The Isomorphism of 3-Qubit Hadamards and (Subscript[E, 8])

J Gregory Moxness

CC BY - NC - SA 4.0 DEED/Attribution-NonCommercial-ShareAlike 4.0 International

www.TheoryOfEverything . org

This paper presents several notable properties of the matrix 𝕌 shown to be related to the isomorphism between $H_4$ and $E_8$. The most significant of these properties is that 𝕌.𝕌 is to rank 8 matrices what the golden ratio is to numbers. That is to say, the difference between it and its inverse is the identity element, albeit with a twist. Specifically, 𝕌.𝕌-(((𝕌.𝕌))^(-1)) is the reverse identity matrix or standard involutory permutation matrix of rank 8. It has the same palindromic characteristic polynomial coefficients as the normalized 3-qubit Hadamard matrix with 8-bit binary basis states, which is known to be isomorphic to E8 through its (8,4) Hamming code.

Initializations (snippets)

General Global Initialization

Standard & Others’ Packages

(* Define conditionally applied doPrint, doQuiet and doOff processing *)
print:=If[doPrint,Print@##]&;

quiet[in_]:=If[doQuiet,Quiet[in],in];
Attributes[quiet]={HoldAll};

off[sym_]:=(
print["doOff=",doOff," off=",sym];
If[doOff,Off@sym]);

<< MultivariateStatistics`; 
<< ComputationalGeometry`;

(**)
If[doSuperLie,<<Quaternions`];

Off[General::munfl];

<<NDSolve`FEM`

(* Tetrahedral Mesh Generator package *)
<<TetGenLink`

Off[Set::shape];

Off[TetGenConvexHull::tetfc];

Off[Volume::nmet];

off[Volume::reg];

off[RegionCentroid::reg];

off[ViewPoint::nlist3];

Off[Export::autofix];

Off[General::stop];

Off[AffineTransform::inpf];

Off[BinaryWrite::nocoerce];

Off[XMLElement::attrhs];

off[HighlightMesh::meshreg];

Off[PropertyValue::pvobj];
(**)

off[ConvexHullMesh::affind]

off[ConvexHullMesh::pts]

Off[Show::gtype]
Off[Show::gcomb]
Off[Show::shx]

Off[TetGenTetrahedralize::reterr]

Off[TetGenDelaunay::tetgpts]

This is for either LieART or GroupMath.
Caution : It uses names that can step on my functions unless modified e.g. {An, Bn, Cn, Dn, E(4,5,6,7,8),F4,G2}

(* SuperLie Algebra **)
If[doSuperLie&&working,
off[Solve::svars];
off[ToExpression::sntx]]
(* This is a replacement rule that corrects my code that uses undefined symbols (which SuperLie replaces) *)
slRep:=If[doSuperLie,ToExpression@"{Global`GPlus->Global`Plus,Global`GTimes->Global`Times,Global`GPower->Global`Power,GPlus->Plus,GTimes->Times,GPower->Power}",xxx->1];

(* LieART **)
If[doLieART&&working,
<<LieART`];

(* GroupTheory **)
If[doGroupTheory&&working,
off[Transpose::nmtx];
<<GroupTheory`];

If[doGroupTheory,
Unprotect[ex,a,b,c,d,F4]];

(* GroupMath **)
If[working&&doGroupMath,
(*)Unprotect@C;**)
Unprotect@BlockDiagonalMatrix;
(*)Clear[a,d,f,(* group names *)
e6,e7,e8,so10,so11,so12,so13,so14,so15,so16,so17,so3,so5,so6,so7,so8,so9,su2,su3,su4,su5,su6,su7,su8,u1];**)
<<GroupMath`;(*)
<<Sym2Int`**)];

Global

first=If[Length@#==0||#==={{}},#,First@Flatten@#]&;
position[x_,y_]:=first@Position[x,y,1,Heads->False];

