I see that the PI workshop includes sessions on such things as Kochen-Specker (related to Bell's Theorem) and "non-locality for qubits".

What do the people there think of the work of Joy Christian ? Particularly, his papers quant-ph/0703179 which was written while he was at Perimeter and 0904.4259 [quant-ph] which was written while he was back at Oxford.

Tony, I have not read these papers and probably never will. If you look at the abstracts for next week you will see that there is a very wide range of views on these questions even within the category theory community.

One of them, by Bill Edwards, seems to be based on a paper by Coecke, Edwards and Spekkens entitled "The Group Theoretic Origin of Non-Locality for Qubits". Since that paper says in part:

"... What is nonlocality? ... The technical definition tells us that “there is no local hidden variable theory.” By Bell’s theorem this means that “some inequality is not satisfied.” ..."

it seems to me that Edwards et al base their work on the validity of Bell's Theorem so therefore they do not consider Joy Christian's "Disproof of Bell's Theorem ..." to be valid

Hi Tony. Right ... sort of. The point is that when logic and dagger monoidal categories are kings, and classical geometry is only emergent, one doesn't think of Bell's theorem as a point worth arguing. One could probably come up with a new definition of the words 'local and hidden' which address some particular concern that you may have, but why bother?

In trying to understand the paper by Coecke, Edwards and Spekkens that seems to be the basis for the talk by Edwards at the PI quantum workshop, it seems to me that a major point of the paper is to distinguish between stabilizer qubit quantum mechanics (Stab) with phase group Z4 and Spekkens's toy theory (Spek) with phase group Z2 x Z2,

which they try to do by showing that Stab must be non-local based on the GHZ version of the Bell-Type Theorems to as to which Joy Christian says in 0904.4259 that he has a "Disproof".

So, IF Joy Christian's GHZ "Disproof" applies to Stab, then IF Joy Christian is correct THEN the paper of Edwards et al is flawed.

As to whether Joy Christian's GHZ disproof applies to Stab, note that the basis of Joy Christian's argument is based on Pauli matrix quantum structure, and that Edwards et al use a version of GHZ that is effectively equivalent to the Pauli matrix version, since they say in part "... GHZ states and Pauli measurements both survive the restriction from full QM to the stabilizer theory [Stab], so the proof applies equally well ...".

So, the answer of "why bother?" is that if Joy Christian is correct, it seems that the paper of Edwards et al is flawed, and if Joy Christian is incorrect, then the paper of Edwards et al may be correct, so it seems useful to determine whether or not Joy Christian is correct.

Tony Smith

PS - Here is more detail of what Edwards et al say:

"... the mathematical difference between the theories is intimately related to one of their key physical differences: the presence or absence of non-locality. ... The GHZ correlations ... can take only two forms, corresponding to thge two four-element groups, Z4 (as in the case of Stab) and Z2 x Z2 (as in the case of Spek) ... The GHZ correlations ... are invoked in one of the most elegant 'no-go' proofs showing that quantum mechanics cannot be explained by a local hidden variable theory ... This no-go proof ... applies to stabilizer theory [Stab] ... The proof begins with a GHZ state. The key ingredients are the probabilities of outcomes ... GHZ states and Pauli measurements both survive the restriction from full QM to the stabilizer theory [Stab], so the proof applies equally well in this case, i.e. it is impossible to model stabilizer theory [Stab] with a hidden variable theory ... A hidden variable interpretation can be constructed for the GHZ state in any Z2 x Z2 MUQT ... we have a concrete example of a local hidden variable theory, Spek, which exhibits exactly these [GHZ] correlations ... we can see that the Z4 type basis structure ... embodies non-locality ... We then conclude that Z4 GHZ states must have non-locality, whereas Z2 x Z2 GHZ states can not ...".

Clifford algebra is the key to the whole premise of Joy Christian's disproof of Bell and Kochen-Specker Theorems. There are many who pronounce their opinion on the issue without understanding the fundementals of the Clifford formalism.

## 6 Comments:

I see that the PI workshop includes sessions on such things as

Kochen-Specker (related to Bell's Theorem)

and

"non-locality for qubits".

