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.

Figure 1

Trade-off between coherence $C\left(\rho \right)$ and disturbance $D(\rho ,\mathcal{E})$ for the weak measurement channel. The figure shows coherence along the $X$ axis and disturbance along the $Y$ axis. Random states were generated and coherence and entropy were calculated using the matlab package [55].

Figure 2

Trade-off between coherence $C\left(\rho \right)$ and disturbance $D(\rho ,\mathcal{E})$ for the depolarizing channel. The figure shows coherence along the $X$ axis and disturbance along the $Y$ axis. Random states were generated and coherence and entropy were calculated using the matlab package [55].

Figure 3

Trade-off between coherence $C\left(\rho \right)$ and disturbance $D(\rho ,\mathcal{E})$ for the amplitude damping channel. The figure shows coherence along the $X$ axis and disturbance along the $Y$ axis. Random states were generated and coherence and entropy were calculated using the matlab package [55].

Figure 4

Trade-off between coherence $C\left(\rho \right)$ and disturbance $D(\rho ,\mathcal{E})$ for the depolarizing channel applied on a qutrit state. The figure shows coherence along the $X$ axis and disturbance along the $Y$ axis. The straight line corresponds to $2C_r\left(\rho \right)+D(\rho ,\mathcal{E})=2{log}_{2}3$. Random states were generated and coherence and entropy were calculated using the matlab package [55].

Figure 5

Trade-off between coherence $C\left(\rho \right)$ and disturbance $D(\rho ,\mathcal{E})$ for the amplitude damping channel applied on a qutrit state. The figure shows coherence along the $X$ axis and disturbance along the $Y$ axis. Random states were generated and coherence and entropy were calculated using the matlab package [55].

Figure 6

Trade-off between geometric measure of coherence and fidelity-based disturbance measure plotted obeying the relation (A3). Random states were generated using the matlab package [55].