Grand Unification of All Discrete Wigner Functions on $d \times d$ Phase Space

• URL: http://arxiv.org/abs/2503.09353v3
• Date: Fri, 08 Aug 2025 06:29:26 GMT
• Title: Grand Unification of All Discrete Wigner Functions on $d \times d$ Phase Space
• Authors: Lucky K. Antonopoulos, Dominic G. Lewis, Jack Davis, Nicholas Funai, Nicolas C. Menicucci,
• Abstract summary: Wigner functions help visualise quantum states and dynamics while supporting quantitative analysis in quantum information.<n>We introduce a stencil-based framework that exhausts all possible $dtimes d$ discrete Wigner functions for a single $d$-dimensional quantum system.
• Score: 0.5242869847419834
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Wigner functions help visualise quantum states and dynamics while supporting quantitative analysis in quantum information. In the discrete setting, many inequivalent constructions coexist for each Hilbert-space dimension. This fragmentation obscures which features are fundamental and which are artefacts of representation. We introduce a stencil-based framework that exhausts all possible $d\times d$ discrete Wigner functions for a single $d$-dimensional quantum system (including a novel one for even $d$), subsuming known forms. We also give explicit invertible linear maps between definitions within the same $d$, enabling direct comparison of operational properties and exposing representation dependence.

