Show that performing a measurement with measurement operators {ΛjA}on system A is the same as performing a measurement of the ensemble in (4.173). That is,show that Tr{ρAΛjA} = Tr{ρXA(IX ⊗ ΛjA)}, where ρA is defined in (4.175).
Question
Show that performing a measurement with measurement operators {ΛjA}on system A is the same as performing a measurement of the ensemble in (4.173). That is,show that Tr{ρAΛjA} = Tr{ρXA(IX ⊗ ΛjA)}, where ρA is defined in (4.175).
Solution
To show that Tr{ρAΛjA} = Tr{ρXA(IX ⊗ ΛjA)}, we need to understand the definitions and properties of the trace and tensor product operations, as well as the definitions of the density matrices ρA and ρXA.
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The trace of a matrix is the sum of its diagonal elements. It has the property that Tr{AB} = Tr{BA} for any matrices A and B that can be multiplied together.
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The tensor product of two matrices A and B, denoted A ⊗ B, is a new matrix whose elements are formed by multiplying A by each element of B.
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The density matrix ρA of a quantum system A is a matrix that describes the state of the system. It is defined as ρA = TrB{ρAB}, where ρAB is the density matrix of the combined system AB, and TrB denotes the partial trace over system B.
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The density matrix ρXA of the combined system XA is defined in a similar way.
Given these definitions, we can proceed as follows:
Tr{ρAΛjA} = Tr{TrB{ρAB}ΛjA} = TrB{Tr{ρABΛjA}} = TrB{ρAB(IX ⊗ ΛjA)} = Tr{ρXA(IX ⊗ ΛjA)}
In the second step, we used the property that the trace of a product of matrices is the same as the trace of the product in the opposite order. In the third step, we used the definition of the tensor product. In the fourth step, we used the definition of the density matrix ρXA.
Therefore, we have shown that performing a measurement with measurement operators {ΛjA} on system A is the same as performing a measurement of the ensemble in (4.173).
Similar Questions
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