Is Partial Trace Cyclic for Density Matrices in Quantum Systems?

[jenga42]

Hello,

I know trace is usually cyclic, but is partial trace cyclic too? Why?

Thanks!

Jenga

 

[jenga42]

Ok... so I know that it isn't cyclic now ... just by picking a random example, but if anyone knows the reason why it's not cyclic, and has a general proof as to why it's not, I'd be very grateful to hear it!

Thanks.

 

[morphism]

What's the definition of a partial trace?

 

[jenga42]

Normal trace is equivalent to the sum of the eigenvalues (or diagonal elements) of a matrix. Partial trace acts only on part of the system, so for a density matrix.. say it's a pure state but entangled,

$$\rho_{AB}=\frac{1}{2}(|01\rangle +|10\rangle )(\langle 01|+\langle 10 |)$$

The partial trace over subsystem B gives the reduced density matrix $$\rho_A$$, so $$Tr_B(\rho_{AB})=\rho_A$$

So

$$\rho_A=_B\langle 0 |\rho_{AB}|0\rangle_B +_B\langle 1 |\rho_{AB}|1\rangle_B$$
$$\rho_A=|1\rangle \langle 1 | + |0\rangle \langle 0 |$$

My question is how do I prove that

$$Tr_B (\rho \sigma) = Tr_B (\sigma \rho)$$

where $$\rho$$ and $$\sigma$$ are both density matrices of a system AB.

Thanks!