Abstract Research on optical modes, such as orbital, temporal, or parity, bring much attention, for these new degrees of freedom allow larger quantum communication alphabets. Each lab usually adapts the Bloch or Poincare sphere to their experiment or light mode. This takes an extra effort and time and produces a plethora of spheres and notations. Yet, we miss a common framework or convention valid among diverse physical-modes. We aim to unite in one representation the best points from many different spheres. Such common-sphere could also help to compare distant experiments, for an intuitive understanding of quantum optical states. We built a common representation by mathematically aligning the Hilbert space and a three dimensional color space. We define a unique color for each one of the three Poincare axes and positive Pauli vectors. Beyond three primary colors and states, our equations associate each Hilbert state to a specific tonality, among the infinite combinations in Color space. These maths achieve a new ability to unequivocally represent any quantum state by its precise combination of colors. Thus, with these equations, quantum states ‘yellow’ or ‘magenta’ are not mere names, rather each one denotes an exact superposition in Hilbert space. To handle disparities between SO3 vs. SU2 space operations, we propose a darkness bit and a Hermite-inspired shape. A simulation of HG modes let us align distinct shapes to quantum optical states. Three examples of applications show our color sphere in practice. First, we apply the Hilbert-Color mapping in Polarization. Then, the same color-space is shown in Orbital Angular Momentum. We also represent location paths in this color-mapping. The simulations and practical comparisons let us refine the proposed color sphere convention. For higher-order and path-to-industry, any sphere section serves as color constellation diagram. One color-space sphere served as common ground to represent coexisting concepts among diverse physical areas. The introduced change diagrams are visual tools to communicate setups and operators. The examples showed a unique notation matches many physical processes. The resulting diagram of superposition of spatially separated optical paths is coherent with a Plate on Polarization or cylinder lens on OAM Hermite Gaussian modes. A unique change diagram describes the three examples. The meaning persist despite the physical implementation. Found also how this color space let us grasp visually some meaning. Thus, the amount of blue in a state representation indicated the degree of its phase shift. Overall, we presented math and visual tools to display and compare experiments. We showed examples in different physical modes, all linked to a unique color sphere.
Abstract Research on optical modes, such as orbital, temporal, or parity, bring much attention, for these new degrees of freedom allow larger quantum communication alphabets. Each lab usually adapts the Bloch or Poincare sphere to their experiment or light mode. This takes an extra effort and time and produces a plethora of spheres and notations. Yet, we miss a common framework or convention valid among diverse physical-modes. We aim to unite in one representation the best points from many different spheres. Such common-sphere could also help to compare distant experiments, for an intuitive understanding of quantum optical states. We built a common representation by mathematically aligning the Hilbert space and a three dimensional color space. We define a unique color for each one of the three Poincare axes and positive Pauli vectors. Beyond three primary colors and states, our equations associate each Hilbert state to a specific tonality, among the infinite combinations in Color space. These maths achieve a new ability to unequivocally represent any quantum state by its precise combination of colors. Thus, with these equations, quantum states ‘yellow’ or ‘magenta’ are not mere names, rather each one denotes an exact superposition in Hilbert space. To handle disparities between SO3 vs. SU2 space operations, we propose a darkness bit and a Hermite-inspired shape. A simulation of HG modes let us align distinct shapes to quantum optical states. Three examples of applications show our color sphere in practice. First, we apply the Hilbert-Color mapping in Polarization. Then, the same color-space is shown in Orbital Angular Momentum. We also represent location paths in this color-mapping. The simulations and practical comparisons let us refine the proposed color sphere convention. For higher-order and path-to-industry, any sphere section serves as color constellation diagram. One color-space sphere served as common ground to represent coexisting concepts among diverse physical areas. The introduced change diagrams are visual tools to communicate setups and operators. The examples showed a unique notation matches many physical processes. The resulting diagram of superposition of spatially separated optical paths is coherent with a Plate on Polarization or cylinder lens on OAM Hermite Gaussian modes. A unique change diagram describes the three examples. The meaning persist despite the physical implementation. Found also how this color space let us grasp visually some meaning. Thus, the amount of blue in a state representation indicated the degree of its phase shift. Overall, we presented math and visual tools to display and compare experiments. We showed examples in different physical modes, all linked to a unique color sphere.
An octant representation of higher-order optical modes that includes Laguerre-Gaussian and Hermite-Gaussian modes is presented. The octant picture captures the high-dimensional nature of three-state optical systems and beyond, with standard Poincaré spheres for orbital angular momentum forming subspaces of the entire state space. This representation enables intuitive manipulation of both classical modes and optical qudits and provides a framework for extending Berry phases and topological invariants to high dimensions.
