Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
QuantumMob`Q3mini`
Basis  |
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Details and Options
Examples
(7)
Basic Examples
(5)
This assumes a system of three unlabelled qubits. Different qubits are distinguished by their positions in the arguments in .
In[1]:=
Basis[3]
Out[1]=
{|0,0,0〉,|0,0,1〉,|0,1,0〉,|0,1,1〉,|1,0,0〉,|1,0,1〉,|1,1,0〉,|1,1,1〉}
This assumes a system of three unlabelled particles which have the Hilbert space dimensions 2, 2, and 3, respectively.
In[2]:=
Basis[{2,2,3}]
Out[2]=
{|0,0,0〉,|0,0,1〉,|0,0,2〉,|0,1,0〉,|0,1,1〉,|0,1,2〉,|1,0,0〉,|1,0,1〉,|1,0,2〉,|1,1,0〉,|1,1,1〉,|1,1,2〉}
In[1]:=
Let[Fermion,c]Let[Boson,a,Top3]Let[Boson,b,Bottom2,Top4]
This gives the standard basis of one Fermion labelled by the symbol c.
In[2]:=
bs=Basis[c]
Out[2]=
{|〉,|〉}
0
c
1
c
In[3]:=
bs//InputForm
Out[3]//InputForm=
{Ket[<|c -> 0|>], Ket[<|c -> 1|>]}
This gives the standard basis of a system of two-flavor Fermions.
In[4]:=
Basis[c@{1,2}]
Out[4]=
,,,
0
c
1
0
c
2
0
c
1
1
c
2
1
c
1
0
c
2
1
c
1
1
c
2
More examples follow.
In[5]:=
Basis[a]Basis[a,b]Basis[{a,b}]
Out[5]=
{|〉,|〉,|〉,|〉}
0
a
1
a
2
a
3
a
Out[5]=
{|〉,|〉,|〉,|〉,|〉,|〉,|〉,|〉,|〉,|〉,|〉,|〉}
0
a
2
b
0
a
3
b
0
a
4
b
1
a
2
b
1
a
3
b
1
a
4
b
2
a
2
b
2
a
3
b
2
a
4
b
3
a
2
b
3
a
3
b
3
a
4
b
Out[5]=
{|〉,|〉,|〉,|〉,|〉,|〉,|〉,|〉,|〉,|〉,|〉,|〉}
0
a
2
b
0
a
3
b
0
a
4
b
1
a
2
b
1
a
3
b
1
a
4
b
2
a
2
b
2
a
3
b
2
a
4
b
3
a
2
b
3
a
3
b
3
a
4
b
In[6]:=
expr=Q[a[1]]+PlusDagger@Pair[a@{1,2}]Basis[expr]
Out[6]=
a
2
a
1
†
a
1
a
1
†
a
1
†
a
2
Out[6]=
,,,,,,,,,,,,,,,
0
a
1
0
a
2
0
a
1
1
a
2
0
a
1
2
a
2
0
a
1
3
a
2
1
a
1
0
a
2
1
a
1
1
a
2
1
a
1
2
a
2
1
a
1
3
a
2
2
a
1
0
a
2
2
a
1
1
a
2
2
a
1
2
a
2
2
a
1
3
a
2
3
a
1
0
a
2
3
a
1
1
a
2
3
a
1
2
a
2
3
a
1
3
a
2
In[1]:=
Let[Qubit,S]
In[2]:=
bs=Basis[{S[1],S[2,$]}]InputForm@First@bs
Out[2]=
,,,
0
S
1
0
S
2
0
S
1
1
S
2
1
S
1
0
S
2
1
S
1
1
S
2
Out[2]//InputForm=
Ket[<|S[1, $] -> 0, S[2, $] -> 0|>]
In[3]:=
KetTrim[bs]
Out[3]=
␣,,,
1
S
2
1
S
1
1
S
1
1
S
2
In[4]:=
Outer[Multiply,Dagger@bs,bs]//MatrixForm
Out[4]//MatrixForm=
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
In[1]:=
Let[Spin,S,Spin1]
In[2]:=
bs=Basis[{S[1],S[2,$]}]
Out[2]=
,,,,,,,,
1
S
1
1
S
2
1
S
1
0
S
2
1
S
1
(-1)
S
2
0
S
1
1
S
2
0
S
1
0
S
2
0
S
1
(-1)
S
2
(-1)
S
1
1
S
2
(-1)
S
1
