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Hubert de Guise

Publications and source records attributed to Hubert de Guise.

At least 19 recordsLinked to original sources
arXiv2026

Semiclassical asymptotics of multiphotonic scattering probabilities with partial indistinguishability

We propose a framework for computing multiphotonic scattering probabilities in a lossless multiport interferometer for arbitrary photon numbers and degrees of indistinguishability. By exploiting a toroidal expansion of multiphotonic states in tensor powers of single-particle states, the framework defines a map from a torus of relative phases to the probability simplex that governs the asymptotic behavior of scattering probabilities in the large-photon limit. Specifically, the probabilities concentrate on the "classically allowed region" defined by the map, and the slowly-varying part of the multiphotonic distribution reproduces a classical measure induced by the map. As a result, we are able to establish a new asymptotic formula for the multiphotonic probabilities in a general scenario of partially indistinguishable photons, while also providing a single-particle picture to explain the asymptotics of known multiphotonic transition amplitudes in the fully indistinguishable case. More broadly, our framework yields new, directly testable consequences in relation to asymptotic photon bunching patterns: it translates features of the classical map -- such as caustics and voids -- into direct predictions about regions of large or exponentially suppressed photon-distribution probability.

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arXiv2026

Unitary and Nonunitary Representations of the Heisenberg-Weyl Lie Algebra

We examine unitary and nonunitary representations of the Heisenberg-Weyl Lie algebra $\mathfrak{hw}_n$, with particular emphasis on tensor products of unitary representations and on indecomposable nonunitary representations. In the unitary setting, the irreducible representations with nontrivial central character are the Schrödinger representations, as classified by the Stone-von Neumann theorem. Although tensor products of these representations are considered in the literature, we give a detailed Lie-algebraic analysis and construct explicit unitary intertwining operators, including the case where the central characters sum to zero. In the nonunitary setting, we consider a natural realization of $\mathfrak{hw}_n$ as a subalgebra of the real symplectic Lie algebra $\mathfrak{sp}_{2n+2}(\mathbb R)$ and prove that every finite-dimensional complex irreducible representation of $\mathfrak{sp}_{2n+2}(\mathbb{R})$ remains indecomposable upon restriction to $\mathfrak{hw}_n$. This yields a large natural family of finite-dimensional, nonunitary indecomposable representations of $\mathfrak{hw}_n$.

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arXiv2024

Linear cross-entropy certification of quantum computational advantage in Gaussian Boson Sampling

Validation of quantum advantage claims in the context of Gaussian Boson Sampling (GBS) currently relies on providing evidence that the experimental samples genuinely follow their corresponding ground truth, i.e., the theoretical model of the experiment that includes all the possible losses that the experimenters can account for. This approach to verification has an important drawback: it is necessary to assume that the ground truth distributions are computationally hard to sample, that is, that they are sufficiently close to the distribution of the ideal, lossless experiment, for which there is evidence that sampling, either exactly or approximately, is a computationally hard task. This assumption, which cannot be easily confirmed, opens the door to classical algorithms that exploit the noise in the ground truth to efficiently simulate the experiments, thus undermining any quantum advantage claim. In this work, we argue that one can avoid this issue by validating GBS implementations using their corresponding ideal distributions directly. We explain how to use a modified version of the linear cross-entropy, a measure that we call the LXE score, to find reference values that help us assess how close a given GBS implementation is to its corresponding ideal model. Finally, we analytically compute the score that would be obtained by a lossless GBS implementation.

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arXiv2024

Qudit non-Clifford interleaved benchmarking

We introduce a scheme to characterise a qudit T gate that has different noise than a set of Clifford gates. We developed our scheme through representation theory and ring theory to generalise non-Clifford interleaved benchmarking to qudit systems. By restricting to the qubit case, we recover the dihedral benchmarking scheme. Our characterisation scheme provides experimental physicists a practical method for characterising universal qudit gate sets and advances randomised benchmarking research by providing the characterisation of a complete qudit library.

