Consider the spin $\frac{1}{2}$ system. A general quantum state could be written as $|\alpha^{(i)}\rangle=c_i|+\rangle+d_i|-\rangle$ and thus we have the density operator $$\rho=\sum_i w_i|\alpha^{(i)}\rangle\langle\alpha^{(i)}|.$$Suppose we know $[S_x]=\text{Tr}(\rho S_x)$, $[S_y]$ and $[S_z]$. I am wondering if there is a quick way of calculating $\rho$ using the fact that the trace of any operator is independent of the basis we choose to evaluate it?
Edit: I should add the following constraints
- $\sum_i w_i=1$
- $c_i^2+d_i^2=1$; $c,d\in\mathbb{C}$
Best Answer
Yes. Note that the set of Pauli matrices and the identity operator on $\mathbb C^2$, $\{\sigma_0:=\mathbb I_2,\sigma_1,\sigma_2\sigma_3\}$, is an orthonormal basis in the space of complex $2\times 2$ matrices $M_2(\mathbb C)$ equipped with an inner product $\langle \cdot,\cdot\rangle: M_2(\mathbb C) \times M_2(\mathbb C) \longrightarrow \mathbb C$ defined by $\langle A,B\rangle := \frac{1}{2}\mathrm{Tr}\, A^\dagger B$.
So every complex $2\times 2$ matrix can be expanded in this orthonormal basis; in particular every density matrix on $\mathbb C^2$:
$$\rho = \sum\limits_{k=0}^3 \langle \sigma_k, \rho\rangle\,\sigma_k \quad .$$
Thus, given the inner product of a density matrix with the Pauli matrices $\sigma_k$ for $k=1,2,3$ (which are trivially related to the $S_k$), we can obtain the coefficients in the above basis expansion and hence can reconstruct $\rho$ immediately. We further have that $$\langle \rho, \sigma_0\rangle = \frac{1}{2}\mathrm{Tr}\, \rho \,\mathbb I_2 = \frac{1}{2}\quad ,$$ by the trace normalization of $\rho$.
Finally, we can ask the following: Given some operator $\tilde \rho$ expanded in this basis, can we deduce from the coefficients of this basis expansion whether $\tilde \rho$ is a density matrix? This is discussed here.