Quantumness certification via non-demolition measurements

TL;DR

This review argues that quantum Non-Demolition Measurements (QNDM) offer a practical, necessary-and-sufficient criterion for certifying quantum behavior by diagnosing macrorealism violations. By encoding multi-time information about an observable into a detector phase, QNDM produces a quasi-probability $ P_{ m QNDM}(ullet)$ whose negativity signals quantum coherence beyond classical trajectories, linking directly to macrorealism per se (MRps) violations and the Leggett-Garg framework. The approach applies to driven and open quantum systems, enabling quantum work and heat statistics without destroying coherence, and shows robustness against noise while outperforming Leggett-Garg inequalities in identifying quantum features. It also enables tracking quantum-to-classical transitions as environmental coupling increases, highlighting practical pathways for foundational tests and quantum information applications. The work points to future directions including deeper connections between negativity and coherence measures, scalability to larger systems, and ties to quantum computation resources such as path coherence and quasi-probability frameworks.

Abstract

The fundamental question of when a static or dynamic system should be deemed intrinsically quantum remains a challenge to address in absolute terms. A rigorous criterion, however, can be established by focusing on the measurable or reconstructible features of the system. This determination transcends mere issues of a system's classical simulability or computational complexity. Instead, the critical requirement lies in the certification (ideally, in real-time) of the emergence and persistence of genuine quantum features, principally entanglement and quantum superposition. Quantum Non-Demolition Measurements (QNDM) serve as the appropriate instrument for this certification, both from a theoretical and experimental standpoint. In this review paper, we demonstrate, with accessible clarity, how the implementation of QNDM can be directly linked to a necessary and sufficient condition for the violation of macrorealism in finite-dimensional systems, establishing a conceptual parallel with Leggett-Garg inequalities. Using concrete examples that detail the detection of negative terms in the quasi-probability density function resulting from QNDM, we introduce the core concepts for certifying genuinely quantum features. As specific examples, we discuss an application where the quantum-to-classical transition due to the interaction with an environment can be tracked by QNDM. Moreover, we argue about the robustness of QNDM protocols in the presence of noise sources and their advantages with respect to the Leggett-Garg inequalities. Because of its straightforward implementation, the QNDM approach can be of direct relevance to both the foundations of quantum mechanics and quantum information theory, where a controlled generation and certification of genuinely quantum resources is a central concern.

Figures (5)

• Figure 1: A pictorial representation of the system's time evolution in terms of the outcomes from sequential measurements performed at three times $(t_0,t_1,t_2)$. In this picture, the trajectories developed by the measured system are ascribable to Feynman paths built over the measurement outcomes. The possible outcomes of the measurements are $a_i$ and $a_l$ at time $t_0$, $a_j$ and $a_m$ at time $t_1$, and $a_k$ at time $t_2$. The sequence of outcomes identifies the paths that the monitored system could follow during its time evolution. The red dashed curve represents a classical (macrorealistic) path where, at any time, we have a single outcome (e.g., $a_i \rightarrow a_m \rightarrow a_k$) so that the observable has a determinate outcome. The blue curves present quantum paths in which the system is in a superposition of states with different outcomes. In this case, the statistics of outcomes got from measuring $\hat{A}$ change as the paths are no longer classical and violate the macrorealism requirement.
• Figure 2: Comparison between the prediction of the LGI and QNDM approach. In the top-left panel, we plot the LG parameter $K$ as a function of $\omega\tau$. The shaded regions are the ones for which the LGI is violated. The other panels show the quasi-probability distribution $\mathcal{P}_{\rm QNDM}(\Delta)$ evaluated at three different values of $\omega\tau$. The distribution $\mathcal{P}_{\rm QNDM}(\Delta)$ always shows negative regions, while only for the green circle point the LGI is violated.
• Figure 3: a) Simulation results for the computation of the LG parameter $K = C_{01} + C_{12} - C_{02}$ in a scenario where the measurement statistical uncertainty is present, with $N_{\text{shots}} = 10^4$. The solid blue line represents the simulated average curve (obtained over $N_{\text{shots}}$ repetitions), while the dashed red line corresponds to the theoretical prediction as given in the example at the end of Sec. \ref{['sec:QNDM_vs_LGIs']}. The blue-shaded region origins from the statistical uncertainty, while the yellow-shaded regions highlight the range of $\omega\tau$ where, considering statistical errors, a violation of the LGI is confidently detected. b) Plot of the quasi-probability density functions $\mathcal{P}_{\rm QNDM}(\Delta)$ for $\delta\lambda = 1$ and $N_{\text{shots}} = 10^{2}$. These curves are obtained by taking the inverse Fourier transform of the simulated data for $\omega\tau=1.5$, and the orange error bars represent the statistical uncertainty. Notice that, in order to extract single probability values from $\mathcal{P}_{\rm QNDM}(\Delta)$, we must integrate it over a fixed interval of $\Delta$. The negative region in $\mathcal{P}_{\rm QNDM}(\Delta)$ is visible for any value of $\omega\tau$. These panels are adapted from figures in Ref. melegari2025.
• Figure 4: a) Simulation results for the computation of the LG parameter $K = C_{01} + C_{12} - C_{02}$ in the presence of both environmental noise and quantum gate errors for $N_{\text{shots}} = 10^4$. b) Plot of $\mathcal{P}_{\rm QNDM}(\Delta)$ for $\delta\lambda = 1$ and $N_{\text{shots}} = 10^{2}$. In all panels, the notation and parameters are as in Fig. \ref{['fig:QNDM_LGI_comparison_noiseless']}. These panels are adapted from figures in Ref. melegari2025.
• Figure 5: QNDM quasi-probability density function $\mathcal{P}_{\Delta U}$ of the internal energy variation $\Delta U$ for a qubit interacting with an environment inducing relaxation. The relaxation strength is parameterized by $p$ that goes from $p=0$ (no dissipation) to $p=1$ (full relaxation process). $\mathcal{P}_{\Delta U}$ is calculated using experimental data obtained from the IBM quantum processor IBMQ-VIGO qiskit2024. The blue dots and the red triangles are the predictions of the QNDM and TPM schemes, respectively. In the experiments, we set $\theta=0.7$, $\phi=1.2$ for the initial state, and $\alpha=1$, $\beta=0.5$ for the system dynamics. The figure is taken from Ref. solinas2021.