use tensor products to combine systems, and say how the dimension grows.

Check yourself