A simple pedagogical discussion and visualization of certain Bell inequalities is presented with the help of a system involving two \textquotedblleft twins\textquotedblright, originally discussed in 2605.03104. These twins answer identically when the same `question' is posed, here exemplified with: Did you like breakfast/lunch/dinner? In analogy, for two entangled particles at distant locations, the `twin-like' quantum state is such that the same outcome for the same measurement (i.e. spin direction) is obtained. In the space $(X,Y,Z)$ of mixed moments (i.e. $X$ refers to breakfast-lunch correlation, being $1$ for same answers and $-1$ for opposite ones), the space of Bell local models is contained in a tetrahedron, while the quantum space is bound by an elliptope that embeds the tetrahedron. The whole no-signalling region in the $(X,Y,Z)$ space is the cube $[-1,1]^{3}.$ The twin scenario applies e.g. to the decay of the Higgs into fermions.
Comments
8 pages, 3 figures. Prepared for the proceedings of the Workshop on Quantum Entanglement, 3-4 July 2026, Jagiellonian University, Krakow, Poland. https://indico.koza.if.uj.edu.pl/event/24/
A simple pedagogical discussion and visualization of certain Bell inequalities is presented with the help of a system involving two \textquotedblleft twins\textquotedblright, originally discussed in 2605.03104. These twins answer identically when the same question' is posed, here exemplified with: Did you like breakfast/lunch/dinner? In analogy, for two entangled particles at distant locations, the twin-like' quantum state is such that the same outcome for the same measurement (i.e. spin direction) is obtained. In the space (X,Y,Z) of mixed moments (i.e. X refers to breakfast-lunch correlation, being 1 for same answers and −1 for opposite ones), the space of Bell local models is contained in a tetrahedron, while the quantum space is bound by an elliptope that embeds the tetrahedron. The whole no-signalling region in the (X,Y,Z) space is the cube [−1,1]3. The twin scenario applies e.g. to the decay of the Higgs into fermions.
We present a geometric characterisation of local models in the bipartite Bell scenario with three measurement settings per site and binary outcomes, i.e. the (3,3,2,2) case. Restricting attention to indistinguishable sites, thus resulting in a `symmetric local' scenario, we introduce a three-dimensional mixed-moment space in which the mixed moments are calculated under off-diagonal measurement settings.
In this reduced representation, the symmetric local region assumes the remarkably simple form of a regular tetrahedron - the 'pyramid'. We prove that only three inequalities are required to characterise the interior of this region. We call them the pyramid inequalities that separate symmetric local (SL) models from their complement, non-symmetric local (SL) models. Remarkably, in this representation, the 'symmetric quantum' region forms an elliptope defined by a single inequality. We also clarify the relation between the symmetry-reduced pyramid representation and the full (3,3,2,2) Bell polytope in the 36-dimensional conditional-probability space, which possesses 684 facet-defining inequalities. The reduction from 684 to three reflects normalisation, symmetry reduction, and projection to the mixed-moment space.
In the pyramid representation, the hierarchy SL⊊SQ⊊NS appears geometrically as a tetrahedron embedded in a somewhat larger curved body of symmetric quantum models, SQ, which in turn is embedded in a cube of no-signalling models, NS. This provides a unified three-dimensional visualisation of symmetric local, symmetric quantum, and symmetric post-quantum correlations. The qualitative advantages of the pyramid representation over the standard one-dimensional CSHS representation of the (2,2,2,2) case are also discussed.
Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest (2,2,2) scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the (2,2,3) scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.
