2-2

David Mermin's "Quantum Mysteries"

Written 7 March 2007

We will need these tools:

P_⊗Q_:=Flatten[Table[Flatten[Table[Part[Outer[Times,P,Q],i,j,k],{j,Dimensions[P][[2]]}]],{i,Dimensions[P][[1]]},{k,Dimensions[Q][[1]]}],1]

(Subscript[σ, 0])=(({{1, 0}, {0, 1}}));
(Subscript[σ, 1])=(({{0, 1}, {1, 0}}));
(Subscript[σ, 2])=(({{0, -I}, {I, 0}}));
(Subscript[σ, 3])=(({{1, 0}, {0, -1}}));

Com[A_,B_]:=A.B-B.A

Null8=(({{0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}}));

Introduction:

Chapter 6 in Asher Peres' Quantum Theory: Concepts & Methods is entitled "Bell's Theorem," and §6-1 bears the subtitle "The dilemma of Einstein, Podolsky and Rosen." At page 152 in that section I encountered a short (and to me not very intelligible) summary of an argument due to N. David Mermin that I had encountered several times previously, but had never had time/motivation to try to trace out in detail. That is exactly what I will attempt to do here.
Mermin's work was inspired by (then unpublished) work of D.M. Greenberger, M.A. Horn & A. Zeilinger (1989) and first appeared under the title "What's wrong with these elements of reality" in Mermin's monthly column in Physics Today 43, 9 (June, 1990), and shortly thereafter, in a somewhat more colorful (but to me equally obscure) form, as "Quantum mysteries revisited," AJP 58, 731 (1990).
EPR imagined themselves to be demonstrating that quantum mechanics was "incomplete" because, in instances of the sort to which they drew attention, it failed to conform to what they considered to be essential basic features of any acceptable theory of physical reality—failed to conform to what has come to be called "local realism."
John Bell, building upon a sharpened version of the EPR model, developed certain inequalities that would have to be honored if quantum mechanics was to be in accordance with a "hidden variable theory" of the sort which EPR appeared to have in mind, and he left to experimentalists the non-trivial task of discovering whether or not those inequalities were in fact satisfied (they turned out not to be).
The EPR/Bell system contemplated a composite 2-particle system. GHZ—in work that Mermin reportedly considered to be "hot stuff" (see page 222 in A. D. Aczel's Entanglement (2001))—argued in reference to a 4-particle system, but without reference to inequalities.  Mermin's wonderful accomplishment was to produce a simpler 3-particle system that leads to the same conclusion. 
Mermin is deep into quantum computation these days, and allows himself to use shorthand notations that, while possibly standard to that field, are very confusing to me. My effort here will be to write out Mermin's argument in mathematical language that I can understand (but that he would doubtless consider to be insanely pedantic and explicit).

Remark added later:

GHZ appears on pages 73–76 in Bell's Theorem, Quantum Theory and Conceptions of the Universe (1989), edited by M. Kafatos. More readily accessible is D.M. Greenberger, M.A. Horn, A. Shimony & A. Zeilinger, "Bell's theorem without inequalities," AJP 58, 1131–1143 (1990).

Description of the system:

We have before us three distinct and distinguishable 2-state systems, named a, b and c. To be physically concrete we may—in the tradition of Bohm and Bell—think of them as spin one-half particles. Though born at the same moment of the same unspecified explosive event, they are now so widely separated that we can in principle consider them to be non-interactive, autonymous physical systems. But to make the argument work we will want ultimately to view them as the component parts of a composite quantum system in a certain highly entangled state (just as Bohm/Bell elected to contentrate on situations in which their two particles were in a singlet state).

Observables of interest:

We have in hand two measurement devices/meters designed to look at system a, two at system b, two at system c (or so we would say if we were concerned only with unentangled states):

(Subscript[A, 1])=((Subscript[σ, 1])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 0]);
(Subscript[A, 2])=((Subscript[σ, 2])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 0]);
(Subscript[B, 1])=((Subscript[σ, 0])⊗(Subscript[σ, 1]))⊗(Subscript[σ, 0]);
(Subscript[B, 2])=((Subscript[σ, 0])⊗(Subscript[σ, 2]))⊗(Subscript[σ, 0]);
(Subscript[C, 1])=((Subscript[σ, 0])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 1]);
(Subscript[C, 2])=((Subscript[σ, 0])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 2]);

The measurement devices can be used in any combination, since they do not interfere with each other (are represented by matrices that commute)

