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arXiv:2609.24169

No violation on a generalisation of Leggett-Garg inequality and Bell-CHSH inequality with extended probability

Sirawit Kajonsombat, Pongwit Srisangyingcharoen

quant-phSubmitted Sep 21, 2026• Announced Sep 22, 2026• 0 citations
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Abstract

A generalisation of the Leggett-Garg inequality (LGI) and the Bell-CHSH inequality is proposed in this work, by replacing a classical probability notion with an extended probability notion. The extended probability serves as an underlying layer of reality to the classical probability. Within the consistent history framework, the underlying layer is so-called a non-settleable history. Hence, the bounds of the LGIs and Bell-CHSH inequalities are extended. The resulting generalised inequalities are satisfied without violation for arbitrary measurement settings. Furthermore, generalised Macro- realism and generalised Local realism are introduced and analysed within the extended probability framework without relying on an explicit measurement. This opens up further possibilities on one of the great pursuits in physics, understanding on how a quantum system can possess classical properties.

Comments
9 pages, 4 figures

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A generalisation of the Leggett-Garg inequality (LGI) and the Bell-CHSH inequality is proposed in this work, by replacing a classical probability notion with an extended probability notion. The extended probability serves as an underlying layer of reality to the classical probability. Within the consistent history framework, the underlying layer is so-called a non-settleable history. Hence, the bounds of the LGIs and Bell-CHSH inequalities are extended. The resulting generalised inequalities are satisfied without violation for arbitrary measurement settings. Furthermore, generalised Macro- realism and generalised Local realism are introduced and analysed within the extended probability framework without relying on an explicit measurement. This opens up further possibilities on one of the great pursuits in physics, understanding on how a quantum system can possess classical properties.
In 1985, Leggett and Garg put forward the concept of macroscopic realism (macrorealism) and, in analogy to Bell's theorem, derived a necessary condition in terms of inequalities, which are now known as the Leggett-Garg inequalities. In this paper, we discuss another necessary condition called no-signaling in time. It solely bases on comparing the probability distribution for a macrovariable at some time for the cases where previously a measurement has or has not been performed. Although the concept is analogous to the no-signaling condition in the case of Bell tests, it can be violated according to quantum mechanical predictions even in situations where no violation of Leggett-Garg inequalities is possible.
The Leggett-Garg inequalities are a set of inequalities obeyed by classical systems but violated in quantum theory. Their violation has been taken as evidence that quantum theory lacks a `realistic' formulation. However in addition to realism the derivation of the Leggett-Garg inequalities relies on another, more obscure assumption of `non-invasive measurability.' The significance of this assumption and the consequences for the interpretation of violations of the Leggett-Garg inequalities have been hotly debated. In this paper we present a pedagogical introduction to the issues, focussing on the significance of non-invasive measurability. We give a simple derivation of the Leggett-Garg inequalities paying particular attention to where and when the two assumptions are used, and we give an example of a realist but not non-invasively measurable hidden variable theory which violates them. We also discuss recent attempts to experimentally implement non-invasive measurements via so called `undetectable measurements' and we show that even undetectable measurements are not necessarily non-invasive.
The Bell inequality constrains the outcomes of measurements on pairs of distant entangled particles. The Bell contradiction states that the Bell inequality is inconsistent with the calculated outcomes of these quantum experiments. This contradiction led many to question the underlying assumptions, viz. so-called realism and locality. The probability model underlying the Bell inequality is generally left implicit. We propose an explicit probability model for the CHSH version of the Bell experiment. This model has only two simultaneously observable detector settings per measurement, and therefore does not assume realism. The quantum expectation now becomes a conditional expectation, given the two detector settings. This probability model is in full agreement with both quantum mechanics and experiments. As a result, the model satisfies the Bell inequality; there are no so-called violations. We extend this model to include a hidden variable. This extended model is not Bell-separable. This non-separability implies that the model is non-deterministic or non-local (or both).
The Leggett-Garg inequality is a widely used test of the ``quantumness'' of a system and involves correlations between measurements realized at different times. According to its widespread interpretation, a violation of the Leggett-Garg inequality disproves macroscopic realism and noninvasiveness. Nevertheless, recent results point out that macroscopic realism is a model-dependent notion and that one should always be able to attribute to invasiveness a violation of a Leggett-Garg inequality. This opens some natural questions: How do we provide such an attribution in a systematic way? How can apparent macroscopic realism violation be recast into a dimensional-independent invasiveness model? The present work answers these questions by introducing an operational model where the effects of invasiveness are controllable through a parameter associated with what is called the measurability of the physical system. Such a parameter leads to different generalized measurements that can be associated with the dimensionality of a system, to measurement errors, or to back action.
The Leggett-Garg inequality (LGI) distinguishes nonmacrorealistic channels from macrorealistic ones by constraining the experimental outcomes of the underlying system. In this work, we propose a class of channels which, initially, cannot violate the LGI (in the form of the temporal Bell inequality) but can violate it after the application of stochastic pre- and post-operations (SPPOs). As a proof-of-principle experiment, we demonstrate the stochastic pre- and post-operations in an amplitude-damping channel with photonic qubits. We denote the above phenomenon as hidden nonmacrorealistic channels. We also discuss the relationship between these hidden nonmacrorealistic channels [in terms of the temporal Clauser-Horne-Shimony-Holt (CHSH) inequality] and the strongly nonlocality-breaking channel, which breaks the hidden spatial CHSH nonlocality for arbitrary states. In general, if the channel satisfies hidden nonmacrorealism, it is not a strongly CHSH nonlocality-breaking channel.
It remains an open question how a realist view of the macroscopic world emerges from a quantum formalism. For testing the macrorealism in the quantum domain, an interesting approach was put forward by Leggett and Garg in 1985, by formulating a suitable inequality valid for any macrorealistic theory. Recently, by following the Wigner idea of local realist inequality, a probabilistic version of standard Leggett-Garg inequalities has also been proposed. While the Wigner form of local realist inequalities is equivalent to the two-party, two-measurements and two-outcomes CHSH inequalities, in this paper we provide a generic proof to demonstrate that the Wigner form of Leggett-Garg inequalities is not only inequivalent to the standard ones, but also stronger than the latter. This is demonstrated by quantifying the amount of disturbance caused by a priori measurement to the subsequent measurements. In this connection, the relation between LGIs and another formulation of macrorealism known as no-signaling in time is examined.