on the logical structure of bell theorems a. broadbent, h. carteret, a-a. méthot, j. walgate

On the logical structure of Bell theorems

A. Broadbent, H. Carteret, A-A. Méthot, J. Walgate

‘Loophole free’ nonlocality tests

A

B

a b

x y

(a,b,x,y)

A B

a b

x y

(a,b,x,y)

Locality Loophole

Bob’s output might be dynamically linked to Alice’s input.

Detection Loophole

Failed measurements might result from a local conspiracy to imitate nonlocality.

Recent work on ‘EPR-Bell Inequalities’

A. Cabello, Phys. Rev. Lett. 88, 060403 (2002)


A. Cabello, Phys. Rev. A. 72, 050101 (2005)

C. Cinelli et al, Phys. Rev. Lett. 95, 240405 (2005)

T. Yang et al, Phys. Rev. Lett. 95, 240406 (2005)


M. Barbieri, Phys. Rev. Lett. 97, 140407 (2006)

A. Cabello, Phys. Rev. A. 73, 022302 (2006)

EPR Inequality A. Cabello, Phys. Rev. Lett. 95, 210401 (2005)

Hyperentangled photon state:

Measurements: Results:

+1 or –1 for each qubit.

e.g. “ +1, +1, –1 , +1 ”

which is equivalent to:

EPR Criterion A. Cabello, Phys. Rev. Lett. 95, 210401 (2005)

If Alice measures ‘Z1 = +1’, then she knows with certainty, without in any way disturbing Bob’s system, that Bob would record ‘Z3 = +1’ were he to measure it.

“

” EPR, (1935)

Measurements: Results:

“

‘Nonlocality’ Game A. Cabello, Phys. Rev. Lett. 95, 210401 (2005)

Hyperentangled photon state:

‘Nonlocality’ Game [1] A. Cabello, Phys. Rev. Lett. 95, 210401 (2005)

+1,+1 ; +1, -1

-1, +1 ; -1, -1

Winning condition:

Possible answers:

“…no classical strategy allows the players to win in more than ¾ of the rounds.” [1]

Any predetermined assignment of values +1 and –1 to these four equations yields:

1 = – 1,

which is thought to be false.


+1,+1 ; +1, -1

-1, +1 ; -1, -1

Winning condition:

Possible answers:

How can this work? By breaking the supposed ‘EPR LER’ criterion.

The criterion is worthless unless we test for it in the experiment.

Alice outputs +1 , +1.

Bob outputs +1 , +1…unless he is asked 4b , in which case he answers +1 , –1.

Result: Alice and Bob always win.


+1,+1 ; +1, -1

-1, +1 ; -1, -1

Winning condition:

Possible answers:

λ1 , λ2 = +1,+1 λ1 , λ2 = +1,-1 λ1 , λ2 = -1,+1 λ1 , λ2 = -1,-1


+1,+1 ; +1, -1

-1, +1 ; -1, -1

Winning condition:

Possible answers:

Alice: λ1 , λ2

Bob: 1b. c , λ1λ2c 2b. c , λ1c 3b. c , λ1λ2c 4b. c , - λ1c

Result: Quantum mechanics!

coin toss+ =

‘EPR Bell Inequalities’

If Alice measures ‘Z2 = +1’, then she knows with certainty, without in any way disturbing Bob’s system, that Bob would record ‘Z4 = +1’ were he to measure it.

Fair enough, perhaps, but this prediction must be tested, not assumed!

So called ‘EPR Bell inequalities’ assume specific quantum behaviour without testing for it, because extra tests are experimentally expensive!

Extra tests would yield a valid experiment, but would reduce the magnitude of the local / nonlocal difference.

Logical structure of nonlocality proofs

Locality Realism+

Limits on (a,b,x,y)

Experiment breaks limits!

Locality Realismor

+

A B

a b

x y

(a,b,x,y)

Logical structure of ‘EPR Bell Inequalities’

Locality Realism+

Limits on (a,b,x,y)

Experiment breaks limits!

Locality Realismor

+

A B

a b

x y

(a,b,x,y)

Locality Realism+ + Quantum Predictions

Locality Realismor Quantum Predictionsor

Conclusions

• This is just one of a family of flawed nonlocality proofs in circulation.

• Specific local models for specific experiments can be hard to find. But there is a very simple way to check a proof: the logic must be watertight: If any additional assumptions are made and not tested, a local model will exist.

• This is true even when the additional assumptions are true! In this case, the additional noncontextuality assumptions are true. Yet making these assumptions allows trivially simply local models to pass the nonlocality test.

on the logical structure of bell theorems a. broadbent, h. carteret, a-a. méthot, j. walgate

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