Quantum entanglement is very much like that, except for one crucial difference. The two marbles are similarly correlated but they exist in a superposition of "the first marble is red and the second marble is blue" and "the first marble is blue and the second marble is red". To help visualize this, imagine a double exposure photograph that exhibits features of two different landscapes at the same time.
Without looking, Alice takes a marble from the box (which I will refer to as the first marble), Bob takes the second marble and they walk to different rooms.[1] When Alice looks at her marble, the superposition disappears and there is a 50% chance of her seeing either a red or a blue marble. If she sees a red marble then, due to the correlation, she immediately knows that Bob has a blue marble. Indeed when Bob looks, his marble is blue. It does not matter if Bob is in the next room, or on the opposite side of the world, or in a galaxy millions of light years away, the marbles will always be the opposite color.
The puzzle for the various interpretations of Quantum Mechanics is to explain how the marbles are always correlated even though the color of Alice's marble is randomly determined at the precise moment that she looks (and not merely unknown to her as in the classical scenario). This is the substance of the EPR Paradox that Einstein (along with colleagues Podolsky and Rosen) devised to challenge the Copenhagen Interpretation. If Alice happens to see a red marble, it seems that Bob's marble somehow "knows" it has to be blue, either by communicating faster than the speed of light or via local rules that determine its color. Einstein presumed the latter mechanism, but this was subsequently proven to be impossible by John Bell some years after Einstein's death.[2]
So how does the Many-World's Interpretation resolve this puzzle? The key insight is that it is not merely the pair of marbles that are in a superposition before Alice looks at her marble. Everything in the environment, including Alice and Bob, are in a superposition along with the marbles. The scenario configurations can be described as follows:
- Alice picks the red marble and Bob picks the blue marble
- Alice picks the blue marble and Bob picks the red marble
- Alice has a marble and Bob has a marble (a superposition)[3]
- Alice sees a red marble and Alice has a red marble and Bob has a blue marble
- Alice sees a blue marble and Alice has a blue marble and Bob has a red marble
Alice then looks and sees a red marble (configuration 4). She is now entangled with all the things on that amplitude path which includes the red marble and Bob who has the blue marble. On the other amplitude path, her twin is entangled with the blue marble and also Bob's twin who has the red marble. There are no amplitude paths back to configuration 3 (or over to configuration 5) from Alice's world branch. This is decoherence - an irreversible transition for Alice from independence to entanglement (correlation) with a single world branch. If she hides the marble and then looks at it again, it will continue to be red.[4]
On this interpretation, there is no mystery to how the marbles correlate. Their correlation is simply the logical consequence of having a red and a blue marble in each world branch just as it is the logical consequence of having a red and a blue marble in the classical scenario. The 50% probability of seeing a red marble merely reflects Alice's knowledge of which of the two world branches that she will find herself in when she looks at her marble.[5]
For my earlier posts in this series, see Visualizing Quantum Mechanics and Interpreting Quantum Mechanics.
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[1] In practice, decoherence effects would prevent actual marbles being put in a superposition, much less picked up and moved by people. However, as a thought experiment, it is analogous to actual experiments of two entangled photons with correlated spins that can be separated by great distances while remaining in a superposition.
[2] See Bell's Theorem which rules out local hidden variables.
[3] The superposition described in configuration 3 can be conceptualized in either of two equivalent forms:
- Alice has the first marble in a superposition of "red marble" and "blue marble" and Bob has the second marble in a superposition of "red marble" and "blue marble"
- A superposition of "Alice has a red marble and Bob has a blue marble" and "Alice has a blue marble and Bob has a red marble"
In this expression, Alice (A), Bob (B) and the marbles (where 1 and 2 represent the first and second marbles while r and b represent red and blue) are each configurations with probability amplitudes (which are complex numbers). In the left-hand-side expression, the full system configuration is the complex product of Alice, Bob and the marble superposition. In turn, the marble superposition is the complex addition of the two distinct marble configurations (one of which is the complex product of the first red marble and the second blue marble while the other is the complex product of the first blue marble and the second red marble).
Notice that the left-hand-side expression is the factorization of the right-hand-side expression. In the left-hand-side expression, Alice and Bob are factored independently of the marble superposition. In the right-hand-side expression, Alice and Bob are, equivalently, part of a more general superposition where their amplitude is found in both branches. In both expressions, a red marble (r1 or r2) is always correlated with a blue marble (b1 or b2).
It's always useful to remember that a complex number can be represented as an arrow (phase vector) with its tail at the origin. To add two arrows, move the second arrow so its tail is at the head of the first arrow. Then draw a new arrow from the origin to the head of the second arrow. To multiply two arrows, sum the two angles and multiply the two lengths to draw a new arrow.
[4] Note that the effects of Alice's observation can't reach Bob faster than the speed of light. So if Bob doesn't look at his own marble, Bob remains independent of either world branch until, for example, Alice walks over and shows Bob her red marble (in accordance with the no-communication theorem). Decoherence is a physical process where Alice and, later, Bob are transformed from independence to correlation with one of the world branches.
[5] When Alice finds herself in configuration 4, it may seem that configuration 2 was not part of her history. However if it were not, then Alice's history would just be the classical scenario where she had picked the red marble from the box at the beginning but just didn't know it yet. However, in our quantum scenario, the marble was in a superposition before she looked which could only be caused by amplitude for Alice flowing through both configurations 1 and 2.
