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arXiv:2408.00088 · Physical Review A

Negativity of nondemolition quasiprobability distribution as a necessary and sufficient condition for macrorealism violation

Paolo Solinas, Stefano Gherardini

Published May 27, 2025• 0 citations
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Abstract

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.

DOI
10.1103/physreva.111.052217

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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.
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 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.
In this paper, we introduce the concept of quasi non-signaling in time (QNSIT), which extends non-signaling in time (NSIT) to quasiprobability distributions. QNSIT requires that the marginals of the quasiprobability distribution under consideration at intermediate times remain unperturbed by the measurement back-action induced by a sequence of quantum measurements. Associated with QNSIT, we propose a quasiprobabilistic characterization of macrorealism, MRqp{\rm MR}_{\rm qp}MRqp​, which requires that both the QNSIT conditions are satisfied and the quasiprobability distribution constructed from the measurement outcomes is positive. We prove that MRqp{\rm MR}_{\rm qp}MRqp​ implies NSIT, whereas the converse does not hold because of hidden superposition paths generated or preserved by the system dynamics. We also prove that, for a two-level system, the MRqp{\rm MR}_{\rm qp}MRqp​ and NSIT conditions coincide. MRqp{\rm MR}_{\rm qp}MRqp​ therefore provides a more sensitive tool for testing macrorealism and can be implemented experimentally through a quantum non-demolition measurement scheme.
In the tests for macrorealism proposed by Leggett and Garg, the temporal correlation functions of a dichotomic variable QQQ must be measured in a noninvasive way to rule out alternative classical explanations of Leggett-Garg inequality violations. Ideal negative measurements in which a null result is argued to be a noninvasive determination of the system's state are often used. From a quantum-mechanical perspective, such a measurement collapses the wave function and will therefore typically be found to be invasive under any experimental check. Here, a simple modified ideal negative measurement protocol is described for measuring the correlation functions, which is argued to be noninvasive from both classical and quantum perspectives and hence the noninvasiveness can then be checked experimentally, thereby permitting a quantitative measure of the degree of clumsiness of the measurement. It is also shown how this procedure may be extended to measure higher-order correlation functions, and a number of higher-order conditions for macrorealism are derived.
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.
Macroscopic realism is a set of assumptions about how we experience the world at a classical level. While the Leggett-Garg inequalities are temporal correlations that are violated by quantum systems not obeying such macrorealism, the no-signaling in time condition is also a necessary condition. This compares measurement outcomes with and without prior measurements. As dynamics and correlations play a central role in these measures, this paper explores the effects of regular versus chaotic dynamics on the violations of macroscopic realism. We observe a close connection between a 3 point out-of-time-order correlator and the conditional probabilities of measurement, and we find unmistakable imprints of chaos on the violations of macrorealism. We provide qualitative semiclassical reasoning for the numerical results involving a kicked top, and for two important initial states that behave very differently.