Related papers

• Spectral Superposition: A Theory of Feature Geometry [1.3837984867394175]
  We develop a theory for studying the geometric structre of features by analyzing the spectra of weight derived matrices.<n>In toy models of superposition, we use this theory to prove that capacity saturation forces spectral localization.
  arXiv  Detail & Related papers  (2026-02-02T15:28:38Z)
• Wigner-Husimi phase-space structure of quasi-exactly solvable sextic potential [0.0]
  We show that the Wigner representation is uniquely responsive to interference effects and displays clear, nonmonotonic entropic behavior as the wells separate.<n>These findings establish a quantitative hierarchy in the ability of $W$, $|W|$, and $H$ to resolve structural changes in a quantum state.
  arXiv  Detail & Related papers  (2025-12-06T04:48:28Z)
• The Origins of Representation Manifolds in Large Language Models [52.68554895844062]
  We show that cosine similarity in representation space may encode the intrinsic geometry of a feature through shortest, on-manifold paths.<n>The critical assumptions and predictions of the theory are validated on text embeddings and token activations of large language models.
  arXiv  Detail & Related papers  (2025-05-23T13:31:22Z)
• Tensor Cross Interpolation of Purities in Quantum Many-Body Systems [0.0]
  In quantum many-body systems the exponential scaling of the Hilbert space with the number of degrees of freedom renders a complete state characterization.<n>Recently, a compact way of storing subregions' purities by encoding them as amplitudes of a fictitious quantum wave function, known as entanglement feature, was proposed.<n>In this work, we demonstrate that the entanglement feature can be efficiently defining using only a amount of samples in the number of degrees of freedom.
  arXiv  Detail & Related papers  (2025-03-21T15:33:00Z)
• Probabilistic size-and-shape functional mixed models [6.424762079392286]
  We show that it is possible to recover the size-and-shape of a square-integrable $mu$ under a functional mixed model.<n>A random object-level unitary transformation then captures size-and-shape emphpreserving deviations of $mu$ from an individual function.
  arXiv  Detail & Related papers  (2024-11-27T15:00:33Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• Function Trees: Transparent Machine Learning [1.3597551064547502]
  Knowing the global properties of such functions can help in understanding the system that produced the data. A function tree is constructed that can be used to rapidly identify and compute all of the function's main and interaction effects.
  arXiv  Detail & Related papers  (2024-03-19T20:23:31Z)
• Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmaps [15.738019181349992]
  We develop nonparametric regression methods for the case when the true regression function is not necessarily smooth. More specifically, our approach is using the fractional Laplacian and is designed to handle the case when the true regression function lies in an $L$-fractional Sobolev space with order $sin (0,1)$.
  arXiv  Detail & Related papers  (2024-02-22T21:47:29Z)
• Interpreting symplectic linear transformations in a two-qubit phase space [0.0]
  For certain discrete Wigner functions, permuting the values of the Wigner function in accordance with a symplectic linear transformation is equivalent to performing a certain unitary transformation on the state. A symplectic linear permutation of the points of the phase space, together with a certain reinterpretation of the Wigner function, is equivalent to a unitary transformation.
  arXiv  Detail & Related papers  (2024-02-15T13:01:26Z)
• Gaussian Entanglement Measure: Applications to Multipartite Entanglement of Graph States and Bosonic Field Theory [50.24983453990065]
  An entanglement measure based on the Fubini-Study metric has been recently introduced by Cocchiarella and co-workers. We present the Gaussian Entanglement Measure (GEM), a generalization of geometric entanglement measure for multimode Gaussian states. By providing a computable multipartite entanglement measure for systems with a large number of degrees of freedom, we show that our definition can be used to obtain insights into a free bosonic field theory.
  arXiv  Detail & Related papers  (2024-01-31T15:50:50Z)
• Universal Representation of Permutation-Invariant Functions on Vectors and Tensors [11.345796608258434]
  A main object of our study is multiset functions -- that is, permutation-invariant functions over inputs of varying sizes. Deep Sets, proposed by citezaheer 2017deep, provides a emphuniversal representation for continuous multiset functions on scalars via a sum-decomposable model. We prove that universal representation is guaranteed for continuous and discontinuous multiset functions though a latent space dimension of $O(ND)$.
  arXiv  Detail & Related papers  (2023-10-20T22:00:59Z)
• Axiomatic characterization of pointwise Shapley decompositions [0.0]
  A common problem in various applications is the additive decomposition of the output of a function with respect to its input variables. In this paper, axioms are developed which fully preserve functional structures and lead to unique decompositions for all Borel measurable functions.
  arXiv  Detail & Related papers  (2023-03-14T10:24:48Z)
• Refining and relating fundamentals of functional theory [0.0]
  We explain why there exist six equivalent universal functionals, prove concise relations among them and conclude that the important notion of $v$-representability is relative to the scope and choice of variable. For systems with time-reversal symmetry, we explain why there exist six equivalent universal functionals, prove concise relations among them and conclude that the important notion of $v$-representability is relative to the scope and choice of variable.
  arXiv  Detail & Related papers  (2023-01-24T18:09:47Z)
• Reinforcement Learning from Partial Observation: Linear Function Approximation with Provable Sample Efficiency [111.83670279016599]
  We study reinforcement learning for partially observed decision processes (POMDPs) with infinite observation and state spaces. We make the first attempt at partial observability and function approximation for a class of POMDPs with a linear structure.
  arXiv  Detail & Related papers  (2022-04-20T21:15:38Z)
• Measuring dissimilarity with diffeomorphism invariance [94.02751799024684]
  We introduce DID, a pairwise dissimilarity measure applicable to a wide range of data spaces. We prove that DID enjoys properties which make it relevant for theoretical study and practical use.
  arXiv  Detail & Related papers  (2022-02-11T13:51:30Z)
• Submodular + Concave [53.208470310734825]
  It has been well established that first order optimization methods can converge to the maximal objective value of concave functions. In this work, we initiate the determinant of the smooth functions convex body $$F(x) = G(x) +C(x)$. This class of functions is an extension of both concave and continuous DR-submodular functions for which no guarantee is known.
  arXiv  Detail & Related papers  (2021-06-09T01:59:55Z)
• Scalable Frank-Wolfe on Generalized Self-concordant Functions via Simple Steps [66.88729048402082]
  Generalized self-concordance is a key property present in the objective function of many learning problems. We show improved convergence rates for various common cases, e.g., when the feasible region under consideration is uniformly convex or polyhedral.
  arXiv  Detail & Related papers  (2021-05-28T15:26:36Z)
• The role of feature space in atomistic learning [62.997667081978825]
  Physically-inspired descriptors play a key role in the application of machine-learning techniques to atomistic simulations. We introduce a framework to compare different sets of descriptors, and different ways of transforming them by means of metrics and kernels. We compare representations built in terms of n-body correlations of the atom density, quantitatively assessing the information loss associated with the use of low-order features.
  arXiv  Detail & Related papers  (2020-09-06T14:12:09Z)
• A Functional Perspective on Learning Symmetric Functions with Neural Networks [48.80300074254758]
  We study the learning and representation of neural networks defined on measures. We establish approximation and generalization bounds under different choices of regularization. The resulting models can be learned efficiently and enjoy generalization guarantees that extend across input sizes.
  arXiv  Detail & Related papers  (2020-08-16T16:34:33Z)
• Piecewise Linear Regression via a Difference of Convex Functions [50.89452535187813]
  We present a new piecewise linear regression methodology that utilizes fitting a difference of convex functions (DC functions) to the data. We empirically validate the method, showing it to be practically implementable, and to have comparable performance to existing regression/classification methods on real-world datasets.
  arXiv  Detail & Related papers  (2020-07-05T18:58:47Z)
• On the Theory of Transfer Learning: The Importance of Task Diversity [114.656572506859]
  We consider $t+1$ tasks parameterized by functions of the form $f_j circ h$ in a general function class $mathcalF circ mathcalH$. We show that for diverse training tasks the sample complexity needed to learn the shared representation across the first $t$ training tasks scales as $C(mathcalH) + t C(mathcalF)$.
  arXiv  Detail & Related papers  (2020-06-20T20:33:59Z)
• Invariant Feature Coding using Tensor Product Representation [75.62232699377877]
  We prove that the group-invariant feature vector contains sufficient discriminative information when learning a linear classifier. A novel feature model that explicitly consider group action is proposed for principal component analysis and k-means clustering.
  arXiv  Detail & Related papers  (2019-06-05T07:15:17Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.