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arXiv:2610.08146

Operational quasiprobability formulation of non-signaling in time for macrorealism violations

Stefano Gherardini, Paolo Solinas

quant-phSubmitted Oct 6, 2026• Announced Oct 7, 2026• 0 citations
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Abstract

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, ${\rm MR}_{\rm qp}$, which requires that both the QNSIT conditions are satisfied and the quasiprobability distribution constructed from the measurement outcomes is positive. We prove that ${\rm MR}_{\rm qp}$ 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 ${\rm MR}_{\rm qp}$ and NSIT conditions coincide. ${\rm MR}_{\rm qp}$ therefore provides a more sensitive tool for testing macrorealism and can be implemented experimentally through a quantum non-demolition measurement scheme.

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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.
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.
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Macroscopic realism, the classical world view that macroscopic objects exist independently of and are not influenced by measurements, is usually tested using Leggett-Garg inequalities. Recently, another necessary condition called no-signaling in time (NSIT) has been proposed as a witness for nonclassical behavior. In this paper, we show that a combination of NSIT conditions is not only necessary but also sufficient for a macrorealistic description of a physical system. Any violation of macroscopic realism must therefore be witnessed by a suitable NSIT condition. Subsequently, we derive an operational formulation for NSIT in terms of positive operator-valued measures and the system Hamiltonian. We argue that this leads to a suitable definition of ``classical'' measurements and Hamiltonians, and we apply our formalism to some generic coarse-grained quantum measurements.
We consider two different types of conditions which were proposed to test macrorealism in the context of a system described by a single dichotomic variable QQQ. This is the view that a macroscopic system evolving in time possesses definite properties which can be determined without disturbing the future or past state. The Leggett-Garg (LG) inequalities, the most commonly studied test, are only necessary conditions for macrorealism, but, building on earlier work J. J. Halliwell, Phys. Rev. A 93, 022123 (2016), it is shown that when the four three-time LG inequalities are augmented with a certain set of two-time inequalities also of the LG form, Fine's theorem applies and these augmented conditions are then both necessary and sufficient. A comparison is carried out with a very different set of necessary and sufficient conditions for macrorealism, namely the no-signaling in time (NSIT) conditions proposed by J. Kofler and C. Brukner, Phys. Rev. A 87, 052115 (2013) and L. Clemente and J. Kofler, Phys. Rev. A 91, 062103 (2015), which ensure that all probabilities for QQQ at one and two times are independent of whether earlier or intermediate measurements are made in a given run, and do not require (but imply) the LG inequalities. We argue that tests based on the LG inequalities have the form of very weak classicality conditions and can be satisfied in the face of moderate interference effects, but those based on NSIT conditions have the form of much stronger coherence witness conditions, satisfied only for zero interference. The two tests differ in their implementation of noninvasive measurability and so are testing different notions of macrorealism: the augmented LG tests are indirect, entailing a combination of the results of different experiments with only compatible quantities measured in each experimental run, in close analogy with Bell tests, and are primarily tests for macrorealism per se; in contrast, the NSIT tests entail sequential measurements of incompatible quantities and are primarily tests for noninvasiveness.
We introduce the terminology \emph{passive realism} to encapsulate the scale-independent assumption that a physical system's pre-existing properties can be passively observed without altering its subsequent dynamics. A necessary condition for passive realism is provided by the no-signalling in time (NSIT) condition, which states that a measurement outcome is independent of whether or not a prior measurement was performed. While closed quantum systems generically violate NSIT due to measurement back action, we investigate whether open-system dynamics can dissipate this disturbance to the environment and restore NSIT. By modeling the time evolution between sequential measurements as an arbitrary quantum channel, we show that such environmental masking is possible only in two special cases. Specifically, we show that if the initial state of a two-time sequential measurement scenario is not maximally mixed, and if the channel governing the dynamics between measurements is not a discard-and-prepare channel, then there necessarily exist binary projective measurements violating NSIT, hence violating passive realism. By providing a constructive procedure for determining these violating measurements---applied here to depolarizing and dephasing channels---we demonstrate that, outside these two cases, system-environment interactions cannot erase the statistical footprint of a quantum measurement for all measurement choices.
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.