Spatial Dimensions in Quantum State Analysis

🚀 New Preprint Out: Systems with Quantum Dimensions I’m excited to share our latest work with Mikolaj Myszkowski and Mattia Damia Paciarini, now on arXiv. What if the number of spatial dimensions were not fixed… but quantum? In this paper, we explore a framework where dimensionality becomes a dynamical quantum variable, leading to surprising structures, emergent symmetries, and scale-dependent effective dimensions. A fresh playground for quantum systems, from gravity to condensed matter and perhaps even quantum computing. Here’s what we found: ✨ Highlights 🔭 Dimensionality as a quantum observable: the number of spatial dimensions is promoted to a quantum operator. 🧠 Superpositions of dimensions: states can simultaneously “live” in different numbers of dimensions. 🎛️ Enhanced symmetries: mixing states across dimensions produces new, unexpected symmetry structures. 🎢 Temperature-dependent dimensions: in the quantum-dimension harmonic oscillator, the effective number of dimensions grows with energy. 🧩 Versatile framework: applicable to quantum gravity, QFT, and condensed matter systems where dimensionality ���flows.” 🌀 Hints toward improved renormalization: QD systems may exhibit better UV behavior. 🎉 This work opens a new avenue for thinking about dimensionality not as a rigid backdrop, but as a quantum-mechanical participant. 🔗 Read the preprint: https://lnkd.in/drpf6PU2 (Systems with Quantum Dimensions, Myszkowski–Paciarini–Sannino) 🙏 Grateful to the Carlsberg Foundation for supporting this research and to our colleagues for inspiring discussions. #quantumdimensions #quantum #theoreticalphysics #theory Carlsberg Foundation Danmarks Grundforskningsfond / The Danish National Research Foundation Department of Mathematics and Computer Science (IMADA), University of Southern Denmark (SDU) Syddansk Universitet - University of Southern Denmark Università degli Studi di Napoli Federico II (UniNa) / University of Naples Federico II Danish Institute for Advanced Study (DIAS) CERN Quantum Theory Center

Hilbert Space ✍️ A Hilbert space is a special kind of mathematical space that helps scientists describe complex systems—especially in physics—by extending the familiar idea of ordinary 3D space into potentially infinite dimensions. Imagine it like a perfectly organized arena where every possible state of a system (such as a particle) is represented as a point or a direction. In this space, each “direction” behaves like a vector, and these vectors can be added together or scaled, just like arrows on a graph. But unlike simple geometry, Hilbert space includes infinitely many directions, allowing it to capture very subtle variations. It also has a built-in way of measuring angles and lengths, which lets scientists determine how similar or different two states are. When applied in quantum physics, a system’s condition is described as a vector in Hilbert space. Different possible states can combine smoothly, forming a superposition. When a measurement is made, the system “projects” onto one of these directions, and the geometry of the space determines the likelihood of each outcome. In essence, Hilbert space acts like a precise mathematical stage where waves, probabilities, and physical states interact in a structured and measurable way, revealing patterns that would otherwise remain hidden.

Hidden Topologies Discovered in Conventional Quantum Entanglement Introduction New physics research reveals that a standard form of quantum entanglement used in laboratories worldwide contains a vast and previously unseen topological structure. The discovery shows that conventional entangled photons can host thousands of distinct topologies in high dimensions, dramatically expanding the toolkit for robust quantum information encoding. Core Discovery Unexpected Depth in Familiar Entanglement • Researchers from the University of the Witwatersrand and Huzhou University found hidden topologies within entangled photons produced by spontaneous parametric downconversion. • The work reports the highest-dimensional topology ever observed in any system: 48 dimensions with more than 17,000 distinct topological signatures. • These signatures form an exceptionally large alphabet for encoding quantum information. How the Topology Emerges • The topology arises from the orbital angular momentum of light, a spatial property long studied in quantum optics. • Measuring the orbital angular momentum of two entangled photons reveals that the entanglement itself has an intrinsic topological structure. • Because orbital angular momentum can take infinitely many values, the associated topology can scale to very high dimensions. Breaking Previous Assumptions • Earlier models assumed that at least two properties of light, such as orbital angular momentum and polarization, were needed to generate topology. • The new results show that orbital angular momentum alone is sufficient. • Beyond two dimensions, topology is no longer described by a single number but by a spectrum of topological values. Practical Advantages • The resources required already exist in most quantum optics laboratories. • No specialized quantum engineering infrastructure is needed. • The topology is naturally embedded in spatial entanglement and was simply overlooked. Implications for Quantum Systems • Topological encoding offers inherent resistance to noise, addressing a key weakness of high-dimensional entanglement. • Revisiting orbital angular momentum entanglement through topology could enable more stable, scalable quantum communication and computing platforms. • The findings open a new experimental pathway for exploring quantum field theory concepts in optical systems. Why This Matters This discovery reframes conventional entanglement as a far richer resource than previously understood. By uncovering thousands of hidden topologies in a widely used optical process, the research unlocks a powerful new method for encoding and protecting quantum information. The result bridges theory and experiment, transforming a familiar laboratory technique into a high-capacity, noise-resilient foundation for future quantum technologies. If this topic resonates, I invite you to connect and continue the conversation. Keith King https://lnkd.in/gHPvUttw

