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## Main Question or Discussion Point

Hi,

I am not able to understand something about partial tracing. We have a quantum state [itex]\rho_{AB}[/itex]. The Hilbert Space is [itex]H_{A}\otimes H_{B}[/itex]. For some observable [itex]A[/itex] in [itex]H_{A}[/itex], we have

[itex]

Tr_{A}(\rho_{A}A)=Tr_{AB}(\rho_{AB}(A\otimes 1))

=\sum\sum<a_{j}, b_{k}|\rho_{AB}(A\otimes 1_{B})|a_{j}, b_{k}>[/itex]

where the summation is over j and k. So here is the question: Why is the first equality true? What exactly is the information conveyed here? I have some idea but its a bit fuzzy so could you help me? Thank you!

And on a related, yet different note, what is the meaning of [itex]Tr_{AB}(\rho_{AB}(A\otimes B))[/itex]. That is, we are measuring with some operator [itex]A[/itex] in [itex]H_{A}[/itex] and [itex]B[/itex] in [itex]H_{B}[/itex]. The trace is some "expectation value" so what information does it have? I think that for non-entagled states, the answer is clear but when the state is an entangled one, then what? Thank you!

I am not able to understand something about partial tracing. We have a quantum state [itex]\rho_{AB}[/itex]. The Hilbert Space is [itex]H_{A}\otimes H_{B}[/itex]. For some observable [itex]A[/itex] in [itex]H_{A}[/itex], we have

[itex]

Tr_{A}(\rho_{A}A)=Tr_{AB}(\rho_{AB}(A\otimes 1))

=\sum\sum<a_{j}, b_{k}|\rho_{AB}(A\otimes 1_{B})|a_{j}, b_{k}>[/itex]

where the summation is over j and k. So here is the question: Why is the first equality true? What exactly is the information conveyed here? I have some idea but its a bit fuzzy so could you help me? Thank you!

And on a related, yet different note, what is the meaning of [itex]Tr_{AB}(\rho_{AB}(A\otimes B))[/itex]. That is, we are measuring with some operator [itex]A[/itex] in [itex]H_{A}[/itex] and [itex]B[/itex] in [itex]H_{B}[/itex]. The trace is some "expectation value" so what information does it have? I think that for non-entagled states, the answer is clear but when the state is an entangled one, then what? Thank you!

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