Com[(Subscript[A, 1]),(Subscript[B, 1])]==Null8
Com[(Subscript[A, 1]),(Subscript[B, 2])]==Null8
Com[(Subscript[A, 2]),(Subscript[B, 1])]==Null8
Com[(Subscript[A, 2]),(Subscript[B, 2])]==Null8
Com[(Subscript[A, 1]),(Subscript[C, 1])]==Null8
Com[(Subscript[A, 1]),(Subscript[C, 2])]==Null8
Com[(Subscript[A, 2]),(Subscript[C, 1])]==Null8
Com[(Subscript[A, 2]),(Subscript[C, 2])]==Null8
Com[(Subscript[B, 1]),(Subscript[C, 1])]==Null8
Com[(Subscript[B, 1]),(Subscript[C, 2])]==Null8
Com[(Subscript[B, 2]),(Subscript[C, 1])]==Null8
Com[(Subscript[B, 2]),(Subscript[C, 2])]==Null8

though the devices that look to any individual particle do interfere with one another:

Com[(Subscript[A, 1]),(Subscript[A, 2])]==Null8
Com[(Subscript[B, 1]),(Subscript[B, 2])]==Null8
Com[(Subscript[C, 1]),(Subscript[C, 2])]==Null8

To emphasize the point just established: each of our  6 observables commute with all members of other families, but not within their own families. Observations performed upon systems (Subscript[𝒮, a]), (Subscript[𝒮, b]) and (Subscript[𝒮, c]), since they do not interfere with one another, can be performed in any order. The observables look at composite states (represented by 8-vectors) and their ±1 meter readings are 4-fold degenerate:

Eigenvalues[(Subscript[A, 1])]
Eigenvalues[(Subscript[A, 2])]
Eigenvalues[(Subscript[B, 1])]
Eigenvalues[(Subscript[B, 2])]
Eigenvalues[(Subscript[C, 1])]
Eigenvalues[(Subscript[C, 2])]

We digress to construct the spectral resolutions of (Subscript[σ, 1]) and (Subscript[σ, 2]):

(Subscript[σ, 1])//MatrixForm
Eigenvalues[(Subscript[σ, 1])]
Eigenvectors[(Subscript[σ, 1])]

(Subscript[u, 1])=(({{(1/(Sqrt[2]))}, {(1/(Sqrt[2]))}}));
(Subscript[d, 1])=(({{(1/(Sqrt[2]))}, {-(1/(Sqrt[2]))}}));

Here are the projectors onto those eigenstates

(Subscript[P, 1u ])=(Subscript[u, 1]).Transpose[(Subscript[u, 1])];
(Subscript[P, 1d ])=(Subscript[d, 1]).Transpose[(Subscript[d, 1])];

(Subscript[P, 1u])//MatrixForm
(Subscript[P, 1d])//MatrixForm

which information we use to obtain the spectral resolution of the hermitian matrix (Subscript[σ, 1]):

(Subscript[σ, 1])==(+1)(Subscript[P, 1u])+(-1)(Subscript[P, 1d])

Ditto for (Subscript[σ, 2]):

(Subscript[σ, 2])//MatrixForm
Eigenvalues[(Subscript[σ, 2])]
Eigenvectors[(Subscript[σ, 2])]

(Subscript[u, 2])=(({{(1/(Sqrt[2]))}, {(I/(Sqrt[2]))}}));
(Subscript[d, 2])=(({{(1/(Sqrt[2]))}, {-(I/(Sqrt[2]))}}));

We verify that those eigenvectors are orthonormal:

Transpose[Conjugate[(Subscript[u, 2])]].(Subscript[u, 2])
Transpose[Conjugate[(Subscript[d, 2])]].(Subscript[d, 2])
Transpose[Conjugate[(Subscript[u, 2])]].(Subscript[d, 2])

(Subscript[P, 2u ])=(Subscript[u, 2]).Transpose[Conjugate[(Subscript[u, 2])]];
(Subscript[P, 2d ])=(Subscript[d, 2]).Transpose[Conjugate[(Subscript[d, 2])]];

(Subscript[P, 2u])//MatrixForm
(Subscript[P, 2d])//MatrixForm

(Subscript[P, 2u]).(Subscript[P, 2u])-(Subscript[P, 2u])//MatrixForm
(Subscript[P, 2d]).(Subscript[P, 2d])-(Subscript[P, 2d])//MatrixForm
(Subscript[P, 2u]).(Subscript[P, 2d])//MatrixForm

(Subscript[σ, 2])==(+1)(Subscript[P, 2u])+(-1)(Subscript[P, 2d])

We are in position now to obtain spectral resolutions of the A/B/C matrices that represent our 6 instruments, this being information basic to the description of actual measurement processes.