What do the people there think of the work of Joy Christian ?

Particularly, his papers

quant-ph/0703179

which was written while he was at Perimeter

and

0904.4259 [quant-ph]

which was written while he was back at Oxford.

Tony Smith

Tony, I have not read these papers and probably never will. If you look at the abstracts for next week you will see that there is a very wide range of views on these questions even within the category theory community.

Thanks for the link to the abstracts.

One of them, by Bill Edwards, seems to be based on a paper by Coecke, Edwards and Spekkens entitled "The Group Theoretic Origin of Non-Locality for Qubits".

Since that paper says in part:

"... What is nonlocality? ...

The technical definition tells us that “there is no local hidden variable theory.”

By Bell’s theorem this means that “some inequality

is not satisfied.” ..."

it seems to me that Edwards et al base their work on the validity of Bell's Theorem

so

therefore they do not consider Joy Christian's "Disproof of Bell's Theorem ..." to be valid

so

I guess that answers my question.

Tony Smith

Hi Tony. Right ... sort of. The point is that when logic and dagger monoidal categories are kings, and classical geometry is only emergent, one doesn't think of Bell's theorem as a point worth arguing. One could probably come up with a new definition of the words 'local and hidden' which address some particular concern that you may have, but why bother?

Kea, as to "why bother?":

In trying to understand the paper by Coecke, Edwards and Spekkens that seems to be the basis for the talk by Edwards at the PI quantum workshop,

it seems to me that a major point of the paper is to distinguish

between

stabilizer qubit quantum mechanics (Stab) with phase group Z4

and

Spekkens's toy theory (Spek) with phase group Z2 x Z2,

which they try to do by showing that Stab must be non-local based on the GHZ version of the Bell-Type Theorems to

as to which

Joy Christian says in 0904.4259 that he has a "Disproof".

So, IF Joy Christian's GHZ "Disproof" applies to Stab,

then

IF Joy Christian is correct THEN the paper of Edwards et al is flawed.

As to whether Joy Christian's GHZ disproof applies to Stab,

note that the basis of Joy Christian's argument is based on Pauli matrix quantum structure,

and

that Edwards et al use a version of GHZ that is effectively equivalent to the Pauli matrix version,

since they say in part

"... GHZ states and Pauli measurements both survive the restriction from full QM to the stabilizer theory [Stab], so the proof applies equally well ...".

So, the answer of "why bother?" is that if Joy Christian is correct,

it seems that the paper of Edwards et al is flawed,

and

if Joy Christian is incorrect,

then the paper of Edwards et al may be correct,

so

it seems useful to determine whether or not Joy Christian is correct.

Tony Smith

PS - Here is more detail of what Edwards et al say:

"... the mathematical difference between the theories is intimately related to one of their key physical differences: the presence or absence of non-locality.

...

The GHZ correlations ... can take only two forms, corresponding to thge two four-element groups,

Z4 (as in the case of Stab)

and

Z2 x Z2 (as in the case of Spek)

...

The GHZ correlations ... are invoked in one of the most elegant 'no-go' proofs showing that quantum mechanics cannot be explained by a local hidden variable theory ...

This no-go proof ... applies to stabilizer theory [Stab] ...

The proof begins with a GHZ state. The key ingredients are the probabilities of outcomes

...

GHZ states and Pauli measurements both survive the restriction from full QM to the stabilizer theory [Stab], so the proof applies equally well in this case,

i.e.

it is impossible to model stabilizer theory [Stab] with a hidden variable theory

...

A hidden variable interpretation can be constructed for the GHZ state in any Z2 x Z2 MUQT ... we have a concrete example of a local hidden variable theory, Spek, which exhibits exactly these [GHZ] correlations

...

we can see that the Z4 type basis structure ... embodies non-locality

...

We then conclude that Z4 GHZ states must have non-locality,

whereas

Z2 x Z2 GHZ states can not ...".

Clifford algebra is the key to the whole premise of Joy Christian's disproof of Bell and Kochen-Specker Theorems. There are many who pronounce their opinion on the issue without understanding the fundementals of the Clifford formalism.

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