Program - Posters

Pierre Botteron - Nonlocal boxes seen as tensors and used to determine if a physical theory is realistic or not

Nonlocal boxes are 2x2x2x2 tensors that generalize the notion of quantum correlations. Some of these tensors violate fundamental principles, leading to the strong conclusion that any physical theory involving these tensors is interpreted as unrealistic. We know from experiments that Quantum Mechanics (QM) is a realistic theory, but we do not know yet if it is the more general theory that accurately describes the world. This raises the following question: if a tensor comes from a more general theory than QM, does it violate some fundamental principle? If so, it would mean that QM is the most general theory describing correlations between distant parties. To this end, we show that some beyond-than-quantum tensors violate the principle of communication complexity and thus are physically unrealistic.

References:

    [1] Physical Review Letters, 132, 070201 (2024) [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.132.070201].

    [2] Quantum 8, 1402 (2024) [https://quantum-journal.org/papers/q-2024-07-10-1402/].

    [3] arXiv:2406.02199 [quant-ph] (2024) [https://arxiv.org/abs/2406.02199].

Nicolas Delporte - Tensor eigenvalues outliers from a field theory perspective

In this talk, I will present a generalization of eigenvalues/eigenvectors to tensors and how they are affected by the presence of a Gaussian noise. We will focus on symmetric and Gaussian random tensors of order 3. I will introduce two eigenvalue distributions: the signed and the genuine, both computed using field theoretical tools (supersymmetry and Schwinger-Dyson equations), yielding an exact expression for the former distribution and an asymptotic one for the latter. I will compare those results to earlier probabilistic methods based on random matrices. Finally, from a saddle-point analysis, I will show how the largest eigenvalue changes as the variance of the noise increases showing the emergence of an outlier after a certain threshold. The work is based on https://arxiv.org/abs/2405.07731, with Naoki Sasakura.

Tomas Espana -  Kendall and generalized correlation matrices

Motivated by the estimation of correlation matrices in high dimensions, we introduce generalized correlation matrices (GCMs), inspired by Daniels’ Γ coefficient. Building on the work of Bandeira et al. (2017), who proved the first result on the limiting spectral distribution of a multivariate U-statistic (i.e. Kendall’s τ), we extend their findings to a family of order-2 and order-3 U-statistics. This extension allows us to derive the limiting spectral distribution of GCMs and of Spearman’s unbiased estimator matrix, under the assumption that the observations are i.i.d. random vectors X1, ..., Xn with i.i.d. components which are absolutely continuous with respect to the Lebesgue measure. We also investigate the use of Kendall’s correlation coefficient for estimating correlations in high-dimensional, heavy-tailed settings. Using multivariate Student synthetic data, we show that the eigenvectors of the empirical Kendall matrix are closer to the true ones than those of the Pearson correlation matrix.

Aabhas Gulati - Bound entanglement in cyclic signed symmetric states

We introduce and study bipartite quantum states in $\M{d} \otimes \M{d}$ which are invariant under the local action of the signed cyclic group. Due to symmetry, these states are sparse and can be parameterized by a triple of vectors. The important semi-definite properties, such as positivity and positivity under partial transpose (PPT) can be simply characterized in terms of these vectors and their discrete fourier transforms. In this article, we use these states to study the separability problem and to construct examples of symmetric bound entangled states. For certain interesting facets, we are able to show the necessary and sufficient conditions for separability. We study in detail an important subclass of local cyclic signed invariant states that are diagonal in the Dicke basis, also sometimes called the diagonal symmetric states. In this case the problem of deciding separability is equivalent to a circulant version of the complete positivity (CP) problem from non-convex optimization. We provide some geometrical results for the PPT and the separable states in this class. In d ≤ 5, we are able to completely characterize the geometry of the sets and construct entanglement witnesses. Finally, we look at invariant states with particular zero patterns on the diagonal. We connect the problem of detecting entanglement to suitably constructed trigonometric polynomials on the parameters of these states. In turn, we also construct families of states that are PPT entangled.

Guglielmo Lami - Anticoncentration and state design of random tensor networks

We investigate quantum random tensor network states where the bond dimensions scale polynomially with the system size N. Specifically, we examine the delocalization properties of random Matrix Product States (RMPS) in the computational basis by deriving an exact analytical expression for the Inverse Participation Ratio (IPR) of any degree, applicable to both open and closed boundary conditions. For bond dimensions X~Y^N, we determine the leading order of the associated overlaps probability distribution and demonstrate its convergence to the Porter-Thomas distribution, characteristic of Haar-random states, as Y increases. Additionally, we provide numerical evidence for the frame potential, measuring the 2-distance from the Haar ensemble, which confirms the convergence of random MPS to Haar-like behavior for X>>N^1/2. We extend this analysis to two-dimensional systems using random Projected Entangled Pair States (PEPS), where we similarly observe the convergence of IPRs to their Haar values for X>>N^1/2. These findings demonstrate that random tensor networks with bond dimensions scaling polynomially in the system size are fully Haar-anticoncentrated and approximate unitary designs, regardless of the spatial dimension.

This is based on the work https://arxiv.org/abs/2409.13023.

Itai Leigh - Discreteness of Asymptotic Tensor Ranks

Tensor parameters that are amortized