- #1

Emil_M

- 46

- 2

## Homework Statement

Suppose [itex] R [/itex] and [itex] Q [/itex] are two quantum systems with the same Hilbert space. Let [itex] |i_R \rangle [/itex] and [itex] |i_Q\rangle [/itex] be orthonormal basis sets for [itex] R [/itex] and [itex] Q [/itex]. Let [itex] A [/itex] be an operator on [itex] R [/itex] and [itex] B [/itex] an operator on [itex] Q [/itex]. Define [itex] |m\rangle := \sum_i |i_R\rangle |i_Q\rangle [/itex]. Show that [itex]

\begin{align}

\mathrm{tr}(A^\dagger B)=\langle m| (A \otimes B)|m\rangle,

\end{align} [/itex]

where the multiplication on the left hand side is of matrices, and it is understood that the matrix elements of [itex] A[/itex] are taken with respect to the basis [itex]|i_R\rangle [/itex] and those for [itex]B [/itex] with respect to the basis [itex]|i_Q\rangle [/itex].

## Homework Equations

See above

## The Attempt at a Solution

I've tried the following

[itex] A=\sum_{i,j} a_{ij} |i_R\rangle \langle j_R| [/itex] and [itex]B=\sum_{k,l} b_{kl} |k_R\rangle \langle l_R| [/itex]

Then

[itex]\begin{align}

A^\dagger B &= \left(\sum_{i,j} a^*_{ji} |i_R\rangle \langle j_R|\right)\left(\sum_{k,l} b_{kl} |k_R\rangle \langle l_R| \right)\\

&=\sum_{ijkl} a^*_{ji}b_{kl} |i_R\rangle |k_Q\rangle \langle j_R| \langle l_Q|

\end{align}

[/itex]

But that doesn't seem to be the right track, is it?

Thanks for your help!