# Physical examples and implementations of covariant/symmetric quantum operations

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In quantum information theory a covariant quantum operation $\Phi:\mathcal{B}(\mathcal{H}_1)\rightarrow \mathcal{B}(\mathcal{H}_2$) is an operation that is invariant under a group action that characterizes the symmetry of the input and output system. In other words, given unitary representations $U_1$ and $U_2$ of $G$ such a map satisfies:
$$U(2)(g)^{\dagger}\Phi(U_1(g)\rho U_1(g)^{\dagger})U_2(g)=\rho$$
for all $\rho\in \mathcal{B}( \mathcal{H}_{1})$ and for all $g\in G$.

One could construct such a map for example by using the dynamical evolution of a system and ensuring that the Hamiltonian H commutes with the unitary matrices $U_1(g)$ and $U_2(g)$: $[U_1(g),H]=[U_2(g),H]=0$. However, how often do such covariant quantum operations appear physically? What would be some non-trivial real physical examples?

asked Sep 18, 2015

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