# If a quantum state is pure why are its observables still probabilistic?

As I understand it, a pure quantum state is one that can be represented as a ket $\lvert\psi\rangle$ in a Hilbert space, and it contains all the information about the state of the system. As such, we have complete information about the state of the system. Why then is the information we extract from the state $\lvert\psi\rangle$ still probabilistic? That is, why are the observables that we measure still associated with a probability? Is it simply that although we have complete information about the state of the system, by "complete information" it is meant that we know all the possible values that each given observable of the state can take?

One reason I ask, is that in the case of an open quantum system, we have to use mixed states in which we have incomplete information about the state of the system. This involves a classical probability since it is the lack of information that causes the probabilistic description of the state of the system, however there is still a "quantum probability" associated with such a system, like in the case of a pure state, since its observables have an intrinsic probabilistic nature.

I find it confusing having the two, how to distinguish between them and why this so-called "quantum probability" arises, particularly when in the case of a closed or isolated quantum system we can use pure states which supposedly contain complete information about the observables of the system?!

• From the perspective of QM, it is true that | psi> contains 'all the information of the system'. Along with operators acting on it, they are the only mathematical objects used to describe the system. But QM gives an inherently probabilistic description of observables, even for pure states - thats just the way the theory is constructed. Apr 15 '16 at 11:13
• A closed quantum system can't be observed and an open quantum system is not described by the wave function. As a result one can't get around measurements being uncertain in nature. That's not the same thing as probabilistic, though. Apr 15 '16 at 11:14
• There always exists uncertainty when a state is a linear combination of stationary states or pure states.
– user36790
Apr 15 '16 at 11:21
• So, by saying that $\lvert\psi\rangle$ contains all the information about the system, is this to be interpreted as knowing all the possible values that the observables of the state can assume (position, momentum, etc.), and the probabilities of these particular values can take, rather than knowing the exact values of position, momentum, etc?! Apr 15 '16 at 13:07

It is a postulate of quantum mechanics that a pure state vector contains all information you can possibly know about a physical state. Note that this is not the same as saying it contains all information you can possibly imagine having about a physical state, particularly since your (and my) imagination is mostly classical - we humans are pretty bad at imagining states with no definite value for an observables.

The idea that this postulate of quantum mechanics might be wrong, that "complete information" means you can assign definite values to all observables and thus get a deterministic outcome for a single measurement, is the idea of hidden variable theories. Local hidden variable theories equivalent to quantum mechanics are excluded by Bell's theorem, non-local ones are possible, but might be even more philosophically dissatisfying.

Now, you also mention mixed states. Mixed states represent lack of knowledge about the quantum state of a system, they mean we are lacking information we could in prinicple have. In priniciple: Measure a system in a way that you obtain the values for a complete set of commuting observables, and you know the pure quantum state it is in. This is different from the probabilities for measurement of a pure state, because standard quantum mechanics says there is nothing even in principle you could know that would make you able to get rid of the probabilities.

However, there is an interesting fact: Quantum mechanics inherently comes with a process to generate the lack of information about subsystems - entanglement. If you have a pure state of a system that has subsystems that is entangled, the states you assign to the subsystems must be mixed states - it is the defining characteristic of entanglement that there are no single pure states you could assign to the subsystems. So this is another very un-classical feature: Having complete knowledge of the state of a system does not imply you have complete knowledge of the states of the subsystems.

As I said at the beginning, there is no "why" we would currently know. but this should not really unsettle you - there are axioms in every physical theory that are taken for granted simply because they produce the correct observations. There is no answer to "Why do Newton's laws hold?" in Newtonian mechanics, or to "Why do electric and magnetic fields obey Maxwell's equations?" in electromagnetism, either.

Technically one cannot say observables are probabilistic, since they are mathematically described by deterministic operators. Now when an observable has different eigenvalues, then the Born rule is used to predict which value the experiment will get, and this is where probabilities arise. The Born rule is a postulate of Quantum Mechanics, historically proposed as an interpretation of the solutions to the Schrödinger equation, in order to fit experiments. See this discussion on the nature of the Born rule.

Nobody knows what the fact that we need the Born rule means ontologically, although it seems clear that it cannot generally be said that an observable has any definite value when it is not being measured, which is confusing indeed.

• So, is the Born rule (as you say) simply an interpretation of the solution one obtains upon solving the Schrodinger equation for a particular quantum mechanical system? What confuses me is that one can have two types of probability, one arising from quantum mechanical effects - the measurement of an observable gives a particular result with some probability associated with it. But also, one can have classical probabilities that arise due to lack of information about the state of the system (such that we can not construct a deterministic evolution)?! Apr 15 '16 at 15:03
• The density matrix is where both types of probabilities are mixed together. It is an effective tool, but not a conceptually clear one. It is because experiments are in practice made on a large number of similar systems, and also because there are uncertainties in actual setups, that classical probabilities are to be taken into account. They have nothing to do with probabilities as per Born's rule, but in the formalism of the density matrix both types of probabilities are handled at the same time. It is not straightforward to get an intuition about the physical meaning of the density matrix. Apr 15 '16 at 15:11
• Is the idea with so-called quantum probabilities that there is an inherent probabilistic nature to the state of a given system, such that even if we have complete information of the state of a given system allowing us to describe as a pure state, the values of the observable quantities themselves have a probability associated with them. In this sense, does "complete knowledge" equate to knowing all the values that any given observable (associated with a given state) can assume, an the probabilities associated with them, at a given instant in time? Apr 15 '16 at 15:40
• I guess you could say so, yes. In my view things are a little bit weirder than that: I do not think there is anything real associated to an observable that is not measured. So I would not even talk about a state as something physical at all times. We can see noncausal correlations in entangled systems so it is very difficult to see a quantum state as anything even remotely close to a classical object. Apr 15 '16 at 15:48
• What is "conceptually not clear" about the density matrix? It describes a lack of classical knowledge on top of a quantum mechanical uncertainty. The lack of information in an open quantum system is a trivial consequence of relativity, QFT is just not a very good tool to analyze it properly, so we do it by an ad-hoc simplification. Apr 15 '16 at 18:14