The following gives the relative weightings of edges emanating from a node:

Graph[MultiwaySystem[{"A""AA","A""A"},"A",4,"EvolutionGraphWeighted"],EdgeLabels"EdgeWeight",EdgeLabelStyleDirective[Red,Large]]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AA","A""A"},"A",4,"EvolutionGraphFull"],EdgeLabels"EdgeWeight",EdgeLabelStyleDirective[Red,Large]]

In[]:=

Out[]=

MultiwaySystem[{"A""AA","A""A"},"A",4,"EvolutionGraphFull"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AA","A""A"},"A",4,"EvolutionGraphFull","IncludeStateWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AA","A""A"},"A",4,"EvolutionGraph","IncludeStateWeights"True],VertexLabels"VertexWeight"]

In[]:=

{(0A)1,(1AA)1,(1A)1,(2AAA)2,(2AA)3,(2A)1,(3AAAA)6,(3AAA)12,(3AA)7,(3A)1,(4AAAAA)24,(4AAAA)60,(4AAA)50,(4AA)15,(4A)1}

»

{(0A)1,(1AA)1,(1A)1,(2AAA)2,(2AA)3,(2A)1,(3AAAA)6,(3AAA)12,(3AA)7,(3A)1,(4AAAAA)24,(4AAAA)60,(4AAA)50,(4AA)15,(4A)1}

»

Out[]=

Graph[MultiwaySystem[{"A""AA","A""A"},"A",4,"StatesGraph","IncludeStateWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AA","A""A"},"A",4,"StatesGraph","IncludeStatePathWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AA","A""A"},"A",4,"EvolutionGraph","IncludeStateWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AA","A""A"},"A",4,"StatesGraph","IncludeStateWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Counts/@MultiwaySystem[{"A""AA","A""A"},"A",5,"AllStatesListUnmerged"]

In[]:=

{A1,AA1,A1,AAA2,AA3,A1,AAAA6,AAA12,AA7,A1,AAAAA24,AAAA60,AAA50,AA15,A1,AAAAAA120,AAAAA360,AAAA390,AAA180,AA31,A1}

Out[]=

Counts[Flatten[MultiwaySystem[{"A""AA","A""A"},"A",4,"AllStatesListUnmerged"]]]

In[]:=

A5,AA26,AAA64,AAAA66,AAAAA24

Out[]=

Graph[MultiwaySystem[{"A""AB","B""A"},"A",5,"StatesGraph","IncludeStatePathWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AB","B""A"},"A",5,"EvolutionGraphFull","IncludeStatePathWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AB","B""A"},"A",5,"EvolutionGraphFull","IncludeStateWeights"True],VertexLabels"VertexWeight"]

In[]:=

{(0A)1,(1AB)1,(2ABB)1,(2AA)1,(3ABBB)1,(3AAB)2,(3ABA)2,(4ABBBB)1,(4AABB)3,(4ABAB)5,(4ABBA)3,(4AAA)4,(5ABBBBB)1,(5AABBB)4,(5ABABB)9,(5ABBAB)9,(5ABBBA)4,(5AAAB)12,(5AABA)10,(5ABAA)12}

»

Out[]=

Counts/@MultiwaySystem[{"A""AB","B"->"A"},"A",5,"AllStatesListUnmerged"]

In[]:=

{A1,AB1,ABB1,AA1,ABBB1,AAB2,ABA2,ABBBB1,AABB3,ABAB5,ABBA3,AAA4,ABBBBB1,AABBB4,ABABB9,ABBAB9,ABBBA4,AAAB12,AABA10,ABAA12}

Out[]=

Counts[Flatten[MultiwaySystem[{"A""AB","B"->"A"},"A",5,"AllStatesListUnmerged"]]]

In[]:=

A1,AB1,ABB1,AA1,ABBB1,AAB2,ABA2,ABBBB1,AABB3,ABAB5,ABBA3,AAA4,ABBBBB1,AABBB4,ABABB9,ABBAB9,ABBBA4,AAAB12,AABA10,ABAA12

Out[]=

Graph[MultiwaySystem[{"A""AB","B""A"},"A",4,"BranchialGraph","IncludeStateWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AB","B""A"},"A",5,"BranchialGraph","IncludeStatePathWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AB","B""A"},"A",5,"BranchialGraphStructure","IncludeStateWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AB","B""A"},"A",5,"StatesGraphStructure","IncludeStateWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

MultiwaySystem[{"A""AA","A""B"},"A",4,"EvolutionGraphFull"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AA","A""B"},"A",4,"StatesGraph","IncludeStateWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

MultiwaySystem[{"A""AB"},"AA",4,"EvolutionGraphFull"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AB"},"AA",4,"EvolutionGraphFull","IncludeStateWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AB"},"AA",4,"WeightedBrachialGraph"]]

In[]:=

Graph[MultiwaySystem[StateEvolutionFunction(getStringStateEvolutionFunction[#1,{AAB}]&),StateEquivalenceFunctiongetStringStateEquivalenceFunction,StateEventFunction(getStringStateEventFunction[#1,{AAB}]&),EventDecompositionFunctiongetStringEventDecompositionFunction,EventApplicationFunctiongetStringEventApplicationFunction,SystemTypeStringSubstitutionSystem,EventSelectionFunctionIdentity,{AA},4,WeightedBrachialGraph]]

