Commutator of a density matrix and a real symmetric matix

In summary, the conversation discusses the relationship between density matrices, a real symmetric matrix, and the trace operator. It is shown that for diagonal terms of an operator, the trace of the hermitian conjugate is equal to the complex conjugate of the trace. This relationship is then applied to the specific case of <<p1|[M,p2]>> and it is explained why this simplifies to (x+iy)-(x-iy) and how Tr{p1*p2*m}=conjugate(Tr{p1*M*p2}). The conversation also references a forum where formulas can be easily shared.
  • #1
Einstein Mcfly
162
3
Let p1,p2 be two density matrices and M be a real, symmetric matrix.
Now,
<<p1|[M,p2]>>=
<<p1|M*p2>>-<<p1|p2*M>>=
Tr{p1*M*p2}-Tr{p1*p2*M}=
2i*Tr{(Im(p1|M*p2))}.
Why is it that this works out as simply as (x+iy)-(x-iy)?
How is Tr{p1*p2*m}=conjugate(Tr{p1*M*p2})? I can't seem to figure
out why this works out so cleanly. Thanks for any comments.
 
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  • #2
For the diagonal terms of an operator A, in some matrix representation,

[tex](A^\dag)_{ii}=(A_{ii})^*[/tex]

and so

[tex]Tr(A^\dag) = Tr(A)^*[/tex].Of course, we mean [tex]A\equiv M^\dag \rho_2[/tex]. Since [tex]Tr(\rho_1)^*=Tr(\rho_1)[/tex] by self-adjointness,

[tex]Tr(\rho_1^\dag \rho_2^\dag M)=Tr(\rho_1^\dag) \dot Tr (M^\dag \rho_2)^* = Tr(\rho_1 M^\dag \rho_2)^*[/tex]
 
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  • #3

Related to Commutator of a density matrix and a real symmetric matix

1. What is the commutator of a density matrix and a real symmetric matrix?

The commutator of two matrices A and B is defined as [A, B] = AB - BA, where AB represents the matrix product of A and B. In the case of a density matrix and a real symmetric matrix, the commutator is a mathematical operation that yields a new matrix as a result.

2. Why is the commutator of a density matrix and a real symmetric matrix important?

The commutator of two matrices is important because it allows us to understand the relationship between the two matrices and how they interact with each other. In the case of a density matrix and a real symmetric matrix, the commutator can give us information about the dynamics of a quantum system.

3. How is the commutator of a density matrix and a real symmetric matrix calculated?

The commutator of two matrices can be calculated by first multiplying the two matrices AB and BA, and then subtracting the two products. In the case of a density matrix and a real symmetric matrix, this calculation can be done using standard matrix multiplication techniques.

4. What are the properties of the commutator of a density matrix and a real symmetric matrix?

The commutator of two matrices has several properties, including linearity, anti-symmetry, and the Jacobi identity. In the case of a density matrix and a real symmetric matrix, the commutator also has the property that it is Hermitian.

5. How is the commutator of a density matrix and a real symmetric matrix used in quantum mechanics?

In quantum mechanics, the commutator of two matrices is used to calculate the uncertainty in a measurement. It is also used to determine the time-evolution of a quantum system and to study the dynamics of entanglement between two quantum systems. In the case of a density matrix and a real symmetric matrix, the commutator is used to analyze the behavior of a quantum system in terms of its density matrix.

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