Bell’s theorem without inequalities and without alignments
description
Transcript of Bell’s theorem without inequalities and without alignments
Bell’s theorem without inequalities and
without alignments
Adán CabelloUniversidad de Sevilla
Spain
• Usual proofs of Bell’s theorem assume that the distant observers who perform spacelike separated measurements share a common reference frame.
• Establishing a perfect alignment between local reference frames requires the transmission of an infinite amount of information.
• Yuval Ne’eman argued that the answer to the puzzle posed by Bell’s theorem was to be found in the implicit assumption that the detectors were aligned.
• For an experiment to show the violation of a Bell’s inequality, perfect alignment is not essential. However, in the proofs of Bell’s theorem without inequalities (GHZ’s, Hardy’s,...) perfect alignment seems to be essential, since these proofs are based on EPR’s “elements of reality”.
Motivation
“If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.”
EPR’s elements of reality
• To prove Bell’s theorem without inequalities without it being necessary that the observers share a reference frame (i.e., without the need that distant local setups be aligned).
• The proof is based on the fact that the required perfect correlations occur for any local rotation of the local setups.
Purpose
“If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, independently of the orientation of the measurement apparatus used, then there exists an element of physical reality corresponding to this physical quantity.”
Rotationally invariant EPR’s elements of reality
Proof
10101001011001012
10
1100210101001011001010011232
11
where
Prepare the 8-qubit state
and
011000 337
1
Proof
F
G
0 0 1 1
0 0 1 1
where j jP 23
112
91',1'
11'|1
11'|1
01,1
GRGRP
GRFRP
GRFRP
FRFRP
BA
AB
BA
BA
Properties:
The local (4-qubit) observables are
This is Alice
Let us suppose that she measures G...
...and obtains the result 1
Then, if Bob (who is spacelike separated from Alice) measures F, he always obtains 1...
...even if Bob rotates his apparatus
He always obtains 1!
Even if Alice has rotated her apparatus!
In any way!
Analogously, if Bob measures G and obtains 1...
...then he can predict that, if Alice measures F, she always obtains 1
Even if Alice rotates her apparatus!
...or Bob!
If Alice and Bob measure G, sometimes (in 8% of the cases) they both obtain 1...
In those cases, what if, instead of measuring G, they had measured F?
If EPR’s elements of reality do exist, then, at least in 8% of the cases, both of them would have obtained F=1
However, they NEVER both obtain 1!!!