Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`QuantumFramework`
QuantumPhaseSpaceTransform  |
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▪
Built-in names for the phase space include:
| | |
"GellMann" | basis of the Gell-Mann matrices | |
"GellMann"[d] | basis of generalized Gell-Mann matrices for dimension d | |
"Wigner" | 2-dimensional basis of the Wigner phase space operators with respect to the computational basis | |
"Wigner"[d] | d-dimensional basis of the Wigner phase space operators with respect to the computational basis | |
"Wootters" | 2-dimensional phase space basis | |
"Wootters"[p] | p-dimensional phase space basis where p is a prime number | |
"WignerMIC" | Phase space basis corresponding to the WignerMICPOVM measurement (Minimal Informationally Complete) | |
"WignerMIC"[d] | WignerMIC basis in the d-dimension | |
"Bloch" | Phase space basis of generalized Bloch coordinates | |
"GellMannMIC" | Phase space basis corresponding to the GellMannMICPOVM (generalized Pauli) measurements | |
"GellMannMIC"[d] | GellMannMIC basis in d-dimension | |
"Tetrahedron" | Phase space basis corresponding to the TetrahedronSICPOVM measurements (Symmetric Informationally Complete), which eigenstates form vertices of a | |
"Tetrahedron"[a,b,c] | A tetrahedron rotated by "U"[a,b,c] | |
"HesseSIC" | ||
"HoggarSIC" | ||
"QBismSIC" | ||
"QBismSIC"[d] | ||
"RandomMIC" | A random MIC basis based on Haar method | |
"RandomMIC"[Method->"Bloch"] | A MIC basis based on random Bloch sphere affine transformation |
Examples
(5)
Basic Examples
(3)
Create a random mixed state:
In[1]:=
SeedRandom[0];ρ=
["RandomMixed"]
 | QuantumState |
Out[1]=
QuantumState
 |  |
|
Transform into Tetrahedron basis and return amplitudes:
In[2]:=
amplitudes=
[ρ,"Tetrahedron"]["Amplitudes"]
 | QuantumPhaseSpaceTransform |
Out[2]=
|〉0.392528,|〉0.300255,|〉0.114132,|〉0.193084
1
2
3
4
Show probabilities from TetrahedronSICPOVM measurement:
In[3]:=
probabilities=
["TetrahedronSICPOVM"][ρ]["Probabilities"]
 | QuantumMeasurementOperator |
Out[3]=
|〉0.392528,|〉0.300255,|〉0.114132,|〉0.193084
1
2
3
4
Check they are the same:
In[4]:=
probabilitiesamplitudes
Out[4]=
True
_________________________________________________________________________________________________________________
Transform a quantum circuit:
In[1]:=
| QuantumPhaseSpaceTransform |
| QuantumCircuitOperator |
Out[1]=
Pay attention to the dimension of wire (which are doubled, compared to the conventional Hilbert space representation).
_________________________________________________________________________________________________________________
Generate a random mixed state:
In[1]:=
SeedRandom[0];ρ=
["RandomMixed"]
| QuantumState |
Out[1]=
QuantumState
| |
|
Show Hadamard operator in the QBismSIC basis:
In[2]:=
| QuantumPhaseSpaceTransform |
| QuantumOperator |
Out[2]=
〈 1 | 〈 2 | 〈 3 | 〈 4 | |
| 1 | 0.5 | 0.5 | 0.5 | -0.5 |
| 2 | 0.5 | -0.5 | 0.5 | 0.5 |
| 3 | 0.5 | 0.5 | -0.5 | 0.5 |
| 4 | -0.5 | 0.5 | 0.5 | 0.5 |
Show the state (vectorized) in the QBismSIC basis:
In[3]:=
| QuantumPhaseSpaceTransform |
Out[3]=
{0.392333,0.272244,0.0961469,0.239276}
Check that the transformation in the phase space returns the same state as the one in the Hilbert space:
In[4]:=
QuantumWeylTransform@
["H"],"QBismSIC"@
[ρ,"QBismSIC"]
["H"][ρ]
| QuantumPhaseSpaceTransform |
 | QuantumOperator |
| QuantumPhaseSpaceTransform |
| QuantumOperator |
Out[4]=
True
Scope
(2)
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