Decoherence-free Subspaces - Alternative: Decoherence-free Subsystems

Alternative: Decoherence-free Subsystems

Consider a quantum system with an N-dimensional system Hilbert space that has a general subsystem decomposition The subsystem is a decoherence-free subsystem with respect to a system-environment coupling if every pure state in remains unchanged with respect to this subsystem under the OSR evolution. This is true for any possible initial condition of the environment. To understand the difference between a decoherence-free subspace and a decoherence-free subsystem, consider encoding a single qubit of information into a two-qubit system. This two-qubit system has a 4-dimensional Hilbert space; one method of encoding a single qubit into this space is by encoding information into a subspace that is spanned by two orthogonal qubits of the 4-dimensional Hilbert space. Suppose information is encoded in the orthogonal state in the following way:

This shows that information has been encoded into a subspace of the two-qubit Hilbert space. Another way of encoding the same information is to encode only one of the qubits of the two qubits. Suppose the first qubit is encoded, then the state of the second qubit is completely arbitrary since:

This mapping is a one-to-many mapping from the one qubit encoding information to a two-qubit Hilbert space. Instead, if the mapping is to, then it is identical to a mapping from a qubit to a subspace of the two-qubit Hilbert space.