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!!!