(* The symbol φ is the main reference for resolving the Golden Ratios. 
It is only evaluated to a symbolic or numeric value using φRep replacement rules *)
Clear[ϕ,σ,Φ,τ,φ];
(* Upper (and lower) case Phi(Φ=(1+(Sqrt[5]))/2) (and phi ϕ=(1-(Sqrt[5]))/2)) represent large (and small) Golden Ratio (respectively) *)
ϕ=1/φ;
(* σ & τ are Galois conjugates, this will get stepped on from the CKM UI *)
σ=-1/φ/.slRep;
τ=Φ=φ;

(* 𝕌Det1f is a scale factor to make Det@𝕌=1 *)𝕌Det1f=2(Sqrt[ϕ]) /.slRep;

(* Create assumptions and replacement rules for φ and ω for automated symbolics *)
φAssumptions:={φ∈(Reals),(φ^(1/2))∈(Reals),(φ^(3/2))∈(Reals),N[((Sqrt[5])+1)/2]≥φ>N[((Sqrt[5])+1)/2]-(chop^2)(*),ω∈(Complexes)**)};
φRep:={φ->((Sqrt[5])+1)/2,ω->(E^(I 2π/3))};

(* GroupMath steps on CircleTimes *)
If[working&&!doGroupMath,
CircleTimes[a_,b_]:=KroneckerProduct[a,b]];

KroneckerSum[a_,b_]/;MatrixQ[a]&&MatrixQ[b]:=Catch@
Module[{n,p,m,q},
{n,p}=Dimensions[a];
{m,q}=Dimensions[b];
If[n≠p||m≠q,Throw[Failed]];
KroneckerProduct[a,IM@m]+
KroneckerProduct[IM@n,b]];

fb:=Fibonacci;

σP@i_:=PauliMatrix@i;
MatrixForm[σP@#⊗IdentityMatrix@#]&/@Range@3

(* Tetration as (^n)a := (a^(a^(a^(...n)))) *)
tetration[a_?NumericQ,0]:=1;
tetration[a_?NumericQ,n_]:=(a^(Sign@n tetration[a,n-Sign@n]));
tetration[a_?ListQ,n_]:=MatrixPower[a,n];
MakeExpression[RowBox[{SuperscriptBox["",n_],a_}],StandardForm]:=
MakeExpression[RowBox[{"tetration","[",a,",",n,"]"}],StandardForm];

n=4;
Table[(^i)2,{i,-n,n}]

(* Fibonacci *)
(^6)(({{1, 1}, {1, 0}}))

Clear[fF,hyperV]
(* Complex Double Factorial *)
fF@k_:=(Sqrt[((2^(k+1))/π)])(k/2)!;

fF@Range[0,8]

(* HyperDimensional volumes for nSpheres (𝕊^(n-1)) *)
hyperV[n_,r_:1]:=((2 (((2π))^((n-1)/2)))/(fF@n))(r^n);
hyperV[#,r]&/@Range@8

hyperV/@Range@8

fixDo@initGlobal;

Section I: Introduction

Introduction

Fig. 1 is the Petrie projection of the Gosset (Subscript[4, 21]) 8-polytope derived from the Split Real Even (SRE) form of the (Subscript[E, 8]) Lie group with unimodular lattice in (ℝ^8) It has 240 vertices and 6,720 edges of 8-dimensional (8D) length (Sqrt[2]). (Subscript[E, 8]) is the largest of the exceptional simple Lie algebras, groups, lattices, and polytopes related to octonions (𝕆 ), (8,4) Hamming codes, and 3-qubit (8 basis state) Hadamard matrix gates. An important and related higher dimensional structure is the (ℝ^24)((ℂ^24)) Leech lattice ((Subscript[Λ, 24])⊃ (Subscript[E, 8]) ♁ (Subscript[E, 8]) ♁ (Subscript[E, 8]) ) with its binary (ternary) Golay code construction.

It has been shown[1] that the matrix 𝕌 in (1) along with its inverse (2) is related to the isomorphism between (Subscript[H, 4]) and (Subscript[E, 8]).

