DFSs can also be characterized as a special class of quantum error correcting codes.
In this representation they are passive error-preventing codes since these subspaces are encoded with information that (possibly) won't require any active stabilization methods.
These subspaces prevent destructive environmental interactions by isolating quantum information.
Decoherence creates problems in this regard by causing loss of coherence between the quantum states of a system and therefore the decay of their interference terms, thus leading to loss of information from the (open) quantum system to the surrounding environment.
The study of DFSs began with a search for structured methods to avoid decoherence in the subject of quantum information processing (QIP).
The methods involved attempts to identify particular states which have the potential of being unchanged by certain decohering processes (i.e. certain interactions with the environment).
Ekert, who studied the consequences of pure dephasing on two qubits that have the same interaction with the environment.
Noteworthy is also independent work by Martin Plenio, Vlatko Vedral and Peter Knight who constructed an error correcting code with codewords that are invariant under a particular unitary time evolution in spontaneous emission.
[2] Shortly afterwards, L-M Duan and G-C Guo also studied this phenomenon and reached the same conclusions as Palma, Suominen, and Ekert.
However, such models were limited since they dealt with the decoherence processes of dephasing and dissipation solely.
To deal with other types of decoherences, the previous models presented by Palma, Suominen, and Ekert, and Duan and Guo were cast into a more general setting by P. Zanardi and M. Rasetti.
They expanded the existing mathematical framework to include more general system-environment interactions, such as collective decoherence-the same decoherence process acting on all the states of a quantum system and general Hamiltonians.
Their analysis gave the first formal and general circumstances for the existence of decoherence-free (DF) states, which did not rely upon knowing the system-environment coupling strength.
Zanardi and Rasetti called these DF states "error avoiding codes".
Subsequently, Daniel A. Lidar proposed the title "decoherence-free subspace" for the space in which these DF states exist.
This observation discerned another prerequisite for the possible use of DF states for quantum computation.
A thoroughly general requirement for the existence of DF states was obtained by Lidar, D. Bacon, and K.B.
[3] A subsequent development was made in generalizing the DFS picture when E. Knill, R. Laflamme, and L. Viola introduced the concept of a "noiseless subsystem".
[1] Knill extended to higher-dimensional irreducible representations of the algebra generating the dynamical symmetry in the system-environment interaction.
Earlier work on DFSs described DF states as singlets, which are one-dimensional irreducible representations.
This work proved to be successful, as a result of this analysis was the lowering of the number of qubits required to build a DFS under collective decoherence from four to three.
[1] The generalization from subspaces to subsystems formed a foundation for combining most known decoherence prevention and nulling strategies.
Consider an N-dimensional quantum system S coupled to a bath B and described by the combined system-bath Hamiltonian as follows:
The Lindblad decohering term determines when the dynamics of a quantum system will be unitary; in particular, when
will remain mutually distinguishable after a decohering process since their respective eigenvalues are degenerate and hence identifiable after action under the error generators.
[4] In this picture DFSs are sets of states (codes rather) whose mutual distinguishability is unaffected by a process
Since DFSs can encode information through their sets of states, then they are secure against errors (decohering processes).
In this way DFSs can be looked at as a special class of QECCs, where information is encoded into states which can be disturbed by an interaction with the environment but retrieved by some reversal process.
This code can be implemented to protect against decoherence and thus prevent loss of information in a small section of the system's Hilbert space.
The errors are caused by interaction of the system with the environment (bath) and are represented by the Kraus operators.