So far every state has been a vector: a system known as completely as quantum mechanics allows. Often we know less. An oven sends out atoms whose spins point up or down at random, half of each, and we do not know which came out. That is a mixed state, an ensemble: a list of states with the probabilities that each was prepared. It is not a superposition. The superposition $(\ket{\uparrow}+\ket{\downarrow})/\sqrt{2}$ is spin up along $x$, and a measurement along $x$ shows it every time; the half-and-half mixture gives up along $x$ only half the time — and half the time along every axis. Probabilities that come from not knowing and amplitudes that interfere are different things, and a mixed state needs a description of its own.

A coin lying under a cup is heads or tails and you merely do not know which: lift the cup and you learn what was already there. A mixture is like that coin. A superposition is not a hidden answer of this kind — measured in the right basis, it gives one definite result every time, which no coin under a cup could promise.

The probability of spin up along an axis in the $x$–$z$ plane at angle $\theta$ from $z$. For the superposition $\tfrac{1}{\sqrt2}(\ket{\uparrow}+\ket{\downarrow})$ it is , reaching certainty along ; for the half-and-half mixture of and it is along every axis.

A mixed state is an ensemble $\{p_{k},\ket{\psi_{k}}\}$: the system was prepared in $\ket{\psi_{k}}$ with probability , and we do not know which.

• A pure state is an ensemble with one member, of probability 1.
• In an ensemble the probabilities of the members add; in a superposition the amplitudes add first and are squared after. Only superpositions interfere.
• The ensemble $\tfrac12\ket{\uparrow},\tfrac12\ket{\downarrow}$ gives spin up with probability along every axis; gives it with probability 1 along .

Every pure state of a spin $\tfrac12$ is spin up along some direction, so a measurement along that direction is certain. For the mixture the probabilities of its two members add, and for every axis they average to one half. No pure state can do that, so the mixture is none of them.

Every state of a spin $\tfrac12$ is spin up along some direction, as the spin chapter showed.

Measured along its own direction, it is up for certain.

The probabilities of the ensemble add, member by member.

With $\ket{\mathbf{n}}$ at polar angle $\theta$: the same for every axis.

So the mixture is not any pure state, superposition or not.

To predict a measurement on an ensemble, work out the probability for each member and average with the weights of the ensemble.

A superposition of up and down along $z$ is spin up along some other axis, and a measurement along that axis is certain. A half-and-half mixture of up and down gives one half along every axis.

The members of an ensemble are alternatives we do not know, not amplitudes: their probabilities are added, never their amplitudes, so they cannot interfere.

A mixed state describes a preparation that is random, a system in thermal equilibrium, or one part of a larger system looked at alone.

Great work! You now know how to:

• describe a system known only with probabilities as an ensemble
• tell a mixture from a superposition by measurement
• prove that an even mixture of up and down is no pure state