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

For decades, physicists have been trying to solve a quiet but profound puzzle: how does the strange, fuzzy world of the very small become the solid, predictable world we see every day? In the realm of atoms and electrons, things do not behave like the objects on our kitchen tables. A particle can exist in multiple states at once, and measuring it seems to change its reality. This disconnect has led to famous tests, such as the Bell-CHSH inequality and the Leggett-Garg inequality, which act like strict gatekeepers. These tests are designed to see if a system follows the rules of classical reality—where things have definite properties whether we look at them or not—or if they remain stubbornly quantum. For a long time, the answer seemed clear: quantum systems break these rules, proving they cannot be described by classical logic.

However, a new perspective suggests that the problem might not be with the quantum system itself, but with the mathematical tools we use to describe it. Traditional probability, the kind used in everyday life and standard physics, insists that the chance of an event happening must be a number between zero and one. It cannot be negative. But in the quantum world, calculations sometimes require numbers that act like negative probabilities to make the math work. A team of researchers from Naresuan University in Thailand has proposed a way to bridge this gap. They suggest replacing the strict rules of classical probability with a broader concept called "extended probability." This approach allows for these strange, negative values to exist as a hidden layer of reality beneath the surface we can observe. By doing so, they have rewritten the rules of the famous tests, creating new versions that quantum systems can pass without breaking a sweat.

The researchers began by looking at the Leggett-Garg inequality, a test that checks if a single system behaves consistently over time. In the classical view, a system should have a definite state at every moment, and measuring it should not disturb that state. Quantum mechanics usually fails this test because the act of measuring seems to force the system into a new state, violating the idea of a pre-existing reality. The team from Thailand took the equations for this test and swapped the standard probability for their extended version. In this new framework, the probability of a sequence of events can include negative values, which act as a buffer against the usual violations. When they ran simulations using this new math, the results were striking. The quantum system no longer broke the rules. Instead, it satisfied a generalized version of the inequality perfectly, regardless of how the measurements were set up.

This success led the authors to rethink what it means for a system to be "real." They introduced the idea of "generalized macrorealism." In the old view, a system is macrorealist only if it has a definite state that we can measure without disturbing it. In the new view, a system can be considered real even if we never measure it, provided it possesses definite properties in this underlying layer of extended probability. The researchers argue that the quantum system is always in a definite state, but our classical tools are too blunt to see it. The "disturbance" we usually blame for the violation is actually just a limitation of using a probability system that forbids negative numbers. By allowing these numbers, the system appears to have a consistent, non-invasive history that we simply couldn't access before.

The team applied the same logic to the Bell-CHSH inequality, which tests how two separate particles are connected across space. Usually, if two particles are entangled, measuring one instantly affects the other, violating the classical idea of locality, which says that what happens here cannot instantly affect something far away. The researchers built a generalized version of this test using extended probability. Just as with the time-based test, the quantum system passed the new inequality without any violation. This suggests that the particles can be described as having independent, definite properties in this extended layer, even when they are far apart. The "spooky action at a distance" that Einstein famously disliked might be an artifact of trying to force quantum behavior into a classical probability box.

To make sense of this, the authors rely on a framework called "consistent histories," which views the universe not as a single timeline, but as a collection of possible stories. In this picture, there are two layers of reality. The deeper layer consists of "non-settleable" histories, which are the full, complex quantum possibilities described by extended probability. These are the stories that cannot be directly observed. The surface layer consists of "settleable" histories, which are the specific outcomes we actually see when we measure something. The transition from the deep layer to the surface layer happens through a process called decoherence, where the interference between different stories cancels out, leaving behind the clean, positive probabilities of classical physics. The researchers propose that the generalized inequalities work because they are looking at the deep layer directly, where the rules are different, rather than forcing the deep layer to pretend it is the surface layer.

The study does not claim to have solved the mystery of quantum mechanics entirely, nor does it suggest that we can now see negative probabilities in our daily lives. Instead, it offers a mathematical demonstration that if we accept extended probability as a valid description of the underlying reality, the conflict between quantum behavior and classical intuition disappears. The simulations show that the bounds of the inequalities are naturally extended to accommodate the quantum world, removing the need for violations. This implies that the quantum system is not "breaking" the laws of reality; rather, our classical laws are just a special, limited case of a broader set of rules. The work suggests that the path to understanding how the quantum world becomes the classical world lies not in discarding quantum mechanics, but in expanding our definition of what a probability can be.

The findings remain a theoretical proposal supported by numerical simulations, not a physical experiment that has been performed in a lab. The researchers have shown that the math works and that the contradictions vanish when the rules are changed. They acknowledge that this approach requires a shift in how we view measurement and reality, moving away from the idea that observation creates the outcome. Instead, they suggest that the outcome exists in a broader, more complex form before we ever look at it. While the idea of negative probability may sound counterintuitive, the paper argues that it is a necessary tool to describe the fundamental layer of nature. If this view holds, it means the universe is more consistent than we thought, and the strange behavior of quantum particles is simply a reflection of a deeper, more flexible reality that we are only beginning to map.