In this work we look for novel classes of Bell's inequalities and methods to produce them. We also find their quantum violations including, if possible, the maximum one. The Jordan bases method that we explain in Chapter 2 is about using a pair of certain type of orthonormal bases whose spans are subspaces related to measurement outcomes of incompatible quantities on the same physical system. Jordan vectors are the briefest way of expressing the relative orientation of any two subspaces. This feature helps us to reduce the dimensionality of the parameter space on which we do searches for optimization. The work is published in [24]. In Chapter 3, we attempt to find a connection between group theory and Bell's theorem. We devise a way of generating terms of a Bell's inequality that are related to elements of an algebraic group. The same group generates both the terms of the Bell's inequality and the observables that are used to calculate the quantum value of the Bell expression. Our results are published in [25][26]. In brief, Bell's theorem is the main tool of a research program that was started by Einstein, Podolsky, Rosen [19] and Bohr [8] in the early days of quantum mechanics in their discussions about the core nature of physical systems. These debates were about a novel type of physical states called superposition states, which are introduced by quantum mechanics and manifested in the apparent inevitable randomness in measurement outcomes of identically prepared systems. Bell's huge contribution was to find a means of quantifying the problem and hence of opening the way to experimental verification by rephrasing the questions as limits on certain combinations of correlations between measurement results of spatially separate systems [7]. Thanks to Bell, the fundamental questions related to the nature of quantum mechanical systems became quantifiable [6]. According to Bell's theorem, some correlations between quantum entangled systems that involve incompatible quantities are not allowed by classical mechanics, a feature that is called as "quantum nonlocality". An experimental observation of those correlations, in other words, a violation of the limits imposed by classical physics, implies the correctness of quantum description and invalidates the classical, local realistic models. The first Bell experiments were proposed by Clauser, Horne, Shimony, and Holt, who invented the most famous Bell's inequality [13]. Later, the Aspect experiments were satisfactory enough for the physics community to be conclusive about the validation of quantum mechanics [1][3][4][2]. Ekert's work on applications of quantum nonlocality to communication resulted in the new field of quantum communication and cryptography, and turned the research program into a practical one [20]. Pitowsky showed a method to find all expressions of limitations due to local realism, all Bell's inequalities, for a given physical scenario. He also proved that the problem is, unfortunately, NP-complete and hence as the scenarios get more complex, they also become computationally intractable [33][34]. Therefore, different methods for the solution of special cases of the problem are necessary. Inequalities found for those special cases can be called classes of Bell's inequalities. For example, Werner and Wolf [41] and Collins, Gisin, Linden, Massar, and Popescu [16] found classes that cover a wide range of scenarios. Our work is a similar kind of effort to produce and study new types of Bell's inequalities.
Qubit-qubit-qutrit quantum correlations in H→ffˉV
We perform an extensive analysis of the quantum correlations carried by the qubit-qubit-qutrit pure state arising in the decay of a massive scalar into a fermion-antifermion pair and a massive gauge boson, H→ffˉV, specialising to the Higgs boson decay h→τ−τ+Z. Working with the exact tree-level spin state and its systematic expansion around the massless-fermion limit, we obtain analytic control over the entire phase space: the bipartite entanglement measures, the genuine 2⊗2⊗3 entanglement structure (the Miyake classification), as well as the Bell-inequality violations and the non-stabiliserness (magic) are all mapped and reproduced by compact formulas. The bipartite measures exhibit a monogamy-like trade-off between the fermion pair and the fermion-boson pairs. The state is genuinely 2⊗2⊗3 entangled over almost the entire phase space, most strongly in the collinear regions. We derive, for the first time, semi-analytical expressions for the tight 4×4×2 Bell inequalities of the 2⊗2⊗3 system, generalising the optimisation previously available only for three qubits, and find that the local-hidden-variable bound is violated over the entire phase space, reaching within a few per cent of the quantum bound at the upper endpoint of the di-tau mass spectrum. We further extend the stabiliser Rényi entropy and the non-local magic to systems with unequal local dimensions, and show that the near-endpoint state carries almost exactly one bit of non-local magic, which peaks at log2727≃1.95 in the collinear regions. The differential decay rate concentrates precisely in the most nonclassical region of the phase space.
Abstract Bell-inequality violations reveal the presence of quantum correlations between two particles that have interacted and then separated. Their generalisation to quantum fields is necessary to study a number of field-theoretic setups, such as cosmological density fluctuations. In this work, we show how Bell operators can be constructed for quantum fields in real space, and for Gaussian states we compute their expectation value in terms of the field power spectra. We then apply our formalism to a scalar field in de-Sitter space-time. We find that, in spite of the tremendous production of entangled particles with opposite wave momenta on large scales, Bell inequalities are not violated in real space. The reason is that, when considering measurements of a field at two distinct locations in real space, one implicitly traces over the configuration of the field at every other location, leading to a mixed bipartite system. This “effective decoherence” effect is responsible for the erasure of quantum features, and casts some doubts on our ability to reveal the quantum origin of cosmological structures. We finally discuss these results in the light of quantum discord.