Equations (1) and (2):

(* Note: the two 120 print outputs confirm by evaluation the (Subscript[H, 4]) & (Subscript[φH, 4]) 600-Cell membership in (Subscript[E, 8]) **)
curr𝕌=9;set𝕌;
2(Sqrt[φ])𝕌//MatrixForm

(* Note: octSym performs a symbolic simplification related to φ ratios *)
octSym/@(2(Sqrt[φ])𝕌Inv)//MatrixForm

The Coxeter-Dynkin diagram for (Subscript[E, 8]) is shown in Fig. 1 along with its Cartan (cmE8) and simple roots matrix (srE8). It has been shown[2] that cmE8=srE8.(srE8^ᵀ), such that we can think of the simple roots as (Sqrt[cmE8]). It was also shown that the SRE (Subscript[E, 8]) vertex coordinates can be derived from the dot product of ±E8roots.srE8. Applying these relationships to 𝕌 gives interesting results as described in Section II.

Fig. 1 (Subscript[E, 8]) Dynkin diagram with it Cartan matrix and simple roots matrix

Defining φ:

(* The symbol φ is the main reference for resolving the Golden Ratios. 
It is only evaluated to a symbolic or numeric value using φRep replacement rules *)
Clear[ϕ,σ,Φ,τ,φ];
(* Upper (and lower) case Phi(Φ=(1+(Sqrt[5]))/2) (and phi ϕ=(1-(Sqrt[5]))/2)) represent large (and small) Golden Ratio (respectively) *)
ϕ=1/φ;
(* σ & τ are Galois conjugates *)
σ=-1/φ/.slRep;
τ=Φ=φ;

(* 𝕌Det1f is a scale factor to make Det@𝕌=1 *)𝕌Det1f=2(Sqrt[ϕ]) /.slRep;

(* φRep are replacement rules used when needing to evaluate φ and ω *)
?φRep

(* Note: slRep is are replacement rules when using the SuperLie' package, which redifines operators such as Plus, Times,... 
*)

?slRep

Section II: Properties of 𝕌

Similar to the relationships between the Cartan matrix, ±roots, weights, heights of (Subscript[E, 8]), we can construct a Cartan matrix cm𝕌=𝕌.𝕌 shown in (3), with 𝕌 playing the role of the simple roots matrix.

Just as the golden ratio φ=(1/2)(1+(Sqrt[5]))≈1.618 generates the integer identity φ-1/φ=1, we now have cm𝕌-((cm𝕌)^(-1)) generating the exchange matrix or standard involutory permutation matrix of rank 8 shown in (4). This has the same palindromic characteristic polynomial coefficients (8) as the normalized 3-qubit Hadamard matrix with 8-bit binary basis states shown in (5), which has been shown by Elkies[3] to be isomorphic to (Subscript[E, 8]) through its (8,4) Hamming code.

Just as ((φ+1/φ)/(2φ-1))=1, ((cm𝕌+((cm𝕌)^(-1)))/(2φ-1))=8x8 Identity Matrix. Of course, we can reverse the rows in cm𝕌, which then swaps the sum and difference operation results of Identity vs. Involutory permutation matrices (respectively). Also as the exponentiation of sum (difference) (φ^n)+(φ^(-n)) results in integer factors on even (odd) n and integer radicand factors on odd (even) n as shown in Fig. 2, by using matrix power operations on ((cm𝕌)^n)± ((cm𝕌)^(-n)) produces the Identity (Involutory) matrices with those same scaling factors. This application of matrix powers to (𝕌^n) instead of ((cm𝕌)^n) puts all even n as the alternating integer (integer radicand) matrices, with odd n shown in (6) and (7).