⚛️ Three tiny 2×2 matrices that explain how a qubit feels the world. 🧭 Let's look at Pauli matrices - why they look the way they do and what they do 🎯 What Pauli wanted to achieve Pauli wanted to have a compact algebraic language for spin-½: a two-state quantum object (up/down) whose state transforms nontrivially under spatial rotations. In short — a minimal set of operators that label, flip, and generate those rotations while producing real measurement outcomes. ⚙️ So the operators must be (real-world → mathematical property) ⚖️ Give two definite outcomes (up / down). → act on a 2-dimensional state space ⇒ 2×2 matrices. 🔍 Produce real measurement results. → observables must be Hermitian ⇒ real eigenvalues. 🔄 Treat “up” vs “down” as exact opposites. → no built-in bias ⇒ traceless (eigenvalues ±1). 🔁 Repeating the same measurement changes nothing. → measure twice along the same axis ⇒ each operator squares to identity. 🔃 Different axis rotations don’t commute (physical rotations). → the operators must satisfy the so-called SU(2) algebra. 🧩 Based on these practical assumptions, Pauli constructed his three matrices σₓ, σᵧ, σ_z as the minimal algebraic package that turns simple physical facts about spin into operators you can compute with. 💡 One of the main practical values Pauli matrices let you compute expectation values — a single real number between +1 and −1 that has a direct physical meaning: it is the average outcome of many measurements of the spin component along one of the chosen axes x, y, or z. ➕ Why Pauli matrices matter for quantum computing They are the language of a qubit. Any single-qubit operation or state can be described with the Pauli matrices. That makes them the natural “vocabulary” engineers and physicists use when designing and analysing qubits. Gates = rotations. Common quantum gates are just rotations generated by Pauli matrices (e.g. rotate around X, Y or Z). So thinking in Pauli terms directly links circuit instructions to physical control pulses. Build any operator from them. Because Pauli matrices form a basis, you can write any 2×2 Hamiltonian, gate, or noise process as a combination of Pauli terms — a powerful bookkeeping tool for design and debugging. Simple error picture. The most basic errors look like Pauli flips (X, Y, Z). That makes it straightforward to describe, detect, and correct errors. Fast state readout & tomography. Measuring Pauli expectation values is exactly what you do in qubit tomography and benchmarking: a few averages give you the Bloch vector and let you reconstruct the state or characterize a gate. Efficient reasoning & simulation. Many quantum control and error-correction techniques exploit the structure of Pauli operators. In short: Pauli matrices turn messy experimental data and control signals into three intuitive numbers and simple algebra — and that’s why they’re everywhere in quantum computing. 🔗 Details & enrollment for our next live sessions here: https://lnkd.in/eaSYghAQ

Geometric Structures of Quantum States based on Representation Theory and Information Geometry by Frédéric Barbaresco (Thales) https://lnkd.in/ePUcDxVY Abstract: We present here the geometric model of Quantum Information, as introduced by Jean-Marie Souriau. He developed the concept of geometric quantisation by introducing the notion of a quantum fibre bundle over a symplectic manifold, which is a coadjoint orbit. This allows for the extension of the definition of the Hamiltonian function, enabling the characterisation of "quantum states" as complex functions defined on the group 𝐺, satisfying two specific inequalities. To achieve this, the Gelfand-Naimark-Segal construction is employed to establish a connection between the quantum state, the unitary representation of the group, and the unit vector in the Hilbert space. These axioms ensure the probabilistic interpretation of quantum mechanics, where the state 𝑚 associates a probability measure with each "observable" of the group 𝐺. In the linear case, this guarantees the Heisenberg uncertainty relations. The convexity of the state space produces so-called "mixed" quantum states, whose existence is essential for quantum thermodynamics, particularly concerning Gibbs states. Based on Information, Geometry, we also introduce a Riemannian metric on the space of quantum states based on how quantum entropy changes, specifically using the second differential of von Neumann entropy. This constructs a natural “distance” based on Fisher Information that quantifies how much quantum information is lost when one state is mixed with a nearby state. In certain limits, the entropy-based metric approximates well-known quantum metrics such as the Bures–Helstrom metric. However, only the entropy-based metric has a direct interpretation in terms of information loss from statistical mixing. The metric arises from a Legendre transform connecting states and observables, paralleling thermodynamic structures (like Massieu functions). This enriches quantum statistical mechanics with information geometry. It connects quantum information geometry with thermodynamics and provides explicit computation for simple systems (e.g., qubits).

