Whether quantum mechanics is predictively complete remains a fundamental question in its foundations. While most investigations have concentrated on two-dimensional systems, high-dimensional systems are more general and hold broader applicability. Extending the non-extensibility theorem by Colbeck and Renner [Nat. Commun.2, 411 (2011)10.1038/ncomms1416], we demonstrate that no alternative theory can enhance the predictive power of quantum mechanics in arbitrary dimensions. By linking the maximal predictive power achievable to experimentally observable correlations, we establish optimal experimental bounds exploiting orbital angular momentum entangled photon pairs. These results impose strict constraints and rule out a broad class of hidden-variable models, including Bell-type and Leggett-type theories, thereby reinforcing the completeness of quantum theory and advancing high-dimensional quantum cryptography.
Whether quantum mechanics is predictively complete remains a fundamental question in its foundations. While most investigations have concentrated on two-dimensional systems, high-dimensional systems are more general and hold broader applicability. Extending the non-extensibility theorem by Colbeck and Renner [Nat. Commun.2, 411 (2011)10.1038/ncomms1416], we demonstrate that no alternative theory can enhance the predictive power of quantum mechanics in arbitrary dimensions. By linking the maximal predictive power achievable to experimentally observable correlations, we establish optimal experimental bounds exploiting orbital angular momentum entangled photon pairs. These results impose strict constraints and rule out a broad class of hidden-variable models, including Bell-type and Leggett-type theories, thereby reinforcing the completeness of quantum theory and advancing high-dimensional quantum cryptography.
The completeness of quantum mechanics in predictive power is a central question in its foundational study. While most investigations focus on two-dimensional systems, high-dimensional systems are more general and widely applicable. Building on the non-extensibility theorem by Colbeck and Renner [Phys. Rev. Lett. 101, 050403 (2008)], which established that no higher theory can enhance the predictive power of quantum mechanics for two-dimensional systems, we extend this result to arbitrarily dimensional systems. We connect maximum potential predictive power achievable by any alternative theory to experimentally observable correlations, and establish optimal experimental bounds across varying dimensions by exploiting two-photon orbital angular momentum entangled states with entanglement concentration. These bounds falsify a broader class of alternative theories, including Bell's and Leggett's models, and those that remain theoretically ambiguous or experimentally unverified. Our findings not only deepen the foundational understanding of quantum mechanics but also hold significant potential for high-dimensional quantum cryptography.
Leggett inequality states that nonlocal hidden-variable models might still be incompatible with the predictions of quantum physics. However, its theoretical and experimental demonstration is only in the scenario of 2-dimensional maximally entangled systems. An open question remains as to whether the Leggett inequality can be violated by nonmaximally entangled states. Here, we answer this question both in theory and experiment. Specifically, from the point of view of geometry, we theoretically map the problem of maximizing the correlation measure in the Leggett inequality to maximizing the sum of an ellipse’s diameter and semi-diameter axes, accordingly, demonstrating that the violation of the Leggett inequality requires a more robust entanglement than that of Bell’s theory. Experimentally, by leveraging the controllable photonic orbital angular momentum entanglement, we demonstrate the violation of Leggett-type inequalities by more than 8.7 and 4.5 standard deviations under concurrence C = 0.95 and 0.9, respectively. Our observations indicate that, the requirement for quantum correlation should be increased to exclude a particular class of non-local hidden variable theories that abide by Leggett’s model, providing insights into the boundaries of quantum correlation and the limitations imposed by non-local hidden variables.
Entanglement in higher dimensions is an attractive concept that is a chal- lenge to realise experimentally. To this end, the entanglement of the orbital angular momentum (OAM) of photons holds promise. The OAM state-space is discrete and theoretically unbounded. In the work that follows, we investi- gate various aspects of OAM entanglement. We show how the correlations in OAM and its conjugate variable, angular position, are determined by phase- matching and the shape of the pump beam in spontaneous parametric down- conversion. We implement tests of quantum mechanics which have been previously done for other variables. We show the Einstein-Podolsky-Rosen paradox for OAM and angle, supporting the incompatibility of quantum me- chanics with locality and realism. We demonstrate violations of Bell-type inequalities, thereby discounting local hidden variables for describing the correlations we observe. We show the Hardy paradox using OAM, again highlighting the nonlocal nature of quantum mechanics. We demonstrate violations of Leggett-type inequalities, thereby discounting nonlocal hidden variables for describing correlations. Lastly, we have looked into the entan- glement of topological vortex structures formed from a special superposition of OAM modes and show violations of Bell-type inequalities confined to a finite, isolated volume.