Equation (3):

cm𝕌:=Chop@octSym[𝕌.𝕌\[Transpose]];
cm𝕌Inv:=Chop@octSym[𝕌Inv.𝕌Inv\[Transpose]];
cm𝕌//MatrixForm
N[%/.φRep]//MatrixForm

Equation (4):

cm𝕌-cm𝕌Inv/.φRep/.slRep//MatrixForm
%==Reverse/@IM@8

((cm𝕌+cm𝕌Inv)/(2φ-1))/.φRep/.slRep//MatrixForm
%==IM@8

Equation (5):

(* Un-Normalized Hadamard *)
(Sqrt[8])HadamardMatrix@(2^3)//MatrixForm

Figure 2 Sum and difference in powers of φ:

Join[{{"n","(φ^n)","(φ^(-n))","(φ^(#))+(φ^(-#))","(φ^(#))-(φ^(-#))"}},
{#,(φ^(#)),(φ^(-#)),Simplify[(φ^(#))+(φ^(-#))/.φRep],Simplify[(φ^(#))-(φ^(-#))/.φRep]}/.slRep&/@Range@5]

Equation (6) and (7):

unitary𝕌=(1/2)(({{0, 0, 0, 0, 0, 0, 0, 2}, {0, 1, -1, 0, 0, -1, -1, 0}, {0, -1, 0, 1, -1, 0, -1, 0}, {0, 0, 1, -1, -1, -1, 0, 0}, {0, 0, -1, -1, -1, 1, 0, 0}, {0, -1, 0, -1, 1, 0, -1, 0}, {0, -1, -1, 0, 0, -1, 1, 0}, {2, 0, 0, 0, 0, 0, 0, 0}})) ;
Det@%
Tr@%%
UnitaryMatrixQ@%%%

Inverse@unitary𝕌==unitary𝕌

unitary𝕌.unitary𝕌

unitary𝕌.Reverse/@unitary𝕌

unitary𝕌+Reverse/@unitary𝕌
Det@%
Tr@%%

unitary𝕌-Reverse/@unitary𝕌
Det@%
Tr@%%

unitary𝕌+unitary𝕌
Det@%
Tr@%%

unitary𝕌-unitary𝕌

(* ((cm𝕌)^n)+((cm𝕌)^(-n)) as ((φ^(#))+(φ^(-#)))IM@8   &   Reverse/@IM@8((φ^(#))-(φ^(-#))) as ((cm𝕌)^n)-((cm𝕌)^(-n)) *)
{#,MatrixPower[cm𝕌,#],Inverse@MatrixPower[cm𝕌,#],
FullSimplify[(MatrixPower[cm𝕌,#]+Inverse@MatrixPower[cm𝕌,#])],FullSimplify[(MatrixPower[cm𝕌,#]+Inverse@MatrixPower[cm𝕌,#])]==Simplify[(φ^(#))+(φ^(-#))/.φRep]IM@8,
FullSimplify[(MatrixPower[cm𝕌,#]-Inverse@MatrixPower[cm𝕌,#])],FullSimplify[(MatrixPower[cm𝕌,#]-Inverse@MatrixPower[cm𝕌,#])]==Simplify[(φ^(#))-(φ^(-#))/.φRep]Reverse/@IM@8
}/.slRep&/@Range@15

(* (𝕌^n)+(𝕌^(-n)) as ((φ^(#))+(φ^(-#)))IM@8   &   Reverse/@IM@8((φ^(#))-(φ^(-#))) as (𝕌^n)-(𝕌^(-n)) - 
Even powers are integer or radical IM@8 & Reverse/@IM@8 (Bosons), 
    Odd powers are mixed (Fermions?) *)
Join[{Style[#,Bold]&/@{"n","(𝕌^n)+(𝕌^(-n))","Det[(𝕌^n)+(𝕌^(-n))]","(𝕌^n)-(𝕌^(-n))","Det[(𝕌^n)-(𝕌^(-n))]"}},
{#,
test=(MatrixPower[𝕌,#]+Inverse@MatrixPower[𝕌,#]/.φRep);
test=If[#<5||EvenQ@#,
(* Show symbolic if < (𝕌^5) **)
(**)octSym/@(**)test,Chop@N@test],Det@Chop@N[test/.φRep],

test=(MatrixPower[𝕌,#]-Inverse@MatrixPower[𝕌,#]/.φRep);
test=If[#<5||EvenQ@#,
(**)octSym/@(**)test,Chop@N[test/.φRep]];
test=If[#<5||EvenQ@#,test,Chop@N@test],Det@Chop@N[test/.φRep]
}/.slRep&/@Range@5]

