Measuring a Quantum Measure Exceeding Unity

Quantum 10, 2215 (2026). https://doi.org/10.22331/q-2026-09-24-2215 The history based formalism known as Quantum Measure Theory (QMT) generalizes the concept of probability-measure so as to incorporate quantum interference. The resulting quantum measure $\mu$ is defined for arbitrary events (sets of histories), not just for observables at a fixed moment of time. Thanks to interference effects, $\mu$ can exceed unity, exhibiting its non-classical nature in a particularly striking manner. Here, in an optical experiment, we illustrate an ancilla based filtering scheme that gives operational meaning to the quantum measure. For a specific photonic event $E$, we report a measured value of $\mu(E)=1.172^{+0.013}_{-0.019}$, which within errors agrees with the theoretical value of $5/4$, while exceeding the maximum value permissible for a classical probability (namely $1$) by $13.32$ upper or $8.89$ lower percentile widths. The directly observed quantity is an ordinary detector probability $p_D\le 1$ (or, with laser light, an equivalent power ratio); the value $\mu(E)\gt1$ is inferred via the calibrated relation $\mu(E)=2p_D$ for our filter. If an unconventional theoretical concept is to play a role in meeting the foundational challenges of quantum theory, it seems important to bring it into contact with experiment as much as possible. Our experiment does this for the quantum measure.

The history based formalism known as Quantum Measure Theory (QMT) generalizes the concept of probability-measure so as to incorporate quantum interference. The resulting quantum measure $\mu$ is defined for arbitrary events (sets of histories), not just for observables at a fixed moment of time. Thanks to interference effects, $\mu$ can exceed unity, exhibiting its non-classical nature in a particularly striking manner. Here, in an optical experiment, we illustrate an ancilla based filtering scheme that gives operational meaning to the quantum measure. For a specific photonic event $E$, we report a measured value of $\mu(E)=1.172^{+0.013}_{-0.019}$, which within errors agrees with the theoretical value of $5/4$, while exceeding the maximum value permissible for a classical probability (namely $1$) by $13.32$ upper or $8.89$ lower percentile widths. The directly observed quantity is an ordinary detector probability $p_D\le 1$ (or, with laser light, an equivalent power ratio); the value $\mu(E)\gt1$ is inferred via the calibrated relation $\mu(E)=2p_D$ for our filter. If an unconventional theoretical concept is to play a role in meeting the foundational challenges of quantum theory, it seems important to bring it into contact with experiment as much as possible. Our experiment does this for the quantum measure.

Featured image: Event-filtering of quantum histories. A coherent optical field explores multiple possible histories through the interferometric system. The event filter selects a chosen set of histories while preserving the interference needed to determine their quantum measure. The measured result, $\mu(E)\approx1.17\gt1$, exceeds the classical probability bound of unity, even though the directly observed detector probability itself remains between zero and one.

Quantum mechanics is usually described in terms of the state of a system at a particular time and the outcomes obtained when we measure it. A complementary viewpoint is to describe a quantum system through its possible histories, meaning the different ways it can evolve between preparation and detection.

Quantum Measure Theory assigns a generalized measure to sets of such histories, incorporating the interference between them. Unlike an ordinary probability, this quantum measure can exceed one. The difficulty is that a general set of histories does not necessarily correspond to a sequence of ordinary measurements, so its quantum measure is not straightforward to access experimentally.

Here we overcome this problem using an ancilla-assisted optical event-filter. In our photonic implementation, polarization is used to tag, select and coherently recombine a chosen set of path histories. The output of the filter is an ordinary detector probability, which always remains between zero and one, but is related by a known factor to the quantum measure of the selected history-event.

For the event studied here, the ideal quantum measure is 1.25. Experimentally, we obtain a median value of approximately 1.17, in close agreement with a realistic theoretical model of the apparatus, which predicts about 1.18, and clearly above the classical probability bound of one.

The significance goes beyond observing a number greater than one. The experiment gives operational access to a quantity associated with an entire quantum history-event, rather than simply with a final measurement outcome. Histories-based formulations are also natural in attempts to formulate quantum physics for spacetime and quantum gravity, although the present work is a quantum-optics experiment and does not test quantum gravity.

If an unconventional theoretical concept is to play a role in meeting the foundational challenges of quantum theory, it seems important to bring it into contact with experiment as much as possible. Our experiment does this for the quantum measure.

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Measuring a Quantum Measure Exceeding Unity

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