Out[]=

Graph[MultiwaySystem[{"A""AB"},"AA",4,"BranchialGraph","IncludeStatePathWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AB"},"AA",4,"BranchialGraph","IncludeStateWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AA","A""A"},"A",4,"EvolutionGraphFull","IncludeStatePathWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AB"},"AA",4,"EvolutionGraphFull","IncludeStatePathWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"AA""AAA"},"AA",4,"EvolutionGraphFull","IncludeStatePathWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"""A","""B"},"",3,"StatesGraph","IncludeStatePathWeights"True,GraphLayout->"LayeredDigraphEmbedding"],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""BBB","BB""A"},"A",8,"StatesGraph","IncludeStatePathWeights"True,GraphLayout->"LayeredDigraphEmbedding"],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""BBB","BB""A"},"A",8,"StatesGraph","IncludeStateWeights"True,GraphLayout->"LayeredDigraphEmbedding"],VertexLabels"VertexWeight"]

In[]:=

Out[]=

## StatesGraph

StatesGraph

MultiwaySystem[{"A""AA","A""A"},"A",4,"EvolutionGraphFull"]

Graph[MultiwaySystem[{"A""AA","A""A"},"A",4,"StatesGraph","IncludeStatePathWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

Graph[MultiwaySystem[{"A""AA","A""A"},"A",4,"StatesGraph","IncludeStateWeights"True],VertexLabels"VertexWeight"]

In[]:=

Out[]=

## Branchial graph

Branchial graph

Table[Graph[MultiwaySystem[{"A""AB","B""A"},"A",t,"BranchialGraphStructure","IncludeStatePathWeights"True],VertexLabels"VertexWeight"],{t,6}]

In[]:=

Out[]=

Each node is a ray in Hilbert space .....

The weight is the norm of the state; the direction of the state is determined by edges in the [[[multiway graph]]] branchial graph

The weight is the norm of the state; the direction of the state is determined by edges in the [[[multiway graph]]] branchial graph

< state1 | state2 > : inner product

What is the interpretation of distance in branchial space?

Insofar as ancestry distance, 1/“entanglement distance”

What is the interpretation of distance in branchial space?

Insofar as ancestry distance, 1/“entanglement distance”

In a Hilbert space, we have elements a and b.

They are represented by complex numbers. We know their norms a* b

They are represented by complex numbers. We know their norms a* b

Each node carries an amplitude, together with entanglement information with other nodes....

The magnitude squared of the amplitude is the path weight of the branchial node

We might want to know the scalar product of a node with another node.....

For this we need “direction” info as well as magnitude

The magnitude squared of the amplitude is the path weight of the branchial node

We might want to know the scalar product of a node with another node.....

For this we need “direction” info as well as magnitude

Unitarity is guaranteed by path counting [and then we normalize]

<Unitarity in relativity nontrivial> [ e.g. BH information paradox ]

<Unitarity in relativity nontrivial> [ e.g. BH information paradox ]

The Hamiltonian is the transformation from one branchial graph to the next....

An operator represents that an event that one does, which changes the branchial graph

An operator represents that an event that one does, which changes the branchial graph

Out[]=

Apply some operator from the outside: that corresponds to some updating event....

Failure of causal invariance = rule not commuting with itself

{"A""AB","A""B"}

MultiwaySystem[{"A""AA","A""B"},"A",3,"StatesGraph"]

In[]:=

Out[]=

Commuting of updating events implies 1-step CI : U1 U2 == U2 U1

In a causal invariant system, eventually things will commute : therefore you perceive definite things to happen

Locally may not commute....

And the way to measure the amount is how far away you end up on the branchial graph

Curvature in multiway graph => noncommuting operators

In a causal invariant system, eventually things will commute : therefore you perceive definite things to happen

Locally may not commute....

And the way to measure the amount is how far away you end up on the branchial graph

Curvature in multiway graph => noncommuting operators

#### Distance in branchial space

Distance in branchial space

Assume a grid branchial graph....

(u-v).(u-v)

{+1,-1,-1,+1}

Each vertex has an associated vector (which is supposed to correspond to a ray in a Hilbert space)

We can to compute the distance between nodes in the Hilbert space metric (which just involves taking the vectors and computing (u-v) .(u - v) )

Each vector has a norm^2 given by the weight on the branchial graph

We can to compute the distance between nodes in the Hilbert space metric (which just involves taking the vectors and computing (u-v) .(u - v) )

Each vector has a norm^2 given by the weight on the branchial graph

With a branchial graph that is an equally weighted grid .... i.e. a pure branching multiway graph

What is the actual quantum inner product

IS the dot product of the vectors attached to different nodes

Geodesic in the multiway graph that goes through a particular point in the branchial graph.....

What is the actual quantum inner product

IS the dot product of the vectors attached to different nodes

Geodesic in the multiway graph that goes through a particular point in the branchial graph.....

Summing paths doesn’t work if the paths have common history.... and they have common history if they’re close on the branchial graph.....

No shared history = disconnected branchial graph

Assume a single critical pair: what is the angle between the vectors