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We are given a generic 2-qubit density matrix

$$\rho=\frac{1}{4}\left[I_4+\Sigma_i a_i \sigma_i \otimes I_2 + \Sigma_i b_i I_2 \otimes \sigma_i + \Sigma_{i,j} c_{ij} \sigma_i \otimes \sigma_j\right]$$

And we want to check if $\rho$ can be unitarily (using a local or global unitary) converted to simpler form

$$\rho'=\frac{1}{4}\left[I_4+\lambda_1 \sigma_z \otimes I_2 + \lambda_2 I_2 \otimes \sigma_z + \lambda_3 \sigma_z \otimes \sigma_z\right].$$

How can we check whether such unitaries exist or not?

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Such a unitary exists for every two-qubit density matrix. By the spectral theorem, for every density matrix $\rho$ there exists a unitary matrix such that $U\rho U^\dagger$ is a real diagonal matrix. Moreover, the matrices $I_4$, $\sigma_z\otimes I_2$, $I_2\otimes\sigma_z$ and $\sigma_z\otimes\sigma_z$ are a basis of the real vector space of diagonal $4\times 4$ matrices with real entries. Consequently, every real diagonal matrix can be written as a linear combination of $I_4$, $\sigma_z\otimes I_2$, $I_2\otimes\sigma_z$ and $\sigma_z\otimes\sigma_z$.

In fact, the basis is orthogonal with respect to Hilbert-Schmidt inner product, so $\lambda_i$ may be easily computed using the inner product.

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