The GHZ experiment cartoon

I'm still a bit obsessed with the GHZ experiment.  This is the one where you prepare a GHZ state $\lvert 000 \rangle + \lvert 111 \rangle$ and send one qubit to each of 3 protagonists: Alice, Bob, and Charlie.  (I will be ignoring shared normalizer constants throughout this post, as I find they don't add anything to the understanding.)

If you do the maths it turns out that when all three choose to measure in the $\lvert + \rangle$, $\lvert - \rangle$ basis (shorthand for $\lvert 0 \rangle + \lvert 1\rangle$ and $\lvert 0 \rangle - \lvert 1\rangle$) then they are guarranteed to get a parity zero result.  On the other hand if only one measures in this basis and the other two measure in the $\lvert +i\rangle$, $\lvert -i\rangle$ basis (shorthand for $\lvert 0 \rangle + i\lvert 1\rangle$ and $\lvert 0 \rangle - i\lvert 1\rangle$) they are guarranteed to get a parity one result.  As I showed in an earlier post this appears to be incompatible with the outcomes being predetermined, unless the qubits share a faster than light communication mechanism.

The cartoon above shows that there is another way to look at it if we accept the multiverse: which is to say that humans can be in a superposition of states as well as qubits.  Let's try to describe the cartoon from the point of view of Alice, the protagonist on the left.

First a GHZ state is prepared.  Next, one qubit is delivered to each of the protagonists and, randomly, they all choose to measure in the $\lvert + \rangle$, $\lvert - \rangle$ basis.  This means we have to change the basis we use to represent the GHZ state, changing it from $\lvert 000 \rangle + \lvert 111 \rangle$ to $\lvert +++ \rangle + \lvert +--\rangle + \lvert -+-\rangle + \lvert --+\rangle$.  In the diagram we've represented each of these 4 summands by a different colour.  Next, Alice splits into two: an Alice that measured zero and an Alice that measured one.  It's not really the case that there are exactly two Alices now, rather there are an infinity, but an infinity which can be divided into two categories.  We are assuming that Bob and Charlie have measured their qubits at more or less the same time, so that Alice cannot possibly know what they have measured yet.  Therefore, from Alice's point of view, Bob and Charlie - and their qubits - currently exist in the Bell state $\lvert ++ \rangle + \lvert -- \rangle$ if she got 0 and the Bell state $\lvert +-\rangle + \lvert -+\rangle$ if she got 1.  Finally, some time passes and Alice becomes entangled with post measurement information from Bob and Charlie.  At this point we can categorize the infinity of Alices into 4 categories, with each category corresponding to one of the summands appearing in the GHZ state expression in the $\lvert +\rangle$, $\lvert - \rangle$ basis.

If two of Alice, Bob, and Charlie choose to measure in the $\lvert +i\rangle$, $\lvert -i\rangle$ basis the picture looks only slightly different as the GHZ state will be represented by a different 4 summands.

The neat thing about this picture is that in addition to there being no randomness (i.e. God does not play dice) there is also no non-locality.  I think Einstein would have liked Everett's interpretation had he lived long enough to see it.

 

 

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