Spectral resolution of the A, B and C meters (of which each comes in two flavors):

From

(Subscript[A, 1])=((Subscript[σ, 1])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 0]);
(Subscript[A, 2])=((Subscript[σ, 2])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 0]);
(Subscript[B, 1])=((Subscript[σ, 0])⊗(Subscript[σ, 1]))⊗(Subscript[σ, 0]);
(Subscript[B, 2])=((Subscript[σ, 0])⊗(Subscript[σ, 2]))⊗(Subscript[σ, 0]);
(Subscript[C, 1])=((Subscript[σ, 0])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 1]);
(Subscript[C, 2])=((Subscript[σ, 0])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 2]);

and the results just obtained, we have

(Subscript[A, 1])==(+1)((Subscript[P, 1u])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 0])+(-1)((Subscript[P, 1d])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 0])
(Subscript[A, 2])==(+1)((Subscript[P, 2u])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 0])+(-1)((Subscript[P, 2d])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 0])
(Subscript[B, 1])==(+1)((Subscript[σ, 0])⊗(Subscript[P, 1u]))⊗(Subscript[σ, 0])+(-1)((Subscript[σ, 0])⊗(Subscript[P, 1d]))⊗(Subscript[σ, 0])
(Subscript[B, 2])==(+1)((Subscript[σ, 0])⊗(Subscript[P, 2u]))⊗(Subscript[σ, 0])+(-1)((Subscript[σ, 0])⊗(Subscript[P, 2d]))⊗(Subscript[σ, 0])
(Subscript[C, 1])==(+1)((Subscript[σ, 0])⊗(Subscript[σ, 0]))⊗(Subscript[P, 1u])+(-1)((Subscript[σ, 0])⊗(Subscript[σ, 0]))⊗(Subscript[P, 1d])
(Subscript[C, 2])==(+1)((Subscript[σ, 0])⊗(Subscript[σ, 0]))⊗(Subscript[P, 2u])+(-1)((Subscript[σ, 0])⊗(Subscript[σ, 0]))⊗(Subscript[P, 2d])

Each instrumental matrix is the weighted sum of two projectors, which we here define:

(Subscript[PA, 1u])=((Subscript[P, 1u])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 0]);
(Subscript[PA, 1d])=((Subscript[P, 1d])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 0]);
(Subscript[PA, 2u])=((Subscript[P, 2u])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 0]);
(Subscript[PA, 2d])=((Subscript[P, 2d])⊗(Subscript[σ, 0]))⊗(Subscript[σ, 0]);
(Subscript[PB, 1u])=((Subscript[σ, 0])⊗(Subscript[P, 1u]))⊗(Subscript[σ, 0]);
(Subscript[PB, 1d])=((Subscript[σ, 0])⊗(Subscript[P, 1d]))⊗(Subscript[σ, 0]);
(Subscript[PB, 2u])=((Subscript[σ, 0])⊗(Subscript[P, 2u]))⊗(Subscript[σ, 0]);
(Subscript[PB, 2d])=((Subscript[σ, 0])⊗(Subscript[P, 2d]))⊗(Subscript[σ, 0]);
(Subscript[PC, 1u])=((Subscript[σ, 0])⊗(Subscript[σ, 0]))⊗(Subscript[P, 1u]);
(Subscript[PC, 1d])=((Subscript[σ, 0])⊗(Subscript[σ, 0]))⊗(Subscript[P, 1d]);
(Subscript[PC, 2u])=((Subscript[σ, 0])⊗(Subscript[σ, 0]))⊗(Subscript[P, 2u]);
(Subscript[PC, 2d])=((Subscript[σ, 0])⊗(Subscript[σ, 0]))⊗(Subscript[P, 2d]);

(Subscript[A, 1])==(+1)(Subscript[PA, 1u])+(-1)(Subscript[PA, 1d])
(Subscript[A, 2])==(+1)(Subscript[PA, 2u])+(-1)(Subscript[PA, 2d])
(Subscript[B, 1])==(+1)(Subscript[PB, 1u])+(-1)(Subscript[PB, 1d])
(Subscript[B, 2])==(+1)(Subscript[PB, 2u])+(-1)(Subscript[PB, 2d])
(Subscript[C, 1])==(+1)(Subscript[PC, 1u])+(-1)(Subscript[PC, 1d])
(Subscript[C, 2])==(+1)(Subscript[PC, 2u])+(-1)(Subscript[PC, 2d])