Orbital angular momentum (OAM) is one of the most recently discovered properties of light, and it is only in the past decade its quantum properties have been the subject of experimental investigations and have found applications. Unlike polarization, which is only bidimensional, orbital angular momentum provides, with relative ease, unprecedented access to a theoretically unbounded discrete state space. The process of spontaneous parametric down-conversion has long been used as a source of two-photon states that can be entangled in several degrees of freedom, including OAM. In this thesis, the properties of the natural OAM spectrum associated with the entangled states produced by parametric down-conversion were investigated. Chapters 2 and 3 describe the production and detection of tunable high-dimensional OAM entanglement in a down-conversion system. By tuning the phase-matching conditions and improving the detection stage, a substantial increase in the half-width of the OAM correlation spectrum was observed. The conjugate variable of OAM, angular position, was also considered when examining high-dimensional states entangled in OAM. In order to efficiently determine their dimension, high-dimensional entangled states were probed by implementing a technique based on phase masks composed of multiple angular sectors, as opposed to narrow single-sector analysers. Presented in chapter 4, this technique allows the measurements of tight angular correlations while maintaining high optical throughput. The states so produced were then used for a number of applications centred around the concept of mutually unbiased bases. One can define sets of mutually unbiased bases for arbitrary subspaces of the OAM state space. Two bases are mutually unbiased if the measurement of a state in one basis provides no information about the state as described in the other basis. Complete measurements in mutually unbiased bases of high-dimensional OAM spaces are presented in chapter 5. Measurements in sets of mutually unbiased bases are integral to quantum science and can be used in a variety of protocols that fully exploit the large size of the OAM state space; we describe their use in efficient quantum state tomography, quantum key distribution and entanglement detection. Caution is however necessary when dealing with state spaces embedded in higher-dimensional spaces, such as that provided by OAM. Experimental tests of Bell-type inequalities allow us to rule out local hidden variable theories in the description of quantum correlations. Correlations inconsistent with the states observed, or even with quantum mechanics, known as super-quantum correlations, have however been recorded previously in experiments that fail to comply with the fair-sampling conditions. Chapter 6 describes an experiment that uses a particular choice of transverse spatial modes for which super-quantum correlations persist even if the detection is made perfectly efficient. The sets of modes carrying OAM allow a complete description of the transverse field. The ability to control and combine additional degrees of freedom provides the possibility for richer varieties of entanglement and can make quantum protocols more powerful and versatile. One such property of light, associated with transverse modes possessing radial nodes in the field distribution, can be accessed within the same type of experimental apparatus used for OAM. In chapter 7, the radial degree of freedom is explored, together with OAM, in the context of Hong-Ou-Mandel interference.
According to Bell's theorem, any model based on local variables cannot reproduce certain quantum correlations. A critical question is whether one could devise an alternative framework, based on nonlocal variables, to reproduce quantum correlations while adhering to fundamental principles. Leggett proposed a nonlocal model, termed crypto-nonlocality, rooted in considerations of the reality of photon polarization, but this property restricted it to being bi-dimensional. In this Letter, we extend the crypto-nonlocal model to higher dimensions and develop a framework for constructing experimentally testable Leggett-type inequalities for arbitrary dimensions. Our investigation into models that yield specific predictions exceeding those of quantum mechanics is intriguing from an information-theoretic perspective and is expected to deepen our understanding of quantum correlations.
It was argued [1] that there can be no extension of quantum mechanics with improved predictive power on a measurement freely chosen, independently of any event that is not in its future light cone. The assumption of measurement choice was criticized [2] to be too strong to be physically necessary and extensions of quantum mechanics were shown [3] to be possible under a more relaxed measurement assumption. Here I point out an error in the criticism and observe that the actual mistake of the no-go theorem lies in an unwarranted assumption implicitly made in the proof of [1]. Hence, quantum mechanics is guaranteed to have the maximal predictive power only in situations of complete certainty and complete uncertainty about measurement outcomes. I then show that the measurement assumption can be further relaxed without affecting the conclusion on the predictive power of quantum mechanics versus alternative theories. I further study the optimal predicative improvement over quantum mechanics of local spin measurements on a pair of entangled qubits by any alternative theory and conjecture a strict upper bound.