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arXiv:2502.17040

Quantum simulations of macrorealism violation via the QNDM protocol

Dario Melegari, M. Cardi, Paolo Solinas

Published Feb 24, 2025• 0 citations
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Abstract

The Leggett-Garg inequalities have been proposed to identify the quantum behaviour of a system; specifically, the violation of macrorealism. They are usually implemented by performing two sequential measurements on quantum systems, calculating the correlators of such measurements and then combining them arriving at Leggett-Garg inequalities. However, this approach only provides sufficient conditions for the violation of macrorealism. Recently, it was proposed an alternative approach that uses non-demolition measurements and gives both a necessary and sufficient condition for the violation of macrorealism. By storing the information in a quantum detector, it is possible to construct a quasi-probability distribution whose negative regions unequivocally identify the quantum behaviour of the system. Here, we perform a detailed comparison between these two approaches. The use of the IBM quantum simulators allows us to evaluate the performance in real-case situations and to include both the statistical and environmental noise. We find that the non-demolition approach is not only able to always identify the quantum features but it requires fewer resources than the standard Leggett-Garg inequalities. In addition, while the efficiency of the latter is strongly affected by the presence of the noise, the non-demolition approach results incredibly robust and its efficiency remains unchanged by the noise. These results make the non-demolition approach a viable alternative to the Leggett-Garg inequalities to identify the violation of macrorealism.