GEOMETRY OF MATTER CONTROLS THE QUANTUM TIMESCALE The longstanding question of how long a quantum transition actually takes has moved to experimentally accessible physics. In the attosecond regime, the transition from an initial bound state to a final photoelectron state is governed not by an external clock but by the internal phase evolution of the electronic wavefunction. The EPFL study demonstrates that this timescale is not universal: it is a symmetry‑dependent property of the material’s electronic structure. The key advance is the use of spin‑ and angle‑resolved photoemission (SARPES) to extract the EWS delay directly from the spin texture of the emitted electrons. In systems with strong spin–orbit coupling, multiple partial waves contribute to the photoemission amplitude. Their interference generates a measurable spin polarization even in nonmagnetic crystals under linearly polarized excitation. Because the spin vector is locked to the relative phase between these channels, it becomes an intrinsic probe of the accumulated phase—and therefore of the transition time—without perturbing the system with an external streaking field. Applying this method across materials of different dimensionality reveals a robust inverse relationship between spatial symmetry and quantum transition time. In 3D Cu with high cubic symmetry, the EWS delay approaches the lower theoretical bound (~26 as). In quasi‑2D TiSe₂ and TiTe₂, the delay increases to ~150 as, independent of correlation strength. This monotonic increase cannot be attributed to electron–electron interactions; instead, it reflects the reduction in the number of symmetry‑allowed propagation channels. The physical picture is that the excited electron occupies a quasi‑stationary state whose lifetime is determined by the density and symmetry of available decay pathways. High‑symmetry 3D lattices support many equivalent channels for constructive interference, enabling rapid phase accumulation and fast emission. As dimensionality is reduced, the Hilbert space of allowed momenta contracts, forcing the electron to undergo a more complex internal phase evolution before escape. The result is a geometry‑induced temporal bottleneck. These findings have several implications for condensed‑matter physics. First, they establish time as a symmetry‑controlled material parameter, not a universal constant of the photoemission process. Second, they impose fundamental constraints on petahertz‑scale electronics, where low‑dimensional nanostructures—despite their technological appeal—will exhibit intrinsically longer response times. Third, the spin‑interference method provides a new route to attosecond‑scale dynamics in systems where strong external fields would destroy fragile quantum phases, including correlated materials and topological states. The results show that spatial symmetry and temporal evolution are deeply entangled. In quantum materials, the structure of space dictates the flow of time.

BREAKING NEWS: Scientists have achieved a major milestone in quantum physics by creating a photon that occupies thirty seven distinct quantum dimensions. This breakthrough demonstrates that individual particles of light can be engineered to store and process far more information than previously thought. In classical physics, a photon is described by simple properties such as wavelength, energy, and polarization. In quantum physics, however, photons can be assigned multiple states at once, forming high dimensional quantum systems that exceed the binary limits of qubits. To create the thirty seven dimensional photon, researchers used advanced optical setups that manipulated the particle’s spatial modes. By shaping the wavefront and allowing it to pass through precisely engineered patterns, they encoded the photon into thirty seven orthogonal states. Each state acts like a separate channel that can carry unique information. This significantly increases the data capacity and computational potential of quantum systems. High dimensional states also have advantages in noise resistance, making them more robust for communication. The experiment relied on interferometry and spatial light modulators to verify that the photon maintained coherent quantum behavior across all thirty seven dimensions. Measurements confirmed that the particle did not collapse into a lower dimensional state and that each encoded mode remained stable. This stability is essential for building quantum devices that depend on multitiered information structures. Applications of high dimensional photons include secure quantum communication, where more dimensions translate into stronger encryption. They may also enhance quantum computing by enabling more complex calculations within a single particle. In quantum teleportation and entanglement research, high dimensional states allow richer and more efficient information transfer. While this achievement is still experimental, it represents a critical step toward scalable quantum technologies. It shows that quantum systems are not limited to simple two state structures but can be expanded to dozens or even hundreds of dimensions with careful engineering. This progress moves the field closer to practical quantum networks and advanced computational platforms. #techmedtime #fblifestyle #quantumphysics #innovation #research

