# Interpretation of density matrix

In Landau’s Statistical Physics (part 1) , section 5, he writes:" In particular, it would be quite incorrect to suppose that the description by means of the density matrix signifies that the subsystem can be found in various ψ states with various probabilities and that the averaging is over these probabilities."

However to my knowledge what Landau opposes is exactly the physical interpretation of a density matrix. What is it that I am missing?

edit: I am not confusing the probabilistic property inherent to a pure quantum state with that of the mixed state. Still, I am under the impression that the density matrix is a characterization of the constitution of the mixed state; for instance, we could use a density matrix to describe an ensemble of systems made up with 70% of state A and 30% of state B (This example comes from Sakurai's Modern Quantum Mechanics(2e), page 180). But is this not what Landau calls incorrect?

Could it be that Landau uses the density matrix to describe a subsystem which is quite determined in a way unknown to us (since we only have incomplete information) and in the above example the matrix is used in a closed and completely described system which is probabilistic in nature? (I am ignoring here the probabilistic property in quantum physics itself as it is present in both cases.)

• Landau explains that in detail in the paragraphs directly after your quote Jul 11, 2019 at 10:26
• I don't have Landau on my hands right now but there are issues with trying to isolate the classical statistical part and the quantum mechanical part of the probabilities described by the density matrix. See: physics.stackexchange.com/questions/98703/…
– user87745
Jul 11, 2019 at 11:02
• @Feynmans Out for Grumpy Cat Given this fact, it is probably right to say Landau's context and Sakurai's example are not of two difference kinds. But then what should I make of Landau's comments?
– X.L
Jul 11, 2019 at 11:13
• @X.L Sorry, I don't exactly understand what you mean. And I will be able to post a more informed comment once I have a look at Landau but I think Landau's comments correspond to exactly the point that has been raised in the question I linked. There seems no unique way to decompose classical and quantum parts of the probabilities in a way Sakurai has decomposed. This is not to say that Sakurai is incorrect. But, what Sakurai is doing is building a density matrix given a classical/quantum decomposition. But given a density matrix, you cannot uniquely construct a classical/quantum decomposition.
– user87745
Jul 11, 2019 at 11:21

I think what he means is with reference to his equation (5.1) $$\psi = \sum_n c_n \psi_n.$$ These $$\psi_n$$ states are in a superposition and this superposition is not the same as saying the system has some probability of being in one state or the other, even though $$|c_n|^2$$ is really the probability of the system being in state $$\psi_n$$ (regular quantum mechanics interference effects). This is manifested in the off diagonal terms of the density matrix. This lead us to the other part of the question.

As you mentioned, density matrix is always interpreted as being a probability of the system of being in some quantum mechanical state. The definition Landau gave for the density matrix is $$w_{mn} = c^*_n c_m,$$ which is clearly a Hermitian matrix and so can be diagonalised, $$w_{\alpha \beta} = c_{\alpha} \delta_{\alpha \beta}.$$ Now in this new basis you can really think of the diagonal terms $$c_{\alpha}$$ as being a probability of the system being in state $$\alpha$$.

It's important to notice that in this diagonal basis you cannot think of the system as being in a superposition of the $$\alpha$$'s, $$\psi \neq \sum_{\alpha} c_{\alpha} \psi_{\alpha}.$$

If this bothers you, I can give you another way to think about this. Instead of equation (5.1) think about the bigger state containing the system and the environment, $$\Psi =\sum_{ij} C_{ij} \theta_i \psi_j,$$ where with Landau, $$\psi_j$$'s are the wavefunctions for the system in consideration, and $$\theta_i$$'s are for the environment. With this, following exactly what Landau has in the book, but for an operator $$f$$ that only act on the system and leave the environment unchanged, we can write the density matrix as, $$w_{j^{\prime} j } = \sum_{i} C^*_{ij^{\prime}} C_{ij}.$$ Notice that this is also Hermitian. Now this sum can be diagonal in $$j$$ and $$j^{\prime}$$ with no problem.

If I interpret what the author had in mind correctly, he would like to distinguish between the uncertainty (and related probability distribution) inherent to quantum mechanics, and the incomplete statistical description given by a density matrix.

In fact, a (non-pure) density matrix describes a (quantum) system on which we do not have complete information due to "external" factors, and not due to a fundamental obstruction. In some sense, it is an admission of partial ignorance by the observer/experimenter (lack of complete information). Contrarily, the wavefunction probabilistic description is fundamental, and a pure quantum state carries the maximal amount of information that it is possible to have in a quantum system.

A classical analogy may also be useful. A density matrix is the statistical analogue to a phase space probability distribution in classical mechanics: we introduce it to describe a system in which we only have statistical information (perhaps due to it being constituted by a very large number of components), and not complete knowledge of the system.

To sum up, Landau is in my opinion saying that a system described by a density matrix is a system on which we have incomplete information because of "our fault", and not because it is really in such a configuration. If we had better instruments, we could in principle investigate and characterize its precise configuration; but since we are not able to do so, we have to content ourselves with the predictions that we can still make assuming only a partial, statistical, knowledge.