How to Calculate Expectation Value Using Trace?

[cscott]

Homework Statement

How do I get the expectation value of operator $\sigma$ using density matrix $\rho$ in a trace: $Tr\left(\sigma\rho\right)$

I have $\sigma$ and $\rho$ in matrix form but how do I get a number out of the trace?

 

[nrqed]

Homework Statement

How do I get the expectation value of operator $\sigma$ using density matrix $\rho$ in a trace: $Tr\left(\sigma\rho\right)$

I have $\sigma$ and $\rho$ in matrix form but how do I get a number out of the trace?

I am not sure I follow your question. Do you know what it means to take the trace of a matrix?

 

[cscott]

I do if it involves just a bra and ket

i.e. $$Tr\left(|a><b|\right) = <b|a>$$

I've been shown $<\Lambda> = Tr\left(\Lambda\rho\right)$

But I have rho and lambda in matrix form and not as a product of bra's and ket's

 

[Hurkyl]

The trace is additive, so

$$\mathrm{Tr}\left( \sum_i | a_i \rangle \langle b_i| \right) = \sum_i \mathrm{Tr}\left( | a_i \rangle \langle b_i| \right)$$

If you have a matrix, this greatly simplifies. It's just the sum of the diagonal entries.

 

[cscott]

The trace is additive, so

$$\mathrm{Tr}\left( \sum_i | a_i \rangle \langle b_i| \right) = \sum_i \mathrm{Tr}\left( | a_i \rangle \langle b_i| \right)$$

If you have a matrix, this greatly simplifies. It's just the sum of the diagonal entries.

Ahh I remember that now.

So I just take the matrix product $\Lambda\rho$ and then sum the diagonal entries to compute the trace?

 

[cscott]

I got the correct answer. Thanks guys.