DOI
10.48550/arxiv.2502.17040

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The Leggett-Garg inequalities have been proposed to identify the quantum behaviour of a system; specifically, the violation of macrorealism. They are usually implemented by performing two sequential measurements on quantum systems, calculating the correlators of such measurements and then combining them arriving at Leggett-Garg inequalities. However, this approach only provides sufficient conditions for the violation of macrorealism. Recently, it was proposed an alternative approach that uses non-demolition measurements and gives both a necessary and sufficient condition for the violation of macrorealism. By storing the information in a quantum detector, it is possible to construct a quasi-probability distribution whose negative regions unequivocally identify the quantum behaviour of the system. Here, we perform a detailed comparison between these two approaches. The use of the IBM quantum simulators allows us to evaluate the performance in real-case situations and to include both the statistical and environmental noise. We find that the non-demolition approach is not only able to always identify the quantum features but it requires fewer resources than the standard Leggett-Garg inequalities. In addition, while the efficiency of the latter is strongly affected by the presence of the noise, the non-demolition approach results incredibly robust and its efficiency remains unchanged by the noise. These results make the non-demolition approach a viable alternative to the Leggett-Garg inequalities to identify the violation of macrorealism.
The Leggett-Garg inequalities have been proposed to identify the quantum behavior of a system; specifically, the violation of macrorealism. They are usually implemented by performing two sequential measurements on quantum systems, calculating the correlators of such measurements and then combining them arriving at Leggett-Garg inequalities. However, this approach only provides sufficient conditions for the violation of macrorealism. Recently, an alternative approach was proposed that uses nondemolition measurements and gives both a necessary and sufficient condition for the violation of macrorealism. By storing the information in a quantum detector, it is possible to construct a quasiprobability distribution whose negative regions unequivocally identify the quantum behavior of the system. Here, we perform a detailed comparison between these two approaches. The use of the IBM quantum simulators allows us to evaluate the performance in real-case situations and to include both the statistical and environmental noise. We find that the nondemolition approach is not only able to always identify the quantum features, but it requires fewer resources than the standard Leggett-Garg inequalities. In addition, while the efficiency of the latter is strongly affected by the presence of the noise, the nondemolition approach results incredibly robust and its efficiency remains unchanged by the noise. These results make the nondemolition approach a viable alternative to the Leggett-Garg inequalities to identify the violation of macrorealism.
The fundamental question of when a static or dynamic system should be deemed intrinsically quantum remains a challenge to address in absolute terms. In this regard, a 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 explain, with accessible clarity, how the implementation of QNDM can be directly linked to a necessary and sufficient condition for the presence of genuinely quantum features in the system's state monitored over time 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 quantumness certification. 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 standard Leggett-Garg inequalities defined by two-time correlators.
Quantum nondemolition measurements define a noninvasive protocol to extract information from a monitored quantum system. This protocol exploits an additional quantum system that is sequentially coupled to the system and eventually measured to reconstruct temporal correlations developed by the system. In reference to a given measurement observable, the protocol's outcomes are well described by a quasiprobability distribution, which can be negative. In this paper, we prove that the presence of negative regions in this distribution defined at three times is a necessary and sufficient condition for the violation of macrorealism. This approach falls within the category of sequential noninvasive measurability and automatically satisfies all no-signaling-in-time conditions at multiple times by design. We show that reconstructing the negativity is a much stronger condition than the violation of the Leggett-Garg inequalities, commonly used for the same task, as they may be satisfied even if the macrorealism is violated. Quantum nondemolition measurements provide a clear and operational tool to identify violations of macrorealism directly. Thus, our result could also be applied to the study of the quantum-to-classical transition and in all those fields where quantumness certification is desirable.
In 1985, Leggett and Garg formulated a class of inequalities for testing the compatibility between macrorealism and quantum mechanics. In this paper, we point out that, based on the same assumptions of macrorealism that are used to derive Leggett-Garg inequalities (LGIs), there is a scope of formulating another class of inequalities different from the standard LGIs. By considering the three-time measurement scenario in a dichotomic system, we first propose an interesting variant of the standard LGIs and show that its quantum violation is larger than the standard LGIs. By extending this formulation to the nnn-time measurement scenario, we found that the quantum violations of variants of the LGIs for a qubit system increase with nnn, and, for a sufficiently large nnn, an algebraic maximum can be reached. We then compare the violations of the standard and variants of the LGIs in an unsharp measurement scenario and show that, for any arbitrary nnn, the violation of the latter is more robust to unsharpness than the former. Furthermore, we examine the relation between the quantum violations of the variants of the LGIs and an another formulation macrorealism, known as no-signaling in time conditions.
Standard Leggett-Garg inequalities (SLGIs) were formulated for testing incompatibility between the classical world view of macrorealism and quantum mechanics. In recent times, various other formulations such as the Wigner form of Leggett-Garg inequalities (WLGIs), entropic Leggett-Garg inequalities (ELGIs), and the no-signaling-in-time (NSIT) condition have also been proposed. It was also recently argued that nonono set of SLGIs can provide the necessary and sufficient conditions for macrorealism, but a suitable conjunction of NSIT conditions provides the same. In this paper, we first provide a comparative study of the various formulations of Leggett-Garg inequalities (LGIs) for testing macrorealism pertaining to the two different unsharp measurements. While the violations of WLGIs are more robust than SLGIs and ELGIs for spin positive operator-valued measures (POVMs), here we demonstrate that for the case of biased POVMs the quantum violations of both SLGIs and ELGIs provide the same robustness as WLGIs. Importantly, the violations of all formulations of LGIs can be achieved for any nonzero value of unsharpness parameter. We have also studied the connection between LGIs and NSIT conditions. Further, we investigate the role of the joint measurability of the POVMs in the violation of LGIs and find that there is nonono generic connection.
The Leggett-Garg inequality holds for any macrorealistic system that is being measured noninvasively. A violation of the inequality can signal that a system does not conform to our primal intuition about the physical world. Alternatively, a violation can simply indicate that a ``clumsy'' experimental technique led to invasive measurements. Here, we consider a recent Leggett-Garg test designed to try to rule out the mundane second possibility. We tailor this Leggett-Garg test to the IBM 5Q Quantum Experience system and find compelling evidence that qubit Q2{Q}_{2}Q2​ of the system cannot be described by noninvasive macrorealism.