It's the quantum version of a correlation. That means that two or more parts of a quantum system have correlated properties. What's strange about it is that the correlation is indeterminate until a measurement is made, after which the correlation is revealed.

The hidden variable approach says no indeterminacy exists. Pure chance is unreal. Chance is deterministic. Like quantum fields are.

By the way. I see youre a math guy. Whats the difference between a function and a distribution?

The second paragraph here is a good description: Generalized functions. I've never worked with these things. Continuous linear functionals are called distributions, also. A linear functional takes a function in a function space and produces a number (real or complex). For example, in the space of complex contours the length of a contour is one such functional.

I aske because of fields being operator valued distributions. Defined at and around points. Like a dirac delta.

From lecture notes by Sourav Chatterjee, Stanford:

Although quantum mechanics has been successful in
explaining many microscopic phenomena which appear to be genuinely ran-
dom (i.e., the randomness does not stem from the lack of information about
initial condition, but it is inherent in the behavior of the particles), it is not
a good theory for elementary particles, mainly for two reasons:

• It does not fit well with special relativity, in that the Schr ̈odinger
equation is not invariant under Lorentz transformations.
• It does not allow creation or annihilation of particles.

Since in lots of interesting phenomena (e.g., in colliders) particles travel at
speeds comparable to the speed of light, and new particles appear after they
collide, these aspects have to be taken into account.

Quantum field theory (QFT) is supposed to describe these phenomena
well, yet its mathematical foundations are shaky or non-existent. The fun-
damental objects in quantum field theory are operator-valued distributions.
An operator-valued distribution is an abstract object, which when integrated
against a test function, yields a linear operator on a Hilbert space instead
of a number.

The fun-
damental objects in quantum field theory are operator-valued distributions.
An operator-valued distribution is an abstract object, which when integrated
against a test function, yields a linear operator on a Hilbert space instead
of a number. — jgill

That's what I mean.