arXiv subjects

Jack Davis

Publications and source records attributed to Jack Davis.

At least 19 recordsLinked to original sources

arXiv2026

Improved GKP magic states from error-corrected non-Gaussian quantum states

Gate teleportation, together with magic state distillation, is a promising route towards fault-tolerant, universal computation. In the context of bosonic quantum computation, Baragiola et al. PRL 123(20).200502 (2019) showed that within the framework of Gottesman--Kitaev--Preskill codes, encoded magic states suitable for distillation can be produced by error correcting Gaussian states, such as the vacuum. Here, we show that applying the same framework to simple non-Gaussian input states can significantly improve the quality of the magic states obtained, reducing the overall resources for the complete distillation procedure. We focus on superpositions of coherent states or Fock states, showing that many can lead to improvements in the generation of high-quality encoded magic states, which in some cases reduces the resources required for magic state distillation by about a factor $3$. We also investigate the primary source of these improvements and find that, unlike what was previously conjectured, the suitability of input states is not fully explained by symmetry arguments. Instead, the best states seem to avoid projection near stabilizer states as a result of the error correction procedure.

quant-ph↗

arXiv2026

Basis-independent stabilizerness and maximally noisy magic states

Absolutely stabilizer states are those that remain convex mixtures of stabilizer states after conjugation by any unitary. Here we give a characterization of such states for multiple qudits of all prime dimensions by introducing a polytope of their allowed spectra. We illustrate this through the examples of one qubit, two qubits, and one qutrit. In particular, the set of absolutely stabilizer states for a single qubit is a ball inscribed in the stabilizer octahedron, but for higher dimensions the geometry is more complicated. For odd-prime-dimensional qudits, we also give a complete characterization of absolutely Wigner-positive states, i.e., states whose Wigner function remains nonnegative after conjugation by any unitary. In so doing, we show there are absolutely Wigner-positive states that are not absolutely stabilizer, which can be seen as a unitarily-invariant version of bound magic. We then study the radii of the largest balls contained in the sets of absolutely stabilizer states and absolutely Wigner-positive states. These radii respectively tell us the lowest possible purity of nonstabilizer and Wigner-negative states. Conversely, we also find the radius of the smallest ball containing the set of absolutely Wigner-positive states, giving a tight purity-based necessary condition thereof.

quant-ph↗

arXiv2025

Characterising the sets of quantum states with non-negative Wigner function

For Hilbert spaces $\mathcal H\subseteq L^2(\mathbb R)$ we consider the convex sets $\mathcal D_+(\mathcal H)$ of Wigner-positive states (WPS), i.e.~density matrices over $\mathcal H$ with non-negative Wigner function. We investigate the topological structure of these sets, namely concerning closure, compactness, interior and boundary (in a relative topology induced by the trace norm). We also study their geometric structure and construct minimal sets of states that generate $\mathcal D_+(\mathcal H)$ through convex combinations. If $\mathcal H$ is finite-dimensional, the existence of such sets follows from a central result in convex analysis, namely the Krein-Milman theorem. In the infinite-dimensional case $\mathcal H=L^2(\mathbb R)$ this is not so, due to lack of compactness of the set $\mathcal D_+(\mathcal H)$. Nevertheless, we prove that a Krein-Milman theorem holds in this case, which allows us to extend most of the results concerning the sets of generators to the infinite-dimensional setting. Finally, we study the relation between the finite and infinite-dimensional sets of WPS, and prove that the former provide a hierarchy of closed subsets, which are also proper faces of the latter. These results provide a basis for an operational characterisation of the extreme points of the sets of WPS, which we undertake in a companion paper. Our work offers a unified perspective on the topological and geometric properties of the sets of WPS in finite and infinite dimensions, along with explicit constructions of minimal sets of generators.

math-ph↗

arXiv2025

Extreme non-negative Wigner functions

Providing an operational characterization of the Wigner-positive states (WPS), i.e., the set of quantum states with non-negative Wigner function, is a longstanding open problem. For pure states, the only WPS are Gaussian states, but the situation is considerably more subtle for mixed states. Here, we approach the problem using convex geometry, reducing the question to the characterization of the extreme points of the set of WPS. We give a constructive method to generate a large class of such extreme WPS, which combines the following steps: (i) we characterize the phase-invariant extreme points of the superset of Wigner-positive quasi-states (WPQS); (ii) we introduce a new quantum map, named Vertigo map, which maps extreme WPQS to extreme WPS while preserving phase invariance; (iii) we identify families of extremality-preserving maps and use them to obtain extreme WPS while relaxing phase invariance. Our construction generates all extreme WPS of low dimension, starting from a specific kind of WPS known as beam-splitter states. Our results build upon new mathematical properties of the set of WPS derived in a companion paper and unveil the remarkable structure of mixed states with non-negative Wigner functions.