Construction of Mermin's input state:

We write

up=(({{1}, {0}}));
dn=(({{0}, {1}}));

to denote the eigenstates of (Subscript[σ, 3])

(Subscript[σ, 3]).up==(+1)up
(Subscript[σ, 3]).dn==(-1)dn

and, with Mermin, look to the several things that can happen when the composite pure state

Ψ=(((up⊗up)⊗up-(dn⊗dn)⊗dn)/(Sqrt[2]));

Ψ//MatrixForm

is presented to A, B and C—in, let us say (though it has been shown not matter), that order. Near the end of this essay I will describe the considerations that serve to motivate the introduction of Ψ.

Remark added later:

I have attributed to Mirman a lot of stuff that he actually took directly from Greenberger, Horn & Zeilinger. In particular, the state that I have just above attributed to Mirman is at page 173 in G. Greenstein & A. G. Zajonc, The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics (2006) called

(Subscript[Ψ, GHZ])

and the argument that they present (with attribution to GHZ) is precisely the argument that Mirman gives, though Mirman seems to want us to think that his argument is a "simplification" of the GHZ argument. Mirman's  contribution appears to have consisted mainly of the introduction of Red and Green into his more vivid presentation of GHZ's argument.

Measurement protocols:

Each of three instruments can be selected from either of two alternatives, so there are (2^3)=8 distinct measurement sequences: these I denote

111
112
121
122
211
212
221
222

where, for example, 121 signals our intention to make our measurements first with  (Subscript[A, 1]), then with  (Subscript[B, 2]), then with  (Subscript[C, 1]).

NOTE: A more attractive notational scheme is provided by the binary numbers

000
001
010
011
100
101
110
111

Mermin restricts his attention to the red cases, in which the number of 1s is odd. He asserts (without argument, but as will emerge) that the other cases are "unremarkable, and of no relevance for the argument that follows."

Data sets:

Select a protocol. An experimental run will produce one or another of eight possible data sets: 
	• the selected A-meter will announce ±1, then
	•the selected B-meter will announce ±1, finally
	• the selected C-meter will announce ±1.

Mermin supposes the meters to have been so constructed that +1 turns on a RED LIGHT, -1 turns on a GREEN LIGHT, and writes

RRR
RRG
RGR
RGG
GRR
GRG
GGR
GGG

to describe the respective data sets.

Expected results of successive ABC measurements:

The computation here is made tedious by the circumstance that it involves sub-sub-cases of sub-cases of cases, and will be made easier by introduction of these definitions:

Prob[P_,ψ_]:=(Transpose[Conjugate[P.ψ]].(P.ψ))[[1]][[1]]
Prepared[P_,ψ_]:=((P.ψ)/(Sqrt[(Transpose[Conjugate[P.ψ]].(P.ψ))[[1]][[1]]]))

The former computes the probability that a projection operator P, when presented with a state ψ, will act to pluck out its component of the incident state. The latter describes the normalized state that the projector has then prepared and shipped out.

Expected results in the case 122:

The A-meter announces R with probability

Prob[(Subscript[PA, 1u]),Ψ]

and ships to B the state

ΨR=Prepared[(Subscript[PA, 1u]),Ψ]

We check the normalization:

(Transpose[Conjugate[ΨR]].ΨR)[[1]][[1]]

In all other cases the A-meter announces G with probability

Prob[(Subscript[PA, 1d]),Ψ]

and ships the state

ΨG=Prepared[(Subscript[PA, 1d]),Ψ]

(Transpose[Conjugate[ΨG]].ΨG)[[1]][[1]]

So far as the A-meter is concerned, we have, in summary

ProbR=(1/2);
ProbG=(1/2);

It is gratifying to observe that

ProbR+ProbG

Pursuing now the R branch (which is to say: the former alternative), the B-meter announces R with probability

Prob[(Subscript[PB, 2u]),ΨR]

and ships the state

ΨRR=Prepared[(Subscript[PB, 2u]),ΨR]

(Transpose[Conjugate[ΨRR]].ΨRR)[[1]][[1]]

