3.4Mixtures and the density matrix

Pure states are the best case: the most that can be known. Real sources are sloppier. A lamp, an oven, a polariser that is slightly off: none of them prepares a pure state. This last lesson builds the tool for describing any preparation at all, and finds that it forgets a surprising amount about where the light came from.

A recipe with a coin in it

Suppose a preparation is not fully pinned down. Each time, with probability $p_1$ it produces the pure state $\mathsf u_1$, with probability $p_2$ the state $\mathsf u_2$, and so on. The $\mathsf u_\alpha$ can be any normalised vectors; they need not be orthogonal. Lesson 1.3 called this a mixture, and the coin-tossing source there was one.

What will a measurement of an observable $A$ give on average? In the pure state $\mathsf u_\alpha$ the mean is $\ip{\mathsf u_\alpha}{A\,\mathsf u_\alpha}$. Averaging again over the coin,

As in the last lesson, the trick is to separate what belongs to the preparation from what belongs to the test. Everything about the preparation can be gathered into one matrix,

a weighted sum of the projectors on the ingredients. It is called the density matrix. In terms of it the mean takes a form that is the same for every observable:

Three facts about $\rho$ follow from how it was built.

• It is Hermitian, and $\Tr\rho = 1$, because the probabilities $p_\alpha$ add up to 1.
• The probability of outcome $\mu$ of any test is $\Tr(\rho P_\mu)$, where $P_\mu$ is the projector on that outcome. So every probability that can be measured is computed from $\rho$.
• A pure state is the special case of a single ingredient, $\rho = \mathsf u\mathsf u^\dagger$, a projector, for which $\rho^2 = \rho$. In general $\Tr\rho^2$ lies between $1/N$ and 1 and equals 1 only for pure states. It is a convenient measure of how pure a preparation is.

Two examples, with components along the horizontal and vertical axes:

The diagonal entries are the same: both give half and half in the horizontal-against-vertical test. The difference is in the off-diagonal corners, which hold the relative phase between the two components. The superposition has it; the mixture does not. That is lesson 1.3 written as two matrices.

Different recipes, the same light

Now work out $\rho$ for a different recipe: toss a coin and make the photon right circular or left circular. The two projectors are

The same matrix as the coin toss between horizontal and vertical. Every probability of every test is $\Tr(\rho P)$, and the two recipes have the same $\rho$. So there is no test, of any kind, that distinguishes them. Not a difficult test that nobody has thought of: none.

This is quite unlike mixing in the ordinary sense. A mixture of chemicals has one composition, and analysis will find it. A quantum mixture that is not pure can be made up in infinitely many ways, and once made, the way is unrecoverable. Someone handed megajoules of the light could never discover whether the coin had chosen between horizontal and vertical or between right and left, even though the two machines that made it look nothing like each other.

A pure state is the exception. It can be “mixed” in only one way, from itself.

K. Completeness of the quantum description. The density matrix $\rho$ completely specifies all the properties of an ensemble of quantum systems.

Rule K is the last of the eleven. It says that $\rho$ is not a summary that leaves something out: there is nothing else to know. The book shows later (chapter 6) that if recipes with the same $\rho$ could be told apart, the ability could be used to send signals faster than light.

One loose end from lesson 2.1. The two boxes there, one with exactly equal numbers of horizontal and vertical photons and one with exactly equal numbers of right and left, could be told apart. There is no conflict. Counting photons into a box is not the same recipe as tossing a coin for each: the photons in a counted box are not independent of one another. Any one photon drawn from either box has $\rho = \tfrac12\mathbb 1$, but the box as a whole is a larger system with a density matrix of its own, and those differ.

The ball of all states

For polarisation, any density matrix can be written with the help of the three Pauli matrices of the last lesson:

The three numbers $s_j$ are the means of the three standard tests with their outcomes labelled $\pm1$: the same $s_1, s_2, s_3$ as in lesson 3.2. For a pure state they locate a point on the surface of the sphere. For a mixture they locate a point inside it. The nearer the centre, the less is known, and the centre itself, $s = 0$, is the random mixture of rule C: ordinary unpolarised light.

Mixing two preparations with weights $\lambda$ and $1 - \lambda$ puts the point on the straight line between theirs, at the matching distance along. That is all the figure above does. And it makes the loss of information visible: a point inside the ball lies on infinitely many chords, and any of them could have been the recipe.

It also tells you how to find an unknown $\rho$: measure the three means, on three batches of photons. Three numbers and you are done. In general a system with $N$ outcomes needs $N + 1$ different tests. And there is a built-in check on the experiment: the eigenvalues of $\rho$ are $\tfrac12(1 \pm |s|)$, and probabilities cannot be negative. If the three measured numbers give a point outside the sphere, the fault is in the apparatus.

Where you have got to

That completes the first three chapters of the book, which Peres calls “gathering the tools”. Look back over what the tools are.

Part I found that outcomes of experiments on single photons and atoms are random, that a superposition is a different thing from a mixture, that a single system cannot be made to reveal its state, and that a measurement does not read off a value that was already there. Part II turned those findings into rules about tests, and found that when paths cannot be told apart it is amplitudes, not probabilities, that add. Part III showed that amplitudes are the components of a vector, that every probability is the squared size of an inner product, that a measurement is a Hermitian matrix, and that any preparation whatever is a density matrix.

The contents page lists the eleven rules, A to K, in one place. You should now be able to say what each one means and which experiment it came from.

What comes next in the book: systems whose tests have infinitely many outcomes, such as the position of a particle, where the uncertainty relation between position and momentum is finally derived properly (chapter 4); systems made of two or more parts, where the really strange consequences live (chapters 5 to 7, including Bell’s theorem); how states change in time (chapter 8); and at the end, a hard look at the measuring apparatus itself (chapter 12).

This lesson follows §3-8. The appendix that closes the chapter, §3-9, is on Dirac’s notation, which the refresher in lesson 3.1 introduced.