Non-orthogonal quantum states? Whassat?

Factual Questions
1

I’m reading an article which says:

I think, but am not certain, that I’ll understand the above if I can be told what “non-orthogonal quantum states” are. Apparently (per the quote) definite perceptions and their quantum superpositions are one example of a set (pair?) of non-orthogonal quantum states. But I am not getting a clear idea as to what it means for them to be an example of that. I am not sure, anyway, what these authors mean by “definite perceptions.” I’ll probably find out when I get to section 3. But meanwhile, maybe someone here can help me out with just understanding “non-orthogonal quantum states” in isolation.

2

Man, when I read that quote, my first instinct was to keep my hand on my wallet. Maybe it makes sense in context, but color me skeptical. That said, you did actually have a factual question that can be answered, so here it is:

In quantum mechanics, the state of a system is represented as a vector in an abstract n-dimensional space, where n is the number of possible distinct states that the object can be in.[sup]1[/sup] For example, the possible states of Schrödinger’s cat can all be represented as a two-dimensional plane, with a state vector pointing in the horizontally corresponding to “alive” and a state vector pointing vertically corresponding to “dead”. After one half-life of Kitty’s radioactive roommate, the prediction is that Kitty’s state will correspond to a vector pointing at a 45-degree angle to the horizontal — in other words, not purely “alive” or “dead” but a sum of both types of vectors.

So a pair of “orthogonal quantum states” are two possible states for a system whose vectors (in this abstract space) are at right angles to each other. “Non-orthogonal state” vectors are not at 90° to each other in this space.

[sup]1[/sup] Note to Hilbertians: go away.

3

Adding to the above, the ‘angle’ between two quantum states relates to their distinguishability. That’s roughly because, if two states are not orthogonal, there is a chance of finding one when the system is in the other; or, to use the cat example, if you have the states |alive> and |alive> + |dead>, then measuring the first, you’ll always find the cat alive, while measuring the second, you’ll find it alive half the time—in which case, there’s no way to tell whether the state before the measurement was |alive> or |alive> + |dead>.

Incidentally, I believe I’ve read the paper the quote is from, or at least one in which a similar claim was made; I can’t say I was convinced by the argumentation.

4

Edited to add: I see this has been pretty much covered anyway.

In its simplest quantum physics says that all the possible states that a system can be in (i.e. its quantum state) at a given time, where each state is encapsulates all the information that it is possible to know[sup]1[/sup] about the system at that time, can be represented by a state vector[sup]2[/sup] and the complete set of state vectors form a Hilbert space[sup]3[/sup].

The way I like to think of this is that each different state or ‘configuration’ of a quantum system is a pointy arrow in some abstract mathematical space.

In a Hilbert space the notion of to two vectors being “at right angles” to each other exists just as it does in the familiar real 3 dimensional vector space (indeed a 3-D real vector space is a Hilbert space), this is known as orthogonality. Of course simply saying this doesn’t tell us much about the implications of orthogonality of quantum states and what orthogonality indicates is a certain kind of independence between two states.

A quantum measurement of quantum mechanics, is represented by a self-adjoint operator[sup]4[/sup] on the Hilbert space and each self-adjoint operator has a set of characteristic real values (eigenvalues) corresponding to the different possible results of a measurement[sup]5[/sup] and each eigenvalue has an associated characteristic vector (eigenvector) corresponding to the state of the system immediately after the measurement has been made which has returned the associated eigenvalue. The eigenvectors of self-adjoint operators (when they exist) are always orthogonal to each other. In other words the different possible quantum states of a system after a measurement are orthogonal to each other.

Given a quantum system in an unknown state and you are told that the system is either in state |a> or state |b> you can make a measurement which reliably determines which state the system is in iff |a> and |b> are orthogonal.
[sup]1[/sup]The notion of a quantum state can be extended somewhat to include states which do not encapsulate all the information that it is in principle possible to know about a system (mixed states), these are represented by density operators on the Hilbert space.

[sup]2[/sup]Technically each quantum state corresponds to a “ray” as multiplying the state vector by a scalar can alter the state vector, but not the quantum state it corresponds to.

[sup]3[/sup]A Hilbert space is just vector space where there is an inner product (i.e. a way of multiplying two vectors to produce a scalar that conforms to certain rules) and that inner product can be used to define a metric (‘distance function’) on the space which is suitably doesn’t have any holes in it (i.e. it is Cauchy-complete).

[sup]4[/sup]An operator on a vector space is a function(al) that takes a vector in the space as it’s argument and maps that to a vector in the same space. Self-adjoint operators are chosen because they have real eigenvalues (when the exist) and the associated eigenvectors are orthogonal.

[sup]5[/sup] Not technically true as the set of characteristic values/eigenvalues can be empty, but this is not actually problem.