Else the B-meter announces G with probability

Prob[(Subscript[PB, 2d]),ΨR]

and ships the state

ΨRG=Prepared[(Subscript[PB, 2d]),ΨR]

(Transpose[Conjugate[ΨRG]].ΨRG)[[1]][[1]]

Pursuing now the G branch of the A-meter's output, when the B-meter announces R we have

Prob[(Subscript[PB, 2u]),ΨG]

ΨGR=Prepared[(Subscript[PB, 2u]),ΨG]

(Transpose[Conjugate[ΨGR]].ΨGR)[[1]][[1]]

while when the B-meter announces G we have

Prob[(Subscript[PB, 2d]),ΨG]

ΨGG=Prepared[(Subscript[PB, 2d]),ΨG]

(Transpose[Conjugate[ΨGG]].ΨGG)[[1]][[1]]

In summary, we have

ProbRR=(1/2);
ProbRG=(1/2);
ProbGR=(1/2);
ProbGG=(1/2);

Tracing now the RR branch to the C-meter: when the C-meter announces R we have

Prob[(Subscript[PC, 2u]),ΨRR]

Also

ΨRRR=Prepared[(Subscript[PC, 2u]),ΨRR]

(Transpose[Conjugate[ΨRRR]].ΨRRR)[[1]][[1]]

but, since we do not plan to make measurements on the C-meter's output, those facts are of no particular relevance, and will henceforth be omitted. Placing the preceding result in the company of its companions, we have

ProbRRR=Prob[(Subscript[PC, 2u]),ΨRR]
ProbRRG=Prob[(Subscript[PC, 2d]),ΨRR]
ProbRGR=Prob[(Subscript[PC, 2u]),ΨRG]
ProbRGG=Prob[(Subscript[PC, 2d]),ΨRG]
ProbGRR=Prob[(Subscript[PC, 2u]),ΨGR]
ProbGRG=Prob[(Subscript[PC, 2d]),ΨGR]
ProbGGR=Prob[(Subscript[PC, 2u]),ΨGG]
ProbGGG=Prob[(Subscript[PC, 2d]),ΨGG]

The probability of any specified sequence of three flashes can now be read from the exhaustive list

ProbR*ProbRR*ProbRRR
ProbR*ProbRR*ProbRRG
ProbR*ProbRG*ProbRGR
ProbR*ProbRG*ProbRGG
ProbG*ProbGR*ProbGRR
ProbG*ProbGR*ProbGRG
ProbG*ProbGG*ProbGGR
ProbG*ProbGG*ProbGGG

Here ProbG*ProbGR*ProbGRG (for example) is to be read as the probability of "green followed by red followed by green." The eight probabilities listed exhaust the possibilities, and gratifyingly add to one.

Mermin directs our attention to the fact that the only possibilities (and they occur with equal likelyhood) are

RRR
RGG
GRG
GGR

—in each of which R occurs an odd number of times. This when our triplex measurement device has been set to the 122 configuration. But from the fact—already established—that the A, B and C matrices commute across family lines it follows that a similar result would have been obtained had we—instead of

(Subscript[A, 1])(Subscript[B, 2])(Subscript[C, 2])

—used either of the following devices:

(Subscript[A, 2])(Subscript[B, 1])(Subscript[C, 2])
(Subscript[A, 2])(Subscript[B, 2])(Subscript[C, 1])

Observe (again with Mermin) that if we looked to the output of only to two of the devices (B and C, say) we would—after drawing the obvious conclusion from a long exploratory series of observations—always be able to infer the state of A:

XRR⟹X=R
XGG⟹X=R
XRG⟹X=G
XGR⟹X=G

Seeming physical implications:

Adopting the voice of EPR, Mermin argues (tentatively) that our ability to make such deductions among objects not in physical communication can only mean that the members of the trio were stamped at birth with instructions—instructions that they carry with them—concerning how they are to behave when they encounter one of our meters. He writes, for example,

(({{R, G, G}, {G, R, R}}))

to indicate that
	• particle a is instructed to register R if it encounters a meter set to position #1 (an (Subscript[A, 1])-meter, or a (Subscript[B, 1])-meter, or a (Subscript[C, 1])-meter) but 	   to register G if it encounters a meter set to position #2; 
	• particle b is instructed to register G if it encounters an #1-meter, but R if it encounters an #2-meter;  
	• particle c is instructed to register G if it encounters an #1-meter, but R if it encounters an #2-meter. 
There are, all together, (2^6)=64 possible instruction sets but, as he argues (I will not reproduce his straightforward argument), only 8 of those 64 conform to the observed facts: they are

