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I was confused about a derivation of Bloch equation in Breuer and Petruccione's The theory of Open Quantum Systems. Can someone gives some hints on derivation from Eq. 3.219 to Eq. 3.226? Specific guidlines are shown as follows.

The first point would be why we can write the density matrix in terms of Bloch vector. Please see the figure below. enter image description here

Why do we choose such a basis?

(2)How do we get the Bloch equations:enter image description here

according to the Lindblad master equation:enter image description here

I would appreciate your hints on how to do the derivations.

Thank you!

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  • $\begingroup$ Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. $\endgroup$
    – Community Bot
    Oct 30, 2023 at 11:42
  • $\begingroup$ Are you planning to tell us the Hamiltonian, or are we supposed to just guess? Can't determine the time evolution without the Hamiltonian... $\endgroup$
    – hft
    Oct 30, 2023 at 14:44
  • $\begingroup$ Also, please don't post pictures of text. Just Tex it out in MathJax: physics.stackexchange.com/help/notation $\endgroup$
    – hft
    Oct 30, 2023 at 14:45

1 Answer 1

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The idea is to expand a density matrix in terms of a set of matrices that can serve as a basis. For an $n\times n$ matrix, one needs $n^2$ such matrices. It turns out that the generators in the fundamental irreducible representation of the SU(n) Lie group (in other words, the associated Lie algebra) are very useful for this purpose. There are always $n^2-1$ generators for SU(n). So, together with the $n\times n$ identity matrix, we get $n^2$ matrices.

These generators have useful properties. They are all trace free. Commutations of different generators again produce generators. As a result, we have $$ \text{tr}\{\tau_i \tau_j\} = \delta_{ij} , $$ where $\tau_i$ represents the $i$-th generator.

Now we can consider an expansion of a density matrix in terms of the set of generator matrices $$ \rho = \sum_{i=0}^{n^2-1} \alpha_i \tau_i . $$ where $\alpha_i$ are the coefficients. To compute these coefficients, we can use the trace $$ \text{tr}\{\tau_i \rho\} = \alpha_i . $$ The vector consisting of these coefficients is called the Bloch vector.

The Pauli matrices are the (unnormalized) generators for SU(2). Therefore, they play an important role in the case of $2\times 2$ density matrices.

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