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Suppose I have a general 2-qubit state written in a basis consisting of tensor products of Pauli matrices:

$\rho=\frac{1}{4}\left[I_2\otimes I_2+\Sigma_{i} a_i \sigma_i\otimes I_2+\Sigma_{i} b_i I_2\otimes\sigma_i+\Sigma_{i,j} c_{ij} \sigma_i\otimes\sigma_j\right]$

Does there exist any unitary that converts $\rho$ into $\rho{'}$, with

$\rho{'}=\frac{1}{4}\left[I_2\otimes I_2+\lambda \sigma_z\otimes\sigma_z\right]$

What is the procedure by which we should search for the unitary?

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No, such a unitary cannot exists for any state $\rho$. To see this, note that the spectrum of $\rho'$ is $(1+\lambda, 1-\lambda, 1-\lambda, 1+\lambda)/4$, in particular, $\rho'$ is full rank, except for $\lambda=\pm1$ when it is rank 2. Now, simply take a rank one state (i.e. a pure state); it cannot be unitarily mapped onto $\rho'$ for any $\lambda$.

Note: You need three parameters to parametrize the spectrum.

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