quant-ph↗

arXiv2025

Quantum recurrences and the arithmetic of Floquet dynamics

The Poincar\'e recurrence theorem shows that conservative systems in a bounded region of phase space eventually return arbitrarily close to their initial state after a finite amount of time. An analogous behavior occurs in certain quantum systems where quantum states can recur after sufficiently long unitary evolution, a phenomenon known as quantum recurrence. Periodically driven (i.e. Floquet) quantum systems in particular exhibit complex dynamics even in small dimensions, motivating the study of how interactions and Hamiltonian structure affect recurrence behavior. While most existing studies treat recurrence in an approximate, distance-based sense, here we address the problem of exact, state-independent recurrences in a broad class of finite-dimensional Floquet systems, spanning both integrable and non-integrable models. Leveraging techniques from algebraic field theory, we construct an arithmetic framework that identifies all possible recurrence times by analyzing the cyclotomic structure of the Floquet unitary's spectrum. This computationally efficient approach yields both positive results, enumerating all candidate recurrence times and definitive negative results, rigorously ruling out exact recurrences for given Hamiltonian parameters. We further prove that rational Hamiltonian parameters do not, in general, guarantee exact recurrence, revealing a subtle interplay between system parameters and long-time dynamics. Our findings sharpen the theoretical understanding of quantum recurrences, clarify their relationship to quantum chaos, and highlight parameter regimes of special interest for quantum metrology and control.

quant-ph↗

arXiv2025

On the complex zeros of the wavefunction

The Schr\"odinger wavefunction is ubiquitous in quantum mechanics, quantum chemistry, and bosonic quantum information theory. Its zero-set for fermionic systems is well-studied and central for determining chemical properties, yet for bosonic systems the zero-set is less understood, especially in the context of characterizing non-classicality. Here we study the zeros of such wavefunctions and give them a novel information-theoretic interpretation. Our main technical result is showing that the wavefunction of most bosonic quantum systems can be extended to a holomorphic function over the complex plane, allowing the application of powerful techniques from complex analysis. As a consequence, we prove a version of Hudson's theorem for the wavefunction and characterize Gaussian dynamics as classical motion of the wavefunction zeros. Our findings suggest that the non-Gaussianity of quantum optical states can be detected by measuring a single quadrature of the electromagnetic field, which we demonstrate in a companion paper [arXiv:2507.23005]. More generally, our results show that the non-Gaussian features of bosonic quantum systems are encoded in the zeros of their wavefunction.

quant-ph↗

arXiv2025

Detecting quantum non-Gaussianity with a single quadrature

Full reconstruction of quantum states from measurement samples is often a prohibitively complex task, both in terms of the experimental setup and the scaling of the sample size with the system. This motivates the relatively easier task of certifying application-specific quantities using measurements that are not tomographically complete, i.e. that provide only partial information about the state related to the application of interest. Here, we focus on simplifying the measurements needed to certify non-Gaussianity in bosonic systems, a resource related to quantum advantage in various information processing tasks. We show that the statistics of a single quadrature measurement, corresponding to standard homodyne detection in quantum optics, can witness arbitrary degrees of non-Gaussianity as quantified by stellar rank. Our results are based on a version of Hudson's theorem for wavefunctions, proved in a companion paper [arXiv:2507.23468], revealing that the zeros in a homodyne distribution are signatures of quantum non-Gaussianity and higher stellar ranks. The validity of our witnesses is supported by a technical result showing that sets of states with bounded energy and finite stellar rank are compact. We provide an analysis of sample complexity, noise robustness, and experimental prospects. Our work drastically simplifies the setup required to detect quantum non-Gaussianity in bosonic quantum states.

quant-ph↗

arXiv2025

Majorization theory for quasiprobabilities

The question of how to characterize disorder arises in a wide variety of settings, including information theory, economics, and quantum thermodynamics. The theory of majorization provides an elegant answer, playing a central role in these fields. However, the existing majorization framework is inapplicable to functions that both take negative values and are defined on infinite spaces. Yet such functions, in the form of quasiprobability distributions, are ubiquitous in fields such as quantum optics, signal analysis, and bosonic quantum computation. Here we develop a notion of majorization that is applicable to such functions, proving that it admits four equivalent characterizations that naturally reduce to the finite case, thereby generalizing a seminal theorem by Hardy, Littlewood, and P\'olya. Moreover, we extend this equivalence to the setting where majorization is considered relative to an arbitrary positive distribution. We give several applications of our results in the context of quantum resource theories. These include deriving families of resource monotones and constraining quantum state conversions. We analytically and numerically study examples using the Wigner and Husimi functions, which feature prominently in quantum optics. Our results provide a comprehensive majorization framework for assessing the disorder of integrable functions over infinite measure spaces.

