Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`QuantumFramework`
EinsteinSummation  |
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Details and Options
Examples
(19)
Basic Examples
(5)
Define two random complex-valued tensors:
In[1]:=
A=SparseArray@RandomComplex[{-1-I,1+I},{2,3,2}];B=SparseArray@RandomComplex[{-1-I,1+I},{3,2}];
Execute a summation as
∑
j
A
ijk
B
jl
In[2]:=
EinsteinSummation[{{i,j,k},{j,l}},{A,B}]
Out[2]=
TensorContractSparseArray
SparseArray
,{{2,4}}
 |  |
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Activate the outcome and get the final tensor:
In[3]:=
einsum=Activate@EinsteinSummation[{{1,2,3},{2,4}},{A,B}]
Out[3]=
SparseArray
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Check the result is the same as running summation manually:
In[4]:=
Table[Sum[A〚i,j,k〛B〚j,l〛,{j,3}],{i,2},{k,2},{l,2}]einsum
Out[4]=
True
Define two symbolic arrays of 3-rank and 2-rank with dimension and :
{,,}
n
1
n
2
n
3
{,}
n
2
n
4
In[1]:=
A=ArraySymbol["A",{n1,n2,n3}];B=ArraySymbol["B",{n2,n4}];
Execute a summation as
∑
j
A
ijk
B
jl
In[2]:=
EinsteinSummation[{{1,2,3},{2,4}},{A,B}]
Out[2]=
TensorContract,{{2,4}}
3
A
2
B
Define two random complex-valued tensors:
In[1]:=
A=SparseArray@RandomComplex[{-1-I,1+I},{2,3}];B=SparseArray@RandomComplex[{-1-I,1+I},{2,3}];
Calculate the inner product of two tensors as
〈A,B〉=Tr[SuperDagger[A]B]=
∑
ij
*
A
ij
B
ij
In[2]:=
einsum=ActivateTensor@EinsteinSummation[{{1,2},{1,2}},{Conjugate@A,B}]
Out[2]=
0.887333-1.34095
Compare the result with the trace:
In[3]:=
einsumTr[ConjugateTranspose[A].B]
Out[3]=
True
Calculate the cross product via Levi-Civita =
(a×b)
i
∑
jk
ϵ
ijk
a
j
b
k
In[1]:=
{a,b}=RandomReal[{-1,1},{2,3}]
Out[1]=
{{0.270589,-0.552533,-0.6784},{0.161889,-0.716632,-0.598085}}
Excuse the cross product using EinsteinSummation:
In[2]:=
einsum=ActivateTensor@EinsteinSummation[{{i,j,k},{j},{k}},{LeviCivitaTensor[3],a,b}]
Out[2]=
{-0.155702,0.0520093,-0.104463}
Compare the result with manually summing over all indices:
In[3]:=
Table[Sum[LeviCivitaTensor[3]〚i,j,k〛a〚j〛b〚k〛,{j,3},{k,3}],{i,3}]einsum
Out[3]=
True
Double trace
∑
ij
T
ij,ij
In[1]:=
EinsteinSummation[{{i,j,i,j}},{ArraySymbol["T",{n1,n2,n3,n4}]}]
Out[1]=
TensorContractInactive[TensorProduct],{{1,3},{2,4}}
4
T
Applications
(14)
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