Light can be polarised horizontally or vertically. It can also be polarised diagonally, and a diagonal wave is exactly a horizontal and a vertical one added in step. For single photons this addition becomes a principle: if a system can be in one state and can be in another, it can be in any combination of the two, and that combination is a new state with definite properties of its own.

North-east is not a coin that is sometimes north and sometimes east. It is a direction of its own, built from north and east. A diagonally polarised photon is the same: sent at a diagonal polariser it passes every time, which no mixture of horizontal and vertical photons would ever do.

The four polarisation states used throughout this chapter. Diagonal is $\tfrac{1}{\sqrt2}\bigl(\ket{H}+\ket{V}\bigr)$ and antidiagonal is $\tfrac{1}{\sqrt2}\bigl(\ket{H}-\ket{V}\bigr)$: the same two ingredients in the same amounts, differing only in a relative sign.

A state of a quantum system is written as a ket, $\ket{\psi}$, a symbol for a vector. If $\ket{A}$ and $\ket{B}$ are possible states, so is every combination $\alpha\ket{A}+\beta\ket{B}$ with complex coefficients. The coefficients are amplitudes: for two perfectly distinguishable states and the probabilities of finding the system in each are and .

• The condition $|\alpha|^{2}+|\beta|^{2}=1$ says the probabilities add to one; a state satisfying it is normalised.
• A global phase changes nothing: $\ket{\psi}$ and are the same physical state, because every probability involves a modulus squared.

A superposition is a sum of states with complex coefficients. It is itself a state, as definite as the ones it is built from.

For a state written in the horizontal and vertical pair, the probability that a horizontal polariser passes the photon is the modulus squared of the coefficient of $\ket{H}$.

Multiplying a whole state by a phase changes every amplitude by the same factor of modulus one, so no probability changes. Physically it is the same state.

Changing the sign of one coefficient, and not the other, is a relative phase. It leaves the probabilities for horizontal and vertical alone but changes the state.

• $\ket{D}=\tfrac{1}{\sqrt2}(\ket{H}+\ket{V})$ passes a diagonal polariser with certainty.

Great work! You now know how to:

• write a superposition as a ket with complex coefficients
• normalise a state and read probabilities from its coefficients
• tell a global phase, which changes nothing, from a relative one, which changes the state
• distinguish a superposition from a mixture