quant-ph↗

arXiv2025

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

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.

quant-ph↗

arXiv2024

Identifying quantum resources in encoded computations

What is the origin of quantum computational advantage? Providing answers to this far-reaching question amounts to identifying the key properties, or quantum resources, that distinguish quantum computers from their classical counterparts, with direct applications to the development of quantum devices. The advent of universal quantum computers, however, relies on error-correcting codes to protect fragile logical quantum information by robustly encoding it into symmetric states of a quantum physical system. Such encodings make the task of resource identification more difficult, as what constitutes a resource from the logical and physical points of view can differ significantly. Here we introduce a general framework which allows us to correctly identify quantum resources in encoded computations, based on phase-space techniques. For a given quantum code, our construction provides a Wigner function that accounts for how the symmetries of the code space are contained within the transformations of the physical space, resulting in an object capable of describing the logical content of any physical state, both within and outside the code space. We illustrate our general construction with the Gottesman--Kitaev--Preskill encoding of qudits with odd dimension. The resulting Wigner function, which we call the Zak-Gross Wigner function, is shown to correctly identify quantum resources through its phase-space negativity. For instance, it is positive for encoded stabilizer states and negative for the bosonic vacuum. We further prove several properties, including that its negativity provides a measure of magic for the logical content of a state, and that its marginals are modular measurement distributions associated to conjugate Zak patches.

quant-ph↗

arXiv2023

Quantum recurrences in the kicked top

The correspondence principle plays an important role in understanding the emergence of classical chaos from an underlying quantum mechanics. Here we present an infinite family of quantum dynamics that never resembles the analogous classical chaotic dynamics irrespective of dimension. These take the form of stroboscopic unitary evolutions in the quantum kicked top that act as the identity after a finite number of kicks. Because these state-independent temporal periodicities are present in all dimensions, their existence represents a universal violation of the correspondence principle. We further discuss the relationship of these periodicities with the quantum kicked rotor, in particular the phenomenon of quantum anti-resonance.

quant-ph↗

arXiv2023

Polytopes of Absolutely Wigner Bounded Spin States

Quasiprobability has become an increasingly popular notion for characterising non-classicality in quantum information, thermodynamics, and metrology. Two important distributions with non-positive quasiprobability are the Wigner function and the Glauber-Sudarshan function. Here we study properties of the spin Wigner function for finite-dimensional quantum systems and draw comparisons with its infinite-dimensional analog, focusing in particular on the relation to the Glauber-Sudarshan function and the existence of absolutely Wigner-bounded states. More precisely, we investigate unitary orbits of mixed spin states that are characterized by Wigner functions lower-bounded by a specified value. To this end, we extend a characterization of the set of absolutely Wigner positive states as a set of linear eigenvalue constraints, which together define a polytope centred on the maximally mixed state in the simplex of spin-$j$ states. The lower bound determines the relative size of such absolutely Wigner bounded (AWB) polytopes and we study their geometric characteristics. In each dimension a Hilbert-Schmidt ball representing a tight purity-based sufficient condition to be AWB is exactly determined, while another ball representing a necessary condition to be AWB is conjectured. Special attention is given to the case where the polytope separates orbits containing only positive Wigner functions from other orbits because of the use of Wigner negativity as a witness of non-classicality. Comparisons are made to absolute symmetric state separability and spin Glauber-Sudarshan positivity, with additional details given for low spin quantum numbers.

quant-ph↗

arXiv2022

Stellar representation of extremal Wigner-negative spin states

The Majorana stellar representation is used to characterize spin states that have a maximally negative Wigner quasiprobability distribution on a spherical phase space. These maximally Wigner-negative spin states generally exhibit a partial but not high degree of symmetry within their star configurations. In particular, for spin $j > 2$, maximal constellations do not correspond to a Platonic solid when available and do not follow an obvious geometric pattern as dimension increases. In addition, they are generally different from spin states that maximize other measures of nonclassicality such as anticoherence or geometric entanglement. Random states ($j \leq 6$) display on average a relatively high amount of negativity, but the extremal states and those with similar negativity are statistically rare in Hilbert space. We also prove that all spin coherent states of arbitrary dimension have non-zero Wigner negativity. This offers evidence that all pure spin states also have non-zero Wigner negativity. The results can be applied to qubit ensembles exhibiting permutation invariance.

