Quantum entanglement vs classical analogy
Quantum entanglement is different from the "classical entanglement" in the following way:
- In your example, each ball has only one property of interest, namely "color $\in$ {white, black}".
- In the traditional examples of quantum entanglement, each ball (particle) has two properties of interest, namely "spin in x-direction $S_x$" and "spin in y-direction $S_y$". Moreover, the properties are complementary, i.e. you can't actually measure them simultaneously to arbitrary precision.
- Still, in the entangled state, each of these properties alone is perfectly (anti-)correlated: If A makes a measurement of $S_x$ and obtains the value, say +1, then B will always obtain -1 if he also measures $S_x$. Similar for -1 and +1 and for the other spin direction $S_y$.
The paradox, now, is the following:
Suppose that Alice measures $S_x$ and obtains, say +1. But observer Bob measures $S_y$, obtaining, say +1. Exhulted, Alice proclaims that she has managed to measure two complementary properties simultaneously! After all, her measurement gave her the spin in x-direction, +1, while Bob's measurement allows her to conclude that the particle also has spin -1 in y-direction.
Imagine her surprise, then, when she tries to confirm her conclusion by measuring $S_y$ herself and obtaining +1 in 50% of the cases.
It is somewhat analogous, but the analogy fails because of Bell's theorem. Bell uses a similar analogy with "Bertlmann's socks", when you see one sock is white, and assuming they match, you learn the other is white instantly.
But let us say Bertlmann has three feet, and tries to wear three matching socks, but he doesn't always succeed. When we see the sock on the left foot, we know that the sock on the middle foot has a 99% probability to be matching. Suppose also that the middle foot is 99% of the time the same as the right foot.
Using these two assumptions, you can conclude that the right foot can only be different from the left foot in at most 2% of the cases. This is intuitive--- the number of cases where there is a mismatch between the left and middle plus the number of cases where there is a mismatch between middle and right is always more than the number of cases where there is a mismatch between left and right. Convince yourself of this. This is called "Bell's inequality".
For entangled paticles, the measurements of "up" and "down" have the property that the quantity associated with the middle-sock can be 99% correlated with the left and right foot, but the left and right foot values are only 96% correlated. This is a violation of Bell's inequality. This means that it is not like socks. If it were like socks, the socks would have to change color in response to what you see, nonlocally, faster than the speed of light.
In classical physics when we have a composite system that consists of two (or more) subsystems, then the state of that system is given by a combination of the states of the individual subsystems. The black-and-white-ball system you mentioned is an example of that; the state of the composite system is that one ball is black and the other white, defined solely by the fact that the state of one subsystem is that the ball is white, and the state of the other that the ball is black.
In quantum mechanics there are cases where things behave like in classical physics, and cases where they don't. When we have a composite system in quantum mechanics, there are two options for its state (wavefunction):
- it can be given just by knowing the states of the individual subsystems, or
- it may be its own thing, i.e. a state that is defined for the composite system and which cannot be decomposed into states of the individual subsystems.
The first option has a classical counterpart, like your example, and has nothing to do with quantum entanglement. A quantum state with the property of option 1 is called a product state in the quantum-physics literature.
The second option is what is called an entangled state, and has no counterpart in classical physics. It displays the idea of entanglement clearly: it is the phenomenon where the state of a system is not given by some combination of the states of the individual subsystems that make it up. In fact, all you can define for entangled systems is the state of the composite system, you cannot assign states to the individual subsystems. Clearly, that can never happen in classical physics.