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arXiv2024

Randomised benchmarking for universal qudit gates

We aim to establish a scalable scheme for characterising diagonal non-Clifford gates for single- and multi-qudit systems; \(d\) is a prime-power integer. By employing cyclic operators and a qudit T gate, we generalise the dihedral benchmarking scheme for single- and multi-qudit circuits. Our results establish a path for experimentally benchmarking qudit systems and are of theoretical and experimental interest because our scheme is optimal insofar as it does not require preparation of the full qudit Clifford gate set to characterise a non-Clifford gate. Moreover, combined with Clifford randomised benchmarking, our scheme is useful to characterise the generators of a universal gate set.

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arXiv2024

Benchmarking of universal qutrit gates

We introduce a characterisation scheme for a universal qutrit gate set. Motivated by the rising interest in qutrit systems, we apply our criteria to establish that our hyperdihedral group underpins a scheme to characterise the performance of a qutrit T gate. Our resulting qutrit scheme is feasible, as it requires resources and data analysis techniques similar to resources employed for qutrit Clifford randomised benchmarking. Combining our T gate benchmarking procedure for qutrits with known qutrit Clifford-gate benchmarking enables complete characterisation of a universal qutrit gate set.

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arXiv2022

Generalized Interference of Fermions and Bosons

Using tools from representation theory, we derive expressions for the coincidence rate of partially-distinguishable particles in an interferometry experiment. Our expressions are valid for either bosons or fermions, and for any number of particles. In an experiment with $n$ particles the expressions we derive contain a term for each partition of the integer $n$; Gamas's theorem is used to determine which of these terms are automatically zero based on the pairwise level of distinguishability between particles. Most sampling schemes (such as Boson Sampling) are limited to completely indistinguishable particles; our work aids in the understanding of systems where an arbitrary level of distinguishability is permitted. As an application of our work we introduce a sampling scheme with partially-distinguishable fermions, which we call Generalized Fermion Sampling.

quant-ph↗
arXiv2022

Angle and angular momentum -- new twist for an old pair

Reaching ultimate performance of quantum technologies requires the use of detection at quantum limits and access to all resources of the underlying physical system. We establish a full quantum analogy between the pair of angular momentum and exponential angular variable, and the structure of canonically conjugate position and momentum. This includes the notion of optimal simultaneous measurement of the angular momentum and angular variable, the identification of Einstein-Podolsky-Rosen-like variables and states, and finally a phase-space representation of quantum states. Our construction is based on close interconnection of the three concepts and may serve as a template for the treatment of other observables. This theory also provides a new testbed for implementation of quantum technologies combining discrete and continuous quantum variables.

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arXiv2021

From polarization multipoles to higher-order coherences

We demonstrate that the multipoles associated with the density matrix are truly observable quantities that can be unambiguously determined from intensity moments. Given their correct transformation properties, these multipoles are the natural variables to deal with a number of problems in the quantum domain. In the case of polarization, the moments are measured after the light has passed through two quarter-wave plates, one half-wave plate, and a polarizing beam splitter for specific values of the angles of the waveplates. For more general two-mode problems, equivalent measurements can be performed.

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arXiv2021

Sum rules in multiphoton coincidence rates

We show that sums of carefully chosen coincidence rates in a multiphoton interferometry experiment can be simplified by replacing the original unitary scattering matrix with a coset matrix containing $0$s. The number and placement of these $0$s reduces the complexity of each term in the sum without affecting the original sum of rates. In particular, the evaluation of sums of modulus squared of permanents is shown to turn in some cases into a sum of modulus squared of determinants. The sums of rates are shown to be equivalent to the removal of some optical elements in the interferometer.

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arXiv2020

$SU(1,1)$ covariant $s$-parametrized maps

We propose a practical recipe to compute the ${s}$-parametrized maps for systems with $SU(1,1)$ symmetry using a connection between the ${Q}$ and ${P} $ symbols through the action of an operator invariant under the group. The particular case of the self-dual (Wigner) phase-space functions, defined on the upper sheet of the two-sheet hyperboloid (or, equivalently, inside the Poincaré disc) are analyzed.

