Locally incoherent witnessing of quantum coherence

Theoretical and experimental studies have suggested the relevance of quantum coherence to the performance of photovoltaic and light-harvesting complex molecular systems. However, there are ambiguities regarding the validity of statements we can make about the coherence in such systems. Here, we analyze the general procedure for coherence detection in quantum systems and show the counterintuitive phenomenon of detecting a quantum system's initial coherence when both the input and output states of the probe interacting with the system are locally completely incoherent. Our analysis yields the necessary and sufficient conditions for valid claims regarding the coherence of directly inaccessible systems. We further provide a proof-of-principle protocol that uses entangled probes to detect quantum coherence satisfying these conditions and discuss its potency for detecting coherence.

Figure 1

A coherence-generating quantum channel with incoherent input and output. In such circumstances, the quantum coherence generated in a directly inaccessible system seems undetectable.

Figure 2

Schematic of our coherence detection protocol. The probe and monitor are initially prepared in the maximally entangled state ${\varrho }_{\mathrm{mp}}^{\left(\mathrm{in}\right)}$. The probe mode then interacts with the directly inaccessible system which is part of the channel $\mathrm{\Lambda }$. We observe the coherence emerging in the state of the monitor conditioned on the outcomes of an incoherent measurement on the probe.

Figure 3

Schematic of our example. Two two-level systems, $\mathrm{p}$ and $\mathrm{m}$, are initially prepared in the maximally entangled state ${\varrho }_{\mathrm{mp}}^{\left(\mathrm{in}\right)}$. The probe mode exchanges excitation with the system $\mathrm{s}$, which is believed to be initially prepared in a superposition of its energy eigenstates at the rate ${\mathcal{g}}_{\mathrm{ps}}$. While the system and probe may suffer from dephasings with rates ${\gamma }_{\mathrm{s}}$ and ${\gamma }_{\mathrm{p}}$, respectively, the monitor is isolated and decoherence free. Observing coherence within the state of the monitor mode conditioned on the outcomes of an incoherent measurement on the probe certifies the coherence of system's initial state.

Figure 4

The evolution of quantum coherence within the state of the system in the first regime. The magnitude of the monitor's off-diagonal density matrix element conditioned on detecting a single excitation in the probe is depicted (a) versus the degree of the system's initial coherence $\epsilon$ at ${\mathcal{g}}_{\mathrm{ps}}t=0.75$ and (b) versus the normalized detection time ${\mathcal{g}}_{\mathrm{ps}}t$ for different values of $\epsilon$.

Figure 5

The evolution of quantum coherence within the state of the system in the second regime. The magnitude of the monitor's off-diagonal density matrix element conditioned on detecting a single excitation in the probe is depicted (a) versus the normalized probe-system interaction duration ${\mathcal{g}}_{\mathrm{ps}}{\tau }_{\mathrm{ps}}$ for the cases where the probe alone decoheres during the interaction, i.e., ${\gamma }_{\mathrm{p}}=0.1$ and ${\gamma }_{\mathrm{s}}=0$ (solid blue line), and where both the probe and the system decohere while interacting, i.e., ${\gamma }_{\mathrm{p}}=0.1$ and ${\gamma }_{\mathrm{s}}=0.1$ (dashed red line), and (b) versus the normalized evolution time of the monitor for ${\gamma }_{\mathrm{s}}={\gamma }_{\mathrm{p}}=0.1$ and two different values of interaction duration. In both graphs the probe continues to dephase after ${\tau }_{\mathrm{ps}}$ until the measurement is performed at an arbitrary time $t$.

Figure 6

Detected coherence for a probe which is partially dephased before its interaction with the system. The magnitude of the monitor's off-diagonal element of the density matrix conditioned on detecting a single excitation in the probe is plotted versus the normalized detection time ${\mathcal{g}}_{\mathrm{ps}}t$. The values of $\epsilon$ are chosen analogous to Fig. 4 in the main text. The solid lines are plotted for $\eta =1$, and the dashed lines correspond to $\eta =0.9$.