(({{R, R, R}, {R, R, R}}))
(({{R, G, G}, {R, G, G}}))
(({{G, R, G}, {G, R, G}}))
(({{G, G, R}, {G, G, R}}))
(({{R, R, R}, {G, G, G}}))
(({{R, G, G}, {G, R, R}}))
(({{G, R, G}, {R, G, R}}))
(({{G, G, R}, {R, R, G}}))

The pay-off:

Mermin would have us look finally to the instrumental configuration 111. Again we have

Prob[(Subscript[PA, 1u]),Ψ]
ΨR=Prepared[(Subscript[PA, 1u]),Ψ]

Prob[(Subscript[PA, 1d]),Ψ]
ΨG=Prepared[(Subscript[PA, 1d]),Ψ]

Results at the next two stages are, however, modified (we have replaced our old instruments with new ones):

Prob[(Subscript[PB, 1u]),ΨR]
ΨRR=Prepared[(Subscript[PB, 1u]),ΨR]

Prob[(Subscript[PB, 1d]),ΨR]
ΨRG=Prepared[(Subscript[PB, 1d]),ΨR]

Prob[(Subscript[PB, 1u]),ΨG]
ΨGR=Prepared[(Subscript[PB, 1u]),ΨG]

Prob[(Subscript[PB, 1d]),ΨG]
ΨGG=Prepared[(Subscript[PB, 1d]),ΨG]

Those results place us in position finally to compute

NewProbRRR=Prob[(Subscript[PC, 1u]),ΨRR]
NewProbRRG=Prob[(Subscript[PC, 1d]),ΨRR]
NewProbRGR=Prob[(Subscript[PC, 1u]),ΨRG]
NewProbRGG=Prob[(Subscript[PC, 1d]),ΨRG]
NewProbGRR=Prob[(Subscript[PC, 1u]),ΨGR]
NewProbGRG=Prob[(Subscript[PC, 1d]),ΨGR]
NewProbGGR=Prob[(Subscript[PC, 1u]),ΨGG]
NewProbGGG=Prob[(Subscript[PC, 1d]),ΨGG]

Here I highlight the cases that display an odd number of reds (which formerly were the only possible outcomes):

ProbR*ProbRR*NewProbRRR
ProbR*ProbRR*NewProbRRG
ProbR*ProbRG*NewProbRGR
ProbR*ProbRG*NewProbRGG
ProbG*ProbGR*NewProbGRR
ProbG*ProbGR*NewProbGRG
ProbG*ProbGG*NewProbGGR
ProbG*ProbGG*NewProbGGG

0

0

0

0

Again, the possible outcomes are equally likely, and their probabilities add to one.

If "instruction sets" were real attributes of such particle triplets then we would expect to obtain an odd number of reds however the ABC meters are set: that is the "local realist" implication of data obtained when we subjected Mermin's carefully-selected Ψ state to the 
	• (Subscript[A, 1])then (Subscript[B, 2])then (Subscript[C, 2])measurement protocol—equivalently, the
	• (Subscript[A, 2])then (Subscript[B, 1])then (Subscript[C, 2])protocol, or the
	• (Subscript[A, 2])then (Subscript[B, 2])then (Subscript[C, 1])protocol.
But it is a conclusion flatly contradicted by results of the 
	• (Subscript[A, 1])then (Subscript[B, 1])then (Subscript[C, 1])protocol.
Here—without fussing with the experimental test of Bell inequalities—the data produced by a  single measurement sequence would serve to render untenable a proposition that EPR hold to be one of the most basic "necessary properties of real things."

Mermin takes us "behind the scenes":

We observed early on that

Com[(Subscript[A, 1]),(Subscript[B, 1])]==Null8
Com[(Subscript[A, 1]),(Subscript[B, 2])]==Null8
Com[(Subscript[A, 2]),(Subscript[B, 1])]==Null8
Com[(Subscript[A, 2]),(Subscript[B, 2])]==Null8
Com[(Subscript[A, 1]),(Subscript[C, 1])]==Null8
Com[(Subscript[A, 1]),(Subscript[C, 2])]==Null8
Com[(Subscript[A, 2]),(Subscript[C, 1])]==Null8
Com[(Subscript[A, 2]),(Subscript[C, 2])]==Null8
Com[(Subscript[B, 1]),(Subscript[C, 1])]==Null8
Com[(Subscript[B, 1]),(Subscript[C, 2])]==Null8
Com[(Subscript[B, 2]),(Subscript[C, 1])]==Null8
Com[(Subscript[B, 2]),(Subscript[C, 2])]==Null8

i.e., that both A matrices commute with both B matrices, which commute with both C matrices, which commute with both A matrices.

