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Partial Trace: Physical Meaning
The physical meaning of partial trace is examined through a toy model and demonstration with it.
Partial trace over a subsystem of the system |
Functions related to this tutorial.
Make sure to load Q3.
In[304]:=
Needs["QuantumMob`Q3`"]
In[305]:=
Let[Qubit,S]Let[Complex,c]
Consider a system of two qubits. The first and second qubit are regarded as the system and environment, respectively.
In[307]:=
bs=Basis[S@{1,2}]
Out[307]=
,,,
0
S
1
0
S
2
0
S
1
1
S
2
1
S
1
0
S
2
1
S
1
1
S
2
In[309]:=
cc=Flatten@Array[c,{2,2}]
Out[309]=
{,,,}
c
1,1
c
1,2
c
2,1
c
2,2
Consider a state vector for the total system.
In[310]:=
ket=bs.cc
Out[310]=
c
1,1
0
S
1
0
S
2
c
1,2
0
S
1
1
S
2
c
2,1
1
S
1
0
S
2
c
2,2
1
S
1
1
S
2
The state of the “system” qubit is determined by the state of the “environment” qubit.
In[321]:=
KetFactor[ket,S[2]]
Out[321]=
⊗++⊗+
0
S
2
c
1,1
0
S
1
c
2,1
1
S
1
1
S
2
c
1,2
0
S
1
c
2,2
1
S
1
The above dependence may also be seen by measuring the “environment” qubit.
In[350]:=
odds=MeasurementOdds[S[2,3]][ket]
Out[350]=
0++++,+,1++++,+
2
c
1,1
2
c
2,1
2
c
1,1
2
c
1,2
2
c
2,1
2
c
2,2
c
1,1
0
S
1
0
S
2
c
2,1
1
S
1
0
S
2
2
c
1,2
2
c
2,2
2
c
1,1
2
c
1,2
2
c
2,1
2
c
2,2
c
1,2
0
S
1
1
S
2
c
2,2
1
S
1
1
S
2
In the above expressions, the normalization of state vector ket is not implied. Imposing the normalization, the above expression reads as follows.
In[352]:=
odds=MeasurementOdds[S[2,3]][ket]/.{Total[Abs[cc]^2]1}//SimplifyThrough
Out[352]=
0+,+,1+,+
2
c
1,1
2
c
2,1
c
1,1
0
S
1
0
S
2
c
2,1
1
S
1
0
S
2
2
c
1,2
2
c
2,2
c
1,2
0
S
1
1
S
2
c
2,2
1
S
1
1
S
2
Note also that the post-measurement states are not normalized; this is typically the case when the state before measurement is not numeric. Let us thus normalize them.
In[356]:=
odds=Map[KetNormalize,odds,{2}]
Out[356]=
0+,+++,1+,+++
2
c
1,1
2
c
2,1
c
1,1
0
S
1
0
S
2
c
1,1
*
c
1,1
c
2,1
*
c
2,1
c
2,1
1
S
1
0
S
2
c
1,1
*
c
1,1
c
2,1
*
c
2,1
2
c
1,2
2
c
2,2
c
1,2
0
S
1
1
S
2
c
1,2
*
c
1,2
c
2,2
*
c
2,2
c
2,2
1
S
1
1
S
2
c
1,2
*
c
1,2
c
2,2
*
c
2,2
The above expression specifies the statistical ensemble of pure states for the “system” qubit. This ensemble can be described by the density operator.
In[358]:=
rho=Total[First[#]*Dyad[Last[#],Last[#],S[1]]&/@KetDrop[odds,S[2]]]//FullGarner
Out[358]=
+++++++
2
c
1,1
2
c
1,2
0
S
1
0
S
1
c
1,1
*
c
2,1
c
1,2
*
c
2,2
0
S
1
1
S
1
c
2,1
*
c
1,1
c
2,2
*
c
1,2
1
S
1
0
S
1
2
c
2,1
2
c
2,2
1
S
1
1
S
1
In[360]:=
rho=Elaborate@Total[First[#]*Dyad[Last[#],Last[#],S[1]]&/@KetDrop[odds,S[2]]]
Out[360]=
1
2
c
1,1
*
c
1,1
c
1,2
*
c
1,2
c
2,1
*
c
2,1
c
2,2
*
c
2,2
1
2
c
2,1
*
c
1,1
c
2,2
*
c
1,2
c
1,1
*
c
2,1
c
1,2
*
c
2,2
x
S
1
1
2
c
2,1
*
c
1,1
c
2,2
*
c
1,2
c
1,1
*
c
2,1
c
1,2
*
c
2,2
y
S
1
1
2
c
1,1
*
c
1,1
c
1,2
*
c
1,2
c
2,1
*
c
2,1
c
2,2
*
c
2,2
z
S
1
You can see this is identical to the result when one takes the partial trace over the “environment” qubit.
In[362]:=
new=Elaborate@PartialTrace[ket,S[2]]
Out[362]=
1
2
c
1,1
*
c
1,1
c
1,2
*
c
1,2
c
2,1
*
c
2,1
c
2,2
*
c
2,2
1
2
c
2,1
*
c
1,1
c
2,2
*
c
1,2
c
1,1
*
c
2,1
c
1,2
*
c
2,2
x
S
1
1
2
c
2,1
*
c
1,1
c
2,2
*
c
1,2
c
1,1
*
c
2,1
c
1,2
*
c
2,2
y
S
1
1
2
c
1,1
*
c
1,1
c
1,2
*
c
1,2
c
2,1
*
c
2,1
c
2,2
*
c
2,2
z
S
1
In[364]:=
new-rho//Simplify
Out[364]=
0
From this example, we conclude that taking the partial trace over a subsystem corresponds to ignoring the subsystem.
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