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arXiv2020

Interferometrically estimating a quadratic form for any immanant of a matrix and its permutations

We devise a multiphoton interferometry scheme for sampling a quadratic function of a specific immanant for any submatrix of a unitary matrix and its row permutations. The full unitary matrix describes a passive, linear interferometer, and its submatrix is used when photons enter in and are detected at subsets of possible input and output channels. Immanants are mathematical constructs that interpolate between the permanent and determinant; contrary to determinants and permanents, which have meaningful physical applications, immanants are devoid of physical meaning classically but here are shown to be meaningful in a quantum setting. Our quadratic form of immanants is sampled by injecting vacuum and single photons into interferometer input ports such that the photon arrival times are entangled, in contrast to previous methods that control arrival times without entangling. Our method works for any number of photons, and we solve explicitly the quadratic form for the two-, three- and four-photon cases.

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arXiv2019

SU(3) Clebsch-Gordan coefficients and some of their symmetries

We discuss the construction and symmetries of su(3) Clebsch-Gordan coefficients arising from the su(3) basis states constructed as triple tensor products of two-dimensional harmonic oscillator states. Because of the su(2) symmetry of the basis states, matrix elements and recursion relations are easily expressed in terms of su(2) technology. As the Weyl group has a particularly simple action on these states, Weyl symmetries of the su(3) coupling coefficients generalizing the well known m-> -m symmetry of su(2) coupling can be obtained, so that any coefficient can be obtained as a sum of Weyl-reflected coefficients lying in the dominant Weyl sector. Some important cases of multiplicity-free decomposition are discussed as examples of applications.

math-ph↗
arXiv2019

Correspondence rules for Wigner functions over SU(3)/U(2)

We present results on the * product for SU(3) Wigner functions over SU(3)/U(2). In particular, we present a form of the so-called correspondence rules, which provide a differential form of the * product A*B and A*B when A is an su(3) generator. For the su(3) Wigner map, these rules must contain second order derivatives and thus substantially differ from the rules of other known cases.

math-ph↗
arXiv2018

State-independent preparation uncertainty relations

The standard state-dependent Heisenberg-Robertson uncertainly-relation lower bound fails to capture the quintessential incompatibility of observables as the bound can be zero for some states. To remedy this problem, we establish a class of tight (i.e., inequalities are saturated)variance-based sum-uncertainty relations derived from the Lie algebraic properties of observables and show that our lower bounds depend only on the irreducible representation assumed carried by the Hilbert space of state of the system. We illustrate our result for the cases of the Weyl-Heisenberg algebra, special unitary algebras up to rank 4, and any semisimple compact algebra. We also prove the usefulness of our results by extending a known variance-based entanglement detection criterion.

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arXiv2018

Coincidence landscapes for polarized bosons

Passive optical interferometry with single photons injected into some input ports and vacuum into others is enriched by admitting polarization, thereby replacing the scalar electromagnetic description by a vector theory, with the recent triad phase being a celebrated example of this richness. On the other hand, incorporating polarization into interferometry is known to be equivalent to scalar theory if the number of channels is doubled. We show that passive multiphoton $m$ channel interferometry described by SU($m$) transformations is replaced by SU($2m$) interferometry if polarization is included and thus that the multiphoton coincidence landscape, whose domain corresponds to various relative delays between photon arrival times, is fully explained by the now-standard approach of using immanants to compute coincidence sampling probabilities. Consequently, we show that the triad phase is manifested simply as SU(6) interferometry with three input photons, with one photon in each of three different input ports. Our analysis incorporates passive polarization multichannel interferometry into the existing scalar-field approach to computing multiphoton coincidence probabilities in interferometry and demystifies the triad phase.

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arXiv2018

Permutational symmetries for coincidence rates in multi-mode multi-photonic interferometry

We obtain coincidence rates for passive optical interferometry by exploiting the permutational symmetries of partially distinguishable input photons, and our approach elucidates qualitative features of multi-photon coincidence landscapes. We treat the interferometer input as a product state of any number of photons in each input mode with photons distinguished by their arrival time. Detectors at the output of the interferometer count photons from each output mode over a long integration time. We generalize and prove the claim of Tillmann et al. [Phys. Rev. X 5 041015 (2015)] that coincidence rates can be elegantly expressed in terms of immanants. Immanants are functions of matrices that exhibit permutational symmetries and the immanants appearing in our coincidence-rate expressions share permutational symmetries with the input state. Our results are obtained by employing representation theory of the symmetric group to analyze systems of arbitrary number of photons in arbitrarily sized interferometers.

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