0
S
2
(-1)
S
1
(-1)
S
2
In[3]:=
bs2=Basis[S[1,1]**S[3,1]-I*S[4,3]**S[1,3]]
Out[3]=
,,,,,,,,,,,,,,,,,,,,,,,,,,
1
S
1
1
S
3
1
S
4
1
S
1
1
S
3
0
S
4
1
S
1
1
S
3
(-1)
S
4
1
S
1
0
S
3
1
S
4
1
S
1
0
S
3
0
S
4
1
S
1
0
S
3
(-1)
S
4
1
S
1
(-1)
S
3
1
S
4
1
S
1
(-1)
S
3
0
S
4
1
S
1
(-1)
S
3
(-1)
S
4
0
S
1
1
S
3
1
S
4
0
S
1
1
S
3
0
S
4
0
S
1
1
S
3
(-1)
S
4
0
S
1
0
S
3
1
S
4
0
S
1
0
S
3
0
S
4
0
S
1
0
S
3
(-1)
S
4
0
S
1
(-1)
S
3
1
S
4
0
S
1
(-1)
S
3
0
S
4
0
S
1
(-1)
S
3
(-1)
S
4
(-1)
S
1
1
S
3
1
S
4
(-1)
S
1
1
S
3
0
S
4
(-1)
S
1
1
S
3
(-1)
S
4
(-1)
S
1
0
S
3
1
S
4
(-1)
S
1
0
S
3
0
S
4
(-1)
S
1
0
S
3
(-1)
S
4
(-1)
S
1
(-1)
S
3
1
S
4
(-1)
S
1
(-1)
S
3
0
S
4
(-1)
S
1
(-1)
S
3
(-1)
S
4
In[1]:=
Let[Qudit,A]
In[2]:=
bs=Basis[A[{1,2}]]
Out[2]=
,,,,,,,,
0
A
1
0
A
2
0
A
1
1
A
2
0
A
1
2
A
2
1
A
1
0
A
2
1
A
1
1
A
2
1
A
1
2
A
2
2
A
1
0
A
2
2
A
1
1
A
2
2
A
1
2
A
2
In[3]:=
Basis[A[{1,2}]]
Out[3]=
,,,,,,,,
0
A
1
0
A
2
0
A
1
1
A
2
0
A
1
2
A
2
1
A
1
0
A
2
1
A
1
1
A
2
1
A
1
2
A
2
2
A
1
0
A
2
2
A
1
1
A
2
2
A
1
2
A
2
In[4]:=
Let[Qudit,B,Dimension5]bs=Basis[B[{1,2}]]
Out[4]=
,,,,,,,,,,,,,,,,,,,,,,,,
0
B
1
0
B
2
0
B
1
1
B
2
0
B
1
2
B
2
0
B
1
3
B
2
0
B
1
4
B
2
1
B
1
0
B
2
1
B
1
1
B
2
1
B
1
2
B
2
1
B
1
3
B
2
1
B
1
4
B
2
2
B
1
0
B
2
2
B
1
1
B
2
2
B
1
2
B
2
2
B
1
3
B
2
2
B
1
4
B
2
3
B
1
0
B
2
3
B
1
1
B
2
3
B
1
2
B
2
3
B
1
3
B
2
3
B
1
4
B
2
4
B
1
0
B
2
4
B
1
1
B
2
4
B
1
2
B
2
4
B
1
3
B
2
4
B
1
4
B
2
In[5]:=
expr=A[1,02]-IA[2,10]bs2=Basis[expr]
Out[5]=
(|2〉〈0|)
1
(|0〉〈1|)
2
Out[5]=
,,,,,,,,
0
A
1
0
A
2
0
A
1
1
A
2
0
A
1
2
A
2
1
A
1
0
A
2
1
A
1
1
A
2
1
A
1
2
A
2
2
A
1
0
A
2
2
A
1
1
A
2
2
A
1
2
A
2
In[6]:=
$N=3;NN=Range[$N];Let[Qudit,A];AA=A[NN,$];
In[7]:=
bs=Basis[AA]
Out[7]=
,,,,,,,,,,,,,,,,,,,,,,,,,,
0
A
1
0
A
2
0
A
3
0
A
1
0
A
2
1
A
3
0
A
1
0
A
2
2
A
3
0
A
1
1
A
2
0
A
3
0
A
1
1
A
2
1
A
3
0
A
1
1
A
2
2
A
3
0
A
1
2
A
2
0
A
3
0
A
1
2
A
2
1
A
3
0
A
1
2
A
2
2
A
3
1
A
1
0
A
2
0
A
3
1
A
1
0
A
2
1
A
3
1
A
1
0
A
2
2
A
3
1
A
1
1
A
2
0
A
3
1
A
1
1
A
2
1
A
3
1
A
1
1
A
2
2
A
3
1
A
1
2
A
2
0
A
3
1
A
1
2
A
2
1
A
3
1
A
1
2
A
2
2
A
3
2
A
1
0
A
2
0
A
3
2
A
1
0
A
2
1
A
3
2
A
1
0
A
2
2
A
3
2
A
1
1
A
2
0
A
3
2
A
1
1
A
2
1
A
3
2
A
1
1
A
2
2
A
3
2
A
1
2
A
2
0
A
3
2
A
1
2
A
2
1
A
3
2
A
1
2
A
2
2
A
3
In[8]:=
EchoTiming[vecBS=Matrix[#,AA]&/@bs;]
⌚
0.010271
Scope
(2)
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