Equation (8):

Expand@CharacteristicPolynomial[HadamardMatrix@(2^3),x]

Simplify@%

Union@Solve[%==0]

Norm/@%%%%

Equation (9):

CharacteristicPolynomial[𝕌,x]
Chop@Simplify[%/.φRep]

Figure 3:

-(Reverse/@schlafli+(1/2)IM@8-(3/2)Reverse/@IM@8/.slRep)
N[%/.φRep]
%==N[cm𝕌Inv/.φRep]

Graphics[{Text[Style[Row@{
Column[{Row@{"Cartan"<>If[cartan===schlafli,"=Schlafli "," "],cartan},Row@{"Group ",group}}],
Column[{Row@If[cartan===schlafli,{""},{" Schlafli ",schlafli/.slRep}],Row@{""}}],
Column[{Row@{" Coxeter ",coxeter},Row@{""}}],
Column[{simpleRoots,Row@{""}}]},
If[rank>6,Tiny,If[rank>4,Small,Medium]],Bold],If[rank<10,{If[cartan===schlafli,8,11]xR,(33.75+1/rank)yR},{If[cartan===schlafli,8,11]xR,30yR}]],
dynkDiag[rank,ptsLoc,dynkin,affine,directed,alg,permName]},ImageSize->rank 65,Background->None]

From [1] we know that 𝕌 produces the folding of (Subscript[E, 8]) to (Subscript[H, 4]) with (𝕌^(-1)) involved in the unfolding back to (Subscript[E, 8]). We also know its palindromic characteristic polynomial coefficients are those shown in (7) with the same form as (6). This gives us a better understanding of why (Subscript[E, 8]) is isomorphic to both the Hadamard matrix and (Subscript[H, 4]). Given that the sum, difference, product, and division of 𝕌 and cm𝕌 generate both the left and right matrix identities of rank 8 suggests a possible connection to Bott periodicity.

Exploring further, if we take seriously the idea of cm𝕌 as a Cartan matrix, it can be visualized with its positive roots, weights, heights, and Hasse diagrams as shown in Appendix A Figs. 5-6. After deleting duplicates generated in the SuperLie[4] analysis of cm𝕌, the cumulative index count up to height 8 is same as that of (Subscript[E, 8]) being 120.

A corresponding Coxeter-Dynkin diagram, with Cartan, Schlafli, and Coxeter matrices is shown in Fig. 3, noting the Schlafli matrix adding fractional 1/2 scaling on the Identity and 3/2 scaling on the Involutory matrices reproduces ((cm𝕌)^(-1)).

If we do the same analysis using the involutory permutation matrix of rank 8 (4) as a Cartan matrix, it shows the only difference is the weights are now integers as opposed to factors of φ. This is shown in Appendix A Figs. 7-8.

Generating as prescribed above the cm𝕌-based vertex coordinates and projecting to 3D using the methods shown in [1] gives somewhat different results than with the folded or unfolded (Subscript[E, 8]). Instead of finding each of 56 possible subsets of 3 dimensions having the same tally of hull groupings with the same hull geometries, 𝕌 groupings rotate into much smaller groups as shown in Fig. 4. A more complete hull breakdown using dimensions {2,3,4} is shown in Appendix B Fig. 9, noting the predominance of regular octahedral and irregular icosahedral hulls.

Fig. 4  Eight orthogonal projections to 3D of cm𝕌-based vertices from the ±roots of cm𝕌

Section IV: Conclusion

This paper presents properties of the matrix 𝕌 shown to be related to the isomorphism between (Subscript[H, 4]) and (Subscript[E, 8]). Significantly, 𝕌.𝕌 is to rank 8 matrices what the golden ratio is to numbers, such that 𝕌.𝕌-(((𝕌.𝕌))^(-1)) is the reverse identity matrix or standard involutory permutation matrix of rank 8, with a possible connection to Bott periodicty. It has the same palindromic characteristic polynomial coefficients as the normalized 3-qubit Hadamard matrix with 8-bit binary basis states known to be isomorphic to E8 through its (8,4) Hamming code. In addition to providing insight into the isomorphisms of (Subscript[E, 8]), taking advantage of this property may open the door to as yet unexplored (Subscript[E, 8])-based Grand Unified Theories or GUTs. It is anticipated that these visualizations and connections will be useful in discovering new insights into unifying the mathematical symmetries as they relate to unification in theoretical physics.