Light carries both spin (polarization) and orbital angular momentum. Combining these degrees of freedom produces hybrid spin-orbit states that live in a high-dimensional Hilbert space, offering greater information capacity and robustness for optical communication, quantum technologies, and metrology. However, generating arbitrary states in these spaces and characterizing them efficiently has remained difficult. Here we show a compact metasurface that generates arbitrary spin-orbit states in a four-dimensional Hilbert space, visualized on a Poincaré hypersphere, with straightforward scalability to higher dimensions. Using a tetratomic unit cell, the single-layer device precisely controls complex amplitude, phase, and polarization. We further introduce an efficient interferometric scheme that reconstructs the full density matrix of any N-dimensional spin-orbit state using only three interferograms. This approach uncovers an intrinsic spin-orbit parity order that governs the symmetry of projected intensity patterns, independent of the weighting of the eigenmodes, and enables controlled mode transformations through higher-order geometric phases. These advances establish a versatile platform for high-dimensional photonic technologies.
Introduction In this chapter, we will be concerned with entangled states of light. Historically, quantum states of light, i.e., photons, have played a dominant role in the experimental study of quantum entanglement. For instance, the first Bell tests and the first realizations of quantum teleportation were performed using two-dimensional polarization-entangled twophoton states [3]. From an experimental point of view these states are readily prepared and manipulated using conventional polarization optics. Recently, however, both theoretical and experimental efforts have shifted towards entanglement in more than two modes , which is often referred to as high-dimensional entanglement. This is much richer than two-dimensional entanglement, as is, for instance, reflected in stronger violations of various measures of nonlocality which reveal the complexity of high-dimensional Hilbert spaces [4–7]. Also, from an applications perspective, it promises a larger channel capacity [8, 9] and improved security for quantum communication [10, 11]. There are several ways to implement high-dimensional two-photon entanglement. One can exploit the temporal [12], frequency [13], or spatial [14–18] degrees of freedom of the optical field. A promising way to implement high-dimensional two-photon entanglement is to use the orbital-angular-momentum (OAM) degree of freedom of light. Light beams that carry OAM have a helical wavefront that spirals around the optical axis during propagation [19]. The winding number m , which characterizes the OAM eigenmodes, can adopt any integer value between −∞ and ∞; since all coaxial OAM modes are orthogonal they can thus span a very high-dimensional Hilbert space.
We propose a geometric hybrid Poincaré sphere (GHPS) as a unified geometrical framework for describing structured photon states with independently controllable spin angular momentum (SAM) and orbital angular momentum (OAM). Unlike the conventional higher-order Poincaré sphere, in which the SAM and OAM are intrinsically coupled through fixed basis states, the GHPS is constructed by defining its poles as direct products of arbitrary orthogonal bases on the Poincaré sphere (PS) and orbital Poincaré sphere (OPS) and by superposing these pole states. Using numerical simulations, we analyze representative GHPS states and show that the GHPS spherical coordinates govern the amplitude ratio and relative phase between the pole bases. This framework enables spatially inhomogeneous polarization distributions and intensity patterns, including nonseparable structures in which polarization and intensity are intrinsically intertwined, and provides a systematic state-space description for the coherent geometrical control of advanced structured light fields.
Difficulties in algorithmic simulation of natural thinking point to the inadequacy of information encodings used to this end. The promising approach to this problem represents information by the qubit states of quantum theory, structurally aligned with major theories of cognitive semantics. The paper develops this idea by linking qubit states with color as fundamental carrier of affective meaning. The approach builds on geometric affinity of Hilbert space of qubit states and color solids, used to establish precise one-to-one mapping between them. This is enabled by original decomposition of qubit in three non-orthogonal basis vectors corresponding to red, green, and blue colors. Real-valued coefficients of such decomposition are identical to the tomograms of the qubit state in the corresponding directions, related to ordinary Stokes parameters by rotational transform. Classical compositions of black, white and six main colors (red, green, blue, yellow, magenta and cyan) are then mapped to analogous superposition of the qubit states. Pure and mixed colors intuitively map to pure and mixed qubit states on the surface and in the volume of the Bloch ball, while grayscale is mapped to the diameter of the Bloch sphere. Herewith, the lightness of color corresponds to the probability of the qubit’s basis state «1», while saturation and hue encode coherence and phase of the qubit, respectively. The developed code identifies color as a bridge between quantum-theoretic formalism and qualitative regularities of the natural mind. This opens prospects for deeper integration of quantum informatics in semantic analysis of data, image processing, and the development of nature-like computational architectures.
We comprehensively review the quantum theory of the polarization properties of light. In classical optics, these traits are characterized by the Stokes parameters, which can be geometrically interpreted using the Poincaré sphere. Remarkably, these Stokes parameters can also be applied to the quantum world, but then important differences emerge: now, because fluctuations in the number of photons are unavoidable, one is forced to work in the three-dimensional Poincaré space that can be regarded as a set of nested spheres. Additionally, higher-order moments of the Stokes variables might play a substantial role for quantum states, which is not the case for most classical Gaussian states. This brings about important differences between these two worlds that we review in detail. In particular, the classical degree of polarization produces unsatisfactory results in the quantum domain. We compare alternative quantum degrees and put forth that they order various states differently. Finally, intrinsically nonclassical states are explored, and their potential applications in quantum technologies are discussed.