Subtleties of witnessing quantum coherence in nonisolated systems

Identifying nonclassicality unambiguously and inexpensively is a long-standing open challenge in physics. The no-signaling-in-time protocol was developed as an experimental test for macroscopic realism, and serves as a witness of quantum coherence in isolated quantum systems by comparing the quantum state to its completely dephased counterpart. We show that it provides a lower bound on a certain resource-theoretic coherence monotone. We go on to generalize the protocol to the case where the system of interest is coupled to an environment. Depending on the manner of the generalization, the resulting witness either reports on system coherence alone, or on a disjunction of system coherence with either (i) the existence of nonclassical system-environment correlations or (ii) non-negligible dynamics in the environment. These are distinct failure modes of the Born approximation in nonisolated systems.

Figure 1

Procedures for extracting coherence witnesses (time runs from top to bottom): (a) In the usual NSIT protocol the system is considered well isolated. (b) Our generalizations include an explicit environment, with each interruption experiment (fast operations $i=\text{I,}\text{II}$ and slow operations $i=\text{III,}\text{IV}$) carrying a cost in terms of the number of subexperiments required. ${\mathcal{E}}_{\mathbb{I}}$ is the identity channel.

Figure 2

(a) In the general case of a nonisolated system, it is not possible to separate preparation and measurement of the system state at $\tau$ as is possible for an isolated system. Instead, a generalized witness compares the effect of a fixed superchannel $\mathcal{S}$ (which represents the combined effect of the correlated system-environment state at $\tau$ and their joint evolution) on an input interruption channel: either the identity channel ${\mathcal{E}}_{\mathbb{I}}$ or the classicalization channel $\mathrm{\Gamma }$ [Eq. (1) or (2)] acting on the system. The associated witness $W^a$ is sensitive to system coherence and/or quantum system-environment correlations at $\tau$. (b) When the Born approximation holds, the environment stays in the same state at all times and therefore need not be actively reset (left panel). ${\mathcal{E}}_{\mathrm{measure}}^{\text{IV}}$ is used to effectively alter the final measurement that is performed. Otherwise, when the Born approximation fails, classical models may explain the discrepancy between $P$ and $P^{\prime }$ (right panel). (c) The Born approximation can be enforced by performing full quantum state tomography of the system at $\tau$. The associated witness $W^c$ is sensitive only to coherence in the reduced state of the system at $\tau$, but requires order $d^2$ experiments due to the implicit full state tomography required.