Short digression:

It should therefore be possible to simultaneously diagonalize all of them. All have identical spectra, of which the spectrum of (Subscript[A, 1]) is (as we also saw early on) typical:

Eigenvalues[(Subscript[A, 1])]

One can create a total of

((8*7*6*5)/(4!))

diagonal 8×8 matrices that have four +1s distributed on the diagonal (and the remaining slots filled with -1s). Our population of diagonalized A/B/C matrices would be drawn from that population, leaving 64 players on the bench. It would be fun/instructive to work out the details, but I won't. End of digression.

From the matrices describing operations that were previously performed sequentially on the respective members of the particle trio let us now construct the products

((Subscript[A, 1]).(Subscript[B, 2]).(Subscript[C, 2])),((Subscript[A, 2]).(Subscript[B, 1]).(Subscript[C, 2])),((Subscript[A, 2]).(Subscript[B, 2]).(Subscript[C, 1]))

These describe measurement operations that would be performed wholistically on the composite trio. We are not surprised to have

Com[((Subscript[A, 1]).(Subscript[B, 2]).(Subscript[C, 2])),((Subscript[A, 2]).(Subscript[B, 1]).(Subscript[C, 2]))]==Null8
Com[((Subscript[A, 2]).(Subscript[B, 1]).(Subscript[C, 2])),((Subscript[A, 2]).(Subscript[B, 2]).(Subscript[C, 1]))]==Null8
Com[((Subscript[A, 2]).(Subscript[B, 2]).(Subscript[C, 1])),((Subscript[A, 1]).(Subscript[B, 2]).(Subscript[C, 2]))]==Null8

and

Eigenvalues[(Subscript[A, 1]).(Subscript[B, 2]).(Subscript[C, 2])]
Eigenvalues[(Subscript[A, 2]).(Subscript[B, 1]).(Subscript[C, 2])]
Eigenvalues[(Subscript[A, 2]).(Subscript[B, 2]).(Subscript[C, 1])]

Looking to

Eigenvectors[(Subscript[A, 1]).(Subscript[B, 2]).(Subscript[C, 2])]
Eigenvectors[(Subscript[A, 2]).(Subscript[B, 1]).(Subscript[C, 2])]
Eigenvectors[(Subscript[A, 2]).(Subscript[B, 2]).(Subscript[C, 1])]

we discover that

Φ=(({{(1/(Sqrt[2]))}, {0}, {0}, {0}, {0}, {0}, {0}, {(1/(Sqrt[2]))}}));

is the only simultaneous eigenvector with a negative unit eigenvalue,

((Subscript[A, 1]).(Subscript[B, 2]).(Subscript[C, 2])).Φ==-Φ
((Subscript[A, 2]).(Subscript[B, 1]).(Subscript[C, 2])).Φ==-Φ
((Subscript[A, 2]).(Subscript[B, 2]).(Subscript[C, 1])).Φ==-Φ

while Mermin's

Ψ=(({{(1/(Sqrt[2]))}, {0}, {0}, {0}, {0}, {0}, {0}, {-(1/(Sqrt[2]))}}))

is the only simultaneous eigenvector with positive unit eigenvalue:

((Subscript[A, 1]).(Subscript[B, 2]).(Subscript[C, 2])).Ψ==+Ψ
((Subscript[A, 2]).(Subscript[B, 1]).(Subscript[C, 2])).Ψ==+Ψ
((Subscript[A, 2]).(Subscript[B, 2]).(Subscript[C, 1])).Ψ==+Ψ

Short digression:

Having introduced

up=(({{1}, {0}}));
dn=(({{0}, {1}}));

we earlier had

Ψ=(((up⊗up)⊗up-(dn⊗dn)⊗dn)/(Sqrt[2]))

We should perhaps not be surprised, therefore, to discover that

Φ==(((up⊗up)⊗up+(dn⊗dn)⊗dn)/(Sqrt[2]))

End of digression.

