Quantum entanglement vs classical analogy

Consider that we have two balls, one white and one black, and two distant observers A and B with closed eyes. We give the first ball to the observer A and the second ball to the observer B. The observers don't know the exact color (state) of their balls, they know only the probability of having one or another color, until they look at them (measure). If the observer A looks at his ball he will see its color, which is white, so he immediately knows the color of the second ball. Lets call this “classical entanglement”.

My question is: What is the difference between this “classical entanglement” and the quantum entanglement, for example, of two entangled electrons with opposite spins states? Can this analogy be used to explain the quantum entanglement?

Answer

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.