Decoherence Free Subspaces and how they stay this way, using the Zeno Effect

You can find straightforward definitions and descriptions for example in this Review of Decoherence Free Subspaces, Noiseless Subsystems, and Dynamical Decoupling.

Formally, an open system is decoherence free if its evolution is unitary, despite non-vanishing couplings to its environment. Information encoded in system states that evolve in such unitary fashion is immune to decoherent environmental effects, although the latter are by no means suppressed.

The typical example is provided by a system S interacting with an environment E according to a total Hamiltonian $$ H = H_Sotimes I_E + I_Sotimes H_E + sum_k{S_kotimes E_k} $$ Here the interaction operators $S_k$, $E_k$ act only on the system and on the environment degrees of freedom, respectively. If it so happens that there exist system states $|psi_Srangle$ that are simultaneous eigenstates of all $S_k$, $$ S_k |psi_Srangle= sigma_k |psi_Srangle $$ then it holds that for any environment state $|phi_Erangle$, $$ H|psi_Sotimesphi_Erangle = left[ H_Sotimes I_E + I_Sotimes left(H_E + sum_k{sigma_k E_k}right)right] |psi_Sotimesphi_Erangle $$ and furthermore $$ e^{-iHt}|psi_Sotimesphi_Erangle = e^{-ileft[ H_Sotimes I_E + I_Sotimes left(H_E + sum_k{sigma_k E_k}right)right] t}|psi_Sotimesphi_Erangle = = left[ e^{-i ;H_Sotimes I_E t}|psi_Srangleright] otimesleft[ e^{-i; I_Sotimes left(H_E + sum_k{sigma_k E_k}right) t} |phi_Erangle right] $$ In other words, after tracing out the environment we find that the system evolves unitarily, decoherence-free, as if all external interactions were absent: $$ rho_S(t) = e^{-i H_S t}|psi_Srangle langle psi_S| e^{i H_S t} $$

We also note that in order for superpositions of distinct decoherence free (DF) system states to share the same unitary evolution, the DF states must belong to the same common eigen-subspace of the couplings ${S_k}_k$. Otherwise we obtain different generators $left(H_E + sum_k{sigma_k E_k}right)$ and tracing out the environment may no longer result in a unitary evolution. This is then a decoherence free subspace (DFS).

At a more general level, the irreps of the algebra generated by the couplings ${S_k}_k$ decomposes the system Hilbert space into a direct sum of formal tensor products of the form ${mathbb C}^n otimes {mathbb C}^d$, corresponding to "noiseless subsystems" living in the ${mathbb C}^n$ component and a "gauge" component ${mathbb C}^d$. That is, information stored in each ${mathbb C}^n$ is again naturally safe from environmental decoherence. The DFS case corresponds to the particular case of a "scalar gauge" with $d = 1$.