This fact clarifies the origin of the "odd number of Rs" rule. For—whatever the initial state of the system—
	• (Subscript[A, 1]) measurement produce ±1 (which we called R/G)
	• subsequent  (Subscript[B, 2]) measurement again produce ±1
	• subsequent  (Subscript[C, 2]) measurement again produce ±1.
If the initial state is Mermin's Ψ then, however one picks the sign-options, it must (as just demonstrated) be the case that 
	(±1)*(±1)*(±1) = +1

Immediately, if one picks a -1 one must pick also another, so the number of +1s (Rs as Mermin calls them) can be only one or three, but is in both cases odd.

Look finally to wholistic measurement the matrix

((Subscript[A, 1]).(Subscript[B, 1]).(Subscript[C, 1]))

We find (note the sign flip) that

((Subscript[A, 1]).(Subscript[B, 1]).(Subscript[C, 1])).Ψ==-Ψ

This Mermin traces to the observation that

((Subscript[A, 1]).(Subscript[B, 2]).(Subscript[C, 2])).((Subscript[A, 2]).(Subscript[B, 1]).(Subscript[C, 2])).((Subscript[A, 2]).(Subscript[B, 2]).(Subscript[C, 1]))==-((Subscript[A, 1]).(Subscript[B, 1]).(Subscript[C, 1]))

and which I trace back to the statements

((Subscript[A, 1]).(Subscript[B, 2]).(Subscript[C, 2])).((Subscript[A, 2]).(Subscript[B, 1]).(Subscript[C, 2])).((Subscript[A, 2]).(Subscript[B, 2]).(Subscript[C, 1]))==(((Subscript[σ, 1])⊗(Subscript[σ, 2]))⊗(Subscript[σ, 2])).(((Subscript[σ, 2])⊗(Subscript[σ, 1]))⊗(Subscript[σ, 2])).(((Subscript[σ, 2])⊗(Subscript[σ, 2]))⊗(Subscript[σ, 1]))

(((Subscript[σ, 1])⊗(Subscript[σ, 2]))⊗(Subscript[σ, 2])).(((Subscript[σ, 2])⊗(Subscript[σ, 1]))⊗(Subscript[σ, 2])).(((Subscript[σ, 2])⊗(Subscript[σ, 2]))⊗(Subscript[σ, 1]))==(((Subscript[σ, 1]).(Subscript[σ, 2]).(Subscript[σ, 2]))⊗((Subscript[σ, 2]).(Subscript[σ, 1]).(Subscript[σ, 2])))⊗((Subscript[σ, 2]).(Subscript[σ, 2]).(Subscript[σ, 1]))

(Subscript[σ, 1]).(Subscript[σ, 2]).(Subscript[σ, 2])==(Subscript[σ, 1])
(Subscript[σ, 2]).(Subscript[σ, 1]).(Subscript[σ, 2])==-(Subscript[σ, 1])
(Subscript[σ, 2]).(Subscript[σ, 2]).(Subscript[σ, 1])==(Subscript[σ, 1])

Which is interesting, but somewhat beside the point—the point being that to achieve

((Subscript[A, 1]).(Subscript[B, 1]).(Subscript[C, 1])).Ψ==-Ψ

the number of -1s (Gs) would have to be odd, forcing the number of +1s (Rs) to be even. But such a condition is flatly disallowed by the exhaustive list of "instruction sets" that we, as EPR local realists, were forced by our initial set of observations to ascribe to particle triples.

Addenda

I have (on the web) stumbled accidentally upon the information that, in a LETTER TO THE EDITOR, J. Savinien, J. Taron & R. Tarrach ("Triviality of GHZ operators of higher spin," J. Phys. A: Math. Gen. 33, No 48 (8 December 2000) L493 – L495) establish that when one passes from spin 1/2 to higher spins 1, 3/2,… "no new contradictions with local realism arise." One simply gets direct sum amplifications of the spin 1/2 theory, which is revealed therefore to be essentially unique.

For a good short account of the meaning of "local realism" see §§15–19 in Chapter 22 of David Bohm's Quantum Theory, which are reproduced as "The paradox of Einstein, Rosen and Podolsky" at page 356 in J. A. Wheeler & W. H. Zurek (editors), Quantum Theory and Measurement (1983).

G. Greenstein & A.G. Zajonc, on pages 177–183 in The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics (2006), describe experimental realizations of GHZ entanglement. The work was done by Zeilinger et al in 1999 and 2000.

This subject was taken up again in Chapter 6, §E of Mirman's Lecture Notes on Quantum Computation (2006). And in several of the essays reprinted in the Quantum Theory section of Boojums All the Way Through (1990).