Quantum computing, partial trace

[emdez]

Homework Statement

Hello, I have to calculate density matrices $\rho _{AB}, \rho _{AC}, \rho _{BC}$ of this state:
$| \psi \rangle _{ABC} = |100 \rangle + | 001 \rangle$

Relevant Equations

$| \psi \rangle _{ABC} = |100 \rangle⟩ + | 001 \rangle$
\begin{align*}
\rho_{AB} &= \text{Tr}_C \, \rho_{ABC} \\
\rho_{AC} &= \text{Tr}_B \, \rho_{ABC} \\
\rho_{BC} &= \text{Tr}_A \, \rho_{ABC}
\end{align*}

I've calculated density matrix
$$ rho_{ABC} = \frac{1}{2} \left( \left| 100 \right\rangle \left\langle 100 \right| + \left| 100 \right\rangle \left\langle 001 \right| + \left| 001 \right\rangle \left\langle 100 \right| + \left| 001 \right\rangle \left\langle 001 \right| \right)$$
and $\rho_{AB} = \frac{1}{2} \left( \left| 10 \right\rangle \left\langle 10 \right| + \left| 00 \right\rangle \left\langle 00 \right| \right) )$. I'm not sure how to calculate other matrices. I don't understand how to calculate partial trace. Could some explain how to do it?

 

[ergospherical]

I'm not very well-versed in this subject, but you must already be close? Taking a "partial trace" over one of the subsystems, e.g. C, means to say:$$\mathrm{Tr}_{C}[ \rho] = \sum_c \langle c | \rho | c \rangle $$where in this case the $|c\rangle$ are a basis for C (and similarly for a, b). This notation "$| abc \rangle$" means the tenor product, $| abc \rangle := |a\rangle \otimes |b \rangle \otimes |c\rangle$ (or just $|a\rangle |b \rangle |c\rangle$) which has its own special properties.

If you look at just this term, for example:$$\sum_c \langle c|100 \rangle \langle 100| c \rangle$$then what you are doing is actually:$$\sum_c \langle c| \bigg{(} |a=1\rangle |b=0\rangle |c=0\rangle \langle a=1 | \langle b=0| \langle c=0| c \rangle \bigg{)}$$The C basis states don't hit the A and B basis states in the tensor product, so you are left with this:$$|a=1\rangle |b=0\rangle \sum_c \langle c|c=0\rangle + \langle a=1 | \langle b=0| \sum_c \langle c=0| c \rangle $$Those sums are both 1 (e.g. $\sum_c \langle c| c=0 \rangle = \langle 0 | 0 \rangle + \langle 1 | 0 \rangle = 1 + 0 = 1$), leaving you with: $$|a=1\rangle |b=0\rangle + \langle a=1 | \langle b=0|$$

and the same for all the other terms. I.e. in practice I think you would just "ignore" the other two positions/subsystems to take the trace quickly).