QuantumInfo.Operators.Unitary

Let $\rho$ be a mixed quantum state of dimension $d$ (an element of `MState d`) and $U$ be a unitary matrix in the unitary group $U(d)$. The function `MState.uConj` computes the conjugation of the state $\rho$ by the unitary $U$, resulting in a new mixed state defined by $U \rho U^\dagger$, where $U^\dagger$ denotes the conjugate transpose of $U$. This operation represents the unitary evolution of the density matrix.

For any mixed quantum state $\rho$ of dimension $d$ and any unitary matrix $U \in U(d)$, the eigenvalue spectrum of the state $U \rho U^\dagger$ (the unitary conjugation of $\rho$) is equal to the eigenvalue spectrum of the original state $\rho$.

For any two mixed quantum states $\rho$ and $\sigma$ of dimension $d$, and any unitary matrix $U$ in the unitary group $U(d)$, the inner product of the states after unitary conjugation by $U$ is equal to the inner product of the original states: $$\langle U \rho U^\dagger, U \sigma U^\dagger \rangle_{\text{Prob}} = \langle \rho, \sigma \rangle_{\text{Prob}}$$ where $\langle \cdot, \cdot \rangle_{\text{Prob}}$ denotes the Hilbert-Schmidt inner product for mixed states.