Bibliography

[1] J. G. Moxness, The isomorphism of h4 and e8, ArXiv eprints math.GR (2023), arXiv:2311.01486 [math.GR].
[2] J. G. Moxness, The 3D Visualization of E8 using an H4 Folding Matrix, www.vixra.org/abs/1411.0130 (2014).
[3] N. D. E. (https://math.stackexchange.com/users/93983/noamd elkies), Isomorphism between e8 lattice and lattice defined by extended hamming code, Mathematics Stack Exchange, uRL:https://math.stackexchange.com/q/572561 (version: 2013-11-19), https://math.stackexchange.com/q/572561.
[4] P. Grozman and D. Leites, Lie superalgebra structures, Czechoslovak Journal of Physics 54, 1313 (2004).

Appendix A:

SuperLie package analysis of cm𝕌 and the rank 8 involution permutation matrix showing the positive roots, weights, heights, and Hasse visualizations up to height 10

Fig. 5 Analysis of cm𝕌 showing the cumulative height group indices, positive roots, weights, and heights

Note: At height 8 the cumulative index count is 120, giving 240 positive and negative roots as in (Subscript[E, 8])

Print[{#,doTime@#;CartanMatrixAlgebra[cmA,{xA,hA,yA},cm,GRange->#]}]&/@Range[rangeG,rangeG];

weights=(*)Global`**)Weight/@GenBasis@cmA;
Length@%
(* Get the symbolic weights with no dups *)
weightsSym=DeleteDuplicates@Chop[octSym/@%%];
Length@%

(* heights==grades *)
grades=(*)Global`**)Grade/@GenBasis@cmA
Length@%

rootVecs=Round[octSym/@Abs[Inverse[cm].#&/@weights]];
rootVecs=DeleteDuplicates@rootVecs;
Length@%

heights=Round@octSym[{#}][[1]]&/@Total/@rootVecs; 
heightTally=Tally@heights
Total[%[[;;#,2]]]&/@Range@Length@heightTally

(* Build the indices into the groups of weights & root vectors *)
indx=0;
indices={1+indx,indx=indx+#[[2]]}&/@heightTally

rootHeights=rootVecs[[#[[1]];;#[[2]]]]&/@indices;
weightHeights=weightsSym[[#[[1]];;#[[2]]]]&/@indices;
Join[{{"Height","Weights","Root Vectors","MatrixPlot"}},
{#,weightHeights[[#]],rootHeights[[#]],MatrixPlot[rootHeights[[#]],ImageSize->200]}&/@Range@Length@indices]

Fig. 6 Analysis of cm𝕌 showing its Hasse visualizations up to height 10

doHasse

Fig. 7 Analysis of cm𝕌-((cm𝕌)^(-1)) showing all integer positive roots, weights, heights

Import@"revIMroots.svg"

Fig. 8 Analysis of cm𝕌-((cm𝕌)^(-1)) showing its Hasse visualizations up to height 10, which are identical to those in Fig. 4

Import@"revIMHasse.svg"

Appendix B:

Orthogonal projection to 3D of the cm𝕌-based vertex coordinates using dimensions {2,3,4}.

These were created using quaternion Weyl orbits directly from the (Subscript[A, 3]), (Subscript[B, 3]),and (Subscript[H, 3]) group symmetries[4] listed in the first column.

Fig. 9 Orthogonal projection to 3D of the cm𝕌-based vertex coordinates using dimensions {2,3,4}, noting the regular octahedral and irregular icosahedral hulls}