A fundamental limit on what neural quantum states can represent Quantum many-body physics faces an irreducible challenge: the Hilbert space grows exponentially with system size. Variational wave functions are the practical workaround—but their reach is constrained. Matrix product states (MPS) are tractable yet bounded by an area law: they cannot efficiently represent critical or volume-law entangled states. This limitation is precisely understood and has shaped decades of method development. Neural quantum states (NQS), introduced by Carleo and Troyer in 2017, were meant to go further. By encoding wave function amplitudes in neural networks, NQS combine the expressivity of deep learning with variational Monte Carlo efficiency. They have since delivered competitive results across condensed matter, quantum chemistry, and nuclear physics. But their theoretical limits remained largely uncharted. Nisarga Paul now proves a fundamental bound. The key insight: a feedforward network with k scalar nonlinearities couples n input spins through at most k+1 independent affine features—linear combinations of spin configurations. Under mild analyticity conditions, this bottleneck translates into an entanglement bound for any subregion A: S ≤ ck log n. Volume-law entanglement requires k growing at least as fast as n/log n. The bound is tight—the Dicke state, representable with a single nonlinearity, achieves S ~ ½ log n exactly—and holds across MLPs, transformers, and single-nonlinearity architectures alike. This is the NQS analog of the MPS area law: a clean, architecture-agnostic expressivity constraint rooted in the structure of the computational graph itself. For quantum simulation in materials discovery, molecular design, or energy research, this result sharpens a concrete design rule: volume-law entangled targets—strongly correlated systems, topological phases—demand NQS architectures where neuron count scales with system size, while ground states with area or logarithmic entanglement scaling may be captured by small-k networks at significantly lower computational cost. Knowing where the ceiling sits is the first step to building efficiently beneath it. Paper: Paul, Physical Review Letters (2026) — © APS | https://lnkd.in/ei-HQKkg #MachineLearning #QuantumPhysics #NeuralNetworks #QuantumManyBody #VariationalMethods #DeepLearning #CondensedMatterPhysics #QuantumChemistry #AIforScience #NeuralQuantumStates #EntanglementEntropy #TensorNetworks #ComputationalPhysics #MaterialsScience #QuantumSimulation

In quantum photonics, classical bitwise logic fails to capture the coherence of field-based systems. Fractal-Wave Algebra (FWA) introduces a paradigm where photons are not particles but multidimensional fields. Each photon contains 37 quantum measurements \(D_1\)–\(D_{37}\), acting as coordinated information channels. Resonance Operators (RO) enforce coherence across these dimensions, forming dynamic clusters that transcend geometry. A photonic AI lattice—e.g., a 16-ring microring chain on IMEC’s iSiPP50G platform—operates via RO-mediated synchronization. With coupling gap 200 nm, \(\kappa = 0.3\), \(\lambda = 1550\) nm, and Q = 10⁵, the system achieves coherence metric CM > 0.95 and error residue ER < 10⁻³. Broadband excitation (\(\Delta\lambda = 10\) nm) confirms self-locking behavior across spectral shifts. FWA operators—ℱ (fractal unfolding), ℛ (resonance coordination), ℰ (encoding), 𝒮 (compression), and 𝒯 (temporal phasing)—govern the evolution of the field. These allow dynamic activation, clustering, and modulation of the 37 dimensions, enabling robust, high-density encoding. RO ensures that perturbations (\(\delta\phi < 0.1\) rad) are compensated across the lattice. This architecture maps mathematical structures (e.g., Riemann zeros) into physical photonic patterns, enabling experimental mathematics and secure quantum processing. The observable output is a projection; true information density arises from coordinated activation of internal field states. Engineers must shift from particle logic to field reasoning: design = field configuration, testing = CM/ER metrics, algorithms = operator orchestration. Small-scale AI agents embedded in this lattice use the 37D space for predictive coordination beyond neural models. TRL‑4 prototypes are achievable via GDSII layouts, MPW submission, and interferometric phase mapping. Public specs (geometry, Q, FSR) remain reproducible; internal field algebra and kernel configurations stay protected. This duality supports both academic dissemination and secure deployment. FWA‑RO photonic AI redefines computation: photons become orchestrated fields, not bits. Coherence, resonance, and operator-driven control enable scalable, resilient, high-dimensional architectures—establishing a new standard for quantum photonics.