quant-ph↗

arXiv2020

Wigner negativity in spin-$j$ systems

The nonclassicality of simple spin systems as measured by Wigner negativity is studied on a spherical phase space. Several SU(2)-covariant states with common qubit representations are addressed: spin coherent, spin cat (GHZ/N00N), and Dicke ($\textsf{W}$). We derive a bound on the Wigner negativity of spin cat states that rapidly approaches the true value as spin increases beyond $j \gtrsim 5$. We find that spin cat states are not significantly Wigner-negative relative to their Dicke state counterparts of equal dimension. We also find, in contrast to several entanglement measures, that the most Wigner-negative Dicke basis element is spin-dependent, and not the equatorial state $| j,0 \rangle$ (or $|j,\pm 1/2 \rangle$ for half-integer spins). These results underscore the influence that dynamical symmetry has on nonclassicality, and suggest a guiding perspective for finding novel quantum computational applications.

quant-ph↗

arXiv2020

Cracking the Black Box: Distilling Deep Sports Analytics

This paper addresses the trade-off between Accuracy and Transparency for deep learning applied to sports analytics. Neural nets achieve great predictive accuracy through deep learning, and are popular in sports analytics. But it is hard to interpret a neural net model and harder still to extract actionable insights from the knowledge implicit in it. Therefore, we built a simple and transparent model that mimics the output of the original deep learning model and represents the learned knowledge in an explicit interpretable way. Our mimic model is a linear model tree, which combines a collection of linear models with a regression-tree structure. The tree version of a neural network achieves high fidelity, explains itself, and produces insights for expert stakeholders such as athletes and coaches. We propose and compare several scalable model tree learning heuristics to address the computational challenge from datasets with millions of data points.

cs.LG↗

arXiv2019

Improved estimate of the collisional frequency shift in Al$^+$ optical clocks

Collisions between background gas particles and the trapped ion in an atomic clock can subtly shift the frequency of the clock transition. The uncertainty in the correction for this effect makes a significant contribution to the total systematic uncertainty budget of trapped-ion clocks. Using a non-perturbative analytic framework that was developed for this problem, we estimate the frequency shift in Al$^+$ ion clocks due to collisions with helium and hydrogen. Our calculations significantly improve the uncertainties in the collisional shift coefficients, and show that the collisional frequency shifts for Al$^+$ are zero to within uncertainty.

physics.atom-ph↗

arXiv2017

Network Inference from a Link-Traced Sample using Approximate Bayesian Computation

We present a new inference method based on approximate Bayesian computation for estimating parameters governing an entire network based on link-traced samples of that network. To do this, we first take summary statistics from an observed link-traced network sample, such as a recruitment network of subjects in a hard-to-reach population. Then we assume prior distributions, such as multivariate uniform, for the distribution of some parameters governing the structure of the network and behaviour of its nodes. Then, we draw many independent and identically distributed values for these parameters. For each set of values, we simulate a population network, take a link-traced sample from that network, and find the summary statistics for that sample. The statistics from the sample, and the parameters that eventually led to that sample, are collectively treated as a single point. We take a Kernel Density estimate of the points from many simulations, and observe the density across the hyperplane coinciding with the statistic values of the originally observed sample. This density function is treat as a posterior estimate of the paramaters of the network that provided the observed sample. We also apply this method to a network of precedence citations between legal documents, centered around cases overseen by the Supreme Court of Canada, is observed. The features of certain cases that lead to their frequent citation are inferred, and their effects estimated by ABC. Future work and extensions are also briefly discussed.

stat.CO↗

arXiv2013

Parallelization of an implicit algorithm for multi-dimensional particle-in-cell simulations

The implicit 2D3V particle-in-cell (PIC) code developed to study the interaction of ultrashort pulse lasers with matter [G. M. Petrov and J. Davis, Computer Phys. Comm. 179, 868 (2008); Phys. Plasmas 18, 073102 (2011)] has been parallelized using MPI (Message Passing Interface). Details on the algorithm implementation are given with emphasis on code optimization by overlapping computations with communications. Performance evaluation has been made on a small Linux cluster with 32 processors for two typical regimes of PIC operation: "particle dominated", for which the bulk of the computation time is spent on pushing particles, and "field dominated", for which computing the fields is prevalent. We found that the MPI implementation of the code offers a significant numerical speedup. In the "particle dominated" regime it is close to the maximum theoretical one, while in the other regime it is about 75-80 % of the maximum speed-up. The code parallelization will allow future implementation of atomic physics and extension to three dimensions.

physics.plasm-ph↗