  We study the dynamics and readout of topological qubits encoded by
zero-energy Majorana bound states in a topological superconductor. We take into
account bosonic modes due to the electromagnetic environment which couple the
Majorana manifold to above-gap continuum quasi-particles. This coupling causes
the degenerate ground state of the topological superconductor to be dressed in
a polaron-like manner by quasi-particle states and bosons, and the system to
become gapless. Topological protection and hence full coherence is only
maintained if the qubit is operated and read out within the low-energy spectrum
of the dressed states. We discuss reduction of fidelity and/or visibility if
this condition is violated by a quantum-dot readout that couples to the bare
(undressed) Majorana modes. For a projective measurement of the bare Majorana
basis, we formulate a Bloch-Redfield approach that is valid for weak
Majorana-environment coupling and takes into account constraints imposed by
fermion-number-parity conservation. Within the Markovian approximation, our
results essentially confirm earlier theories of finite-temperature decoherence
based on Fermi's golden rule. However, the full non-Markovian dynamics reveals,
in addition, the fidelity reduction by a projective measurement. Using a
spinless nanowire model with $p$-wave pairing, we provide quantitative results
characterizing these effects.
