Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`QuantumFramework`
QuantumMeasurementOperator  |
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Details and Options
Examples
(12)
Basic Examples
(4)
In[1]:=
 | QuantumMeasurementOperator |
Out[1]=
QuantumMeasurementOperator
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Return its traditional form
In[2]:=
 | QuantumMeasurementOperator |
Out[2]//TraditionalForm=
|〉〈|+|〉〈|
ℰ
0
+
+
ℰ
1
−
−
Note that above measurement resembles a spider-like transformation, with one input (bra) and two outputs (kets) where one of the outputs (denoted by ) represents the environment (aka detector). Diagram representation of quantum measurement operator shows the detector/environment as the extra qudit:
ℰ
i
In[3]:=
| QuantumCircuitOperator |
 | QuantumMeasurementOperator |
Out[3]=
When focused solely on measurement outcomes, one can readily compute them directly from the quantum state by appropriately transforming it into the relevant basis.
Generate a random pure state in the computational basis. Determine measurement outcomes by applying the quantum measurement operator associated with the X basis. Additionally, calculate measurement results directly from the state transformed into the X basis:
In[4]:=
state=
["RandomPure"];
[state,"X"]["Probabilities"]
["X"][state]["ProbabilitiesList"]
| QuantumState |
| QuantumState |
 | QuantumMeasurementOperator |
Out[4]=
True
In[1]:=
| QuantumMeasurementOperator |
Out[1]=
QuantumMeasurementOperator
 |  |
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Corresponding spectral representation:
In[2]:=
TraditionalForm[%]
Out[2]//TraditionalForm=
|〉〈|+|〉〈|+|〉〈|+|〉〈|
ℰ
-1
+
Ψ
+
Ψ
ℰ
2
-
Φ
-
Φ
ℰ
3
-
Ψ
-
Ψ
ℰ
4
+
Φ
+
Φ
A one-qudit measurement operator can act on system of many qudits.
In[1]:=
qmo=
["X",{2}]
| QuantumMeasurementOperator |
Out[1]=
QuantumMeasurementOperator
|  |
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In[2]:=
mea=qmo
[{"RandomPure",2}]
| QuantumState |
Out[2]=
QuantumMeasurement
|  |
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In[3]:=
mea["Probabilities"]
Out[3]=
|〉0.327595,|〉0.672405
+
−
Additional measurement operator can be applied on a quantum measurement:
In[4]:=
| QuantumMeasurementOperator |
Out[4]=
|0〉0.235724,|0〉0.5117,|1〉0.0918707,|1〉0.160705
+
−
+
−
One can also input any set of operators, with and =, to generalize measurements as positive operator-valued measures (POVMs):
{}
E
m
E
m
M
m
M
m
In[1]:=
povm=,0,{0,0},,,,,,-,-,;
2
3
1
6
1
2
3
1
2
3
1
2
1
6
1
2
3
1
2
3
1
2
Test each element of POVM is explicitly positive semi-definite:
In[2]:=
PositiveSemidefiniteMatrixQ/@povm
Out[2]=
{True,True,True}
Test the complete relation of POVM elements:
In[3]:=
(Plus@@povm)IdentityMatrix[2]
Out[3]=
True
Define the quantum measurement using POVM, and apply it on a quantum state:
In[4]:=
| QuantumMeasurementOperator |
Out[4]=
QuantumMeasurementOperator
|  |
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In[5]:=
| QuantumMeasurementOperator |
| QuantumState |
Out[5]=
Scope
(2)
Generalizations & Extensions
(2)
Applications
(3)
Possible Issues
(1)
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""