Trade-off relation for coherence and disturbance

The measurement process necessarily disturbs the state of a quantum system, unless the state is an eigenstate of the observable being measured. Once we perform a complete measurement in a given basis, the system undergoes decoherence and loses its coherence. If there is no disturbance, the state retains all of its coherence. It is therefore natural to ask if there is a trade-off between disturbance caused to a state and its coherence. We present coherence disturbance trade-off relations using the relative entropy of coherence for measurement channel as well as for general channels (completely positive trace-preserving maps). For bipartite states we prove a trade-off relation between the quantum coherence, entanglement, and disturbance. Similar relation also holds for quantum coherence, quantum discord, and disturbance for any bipartite state. We illustrate the trade-off between the coherence and the disturbance for single-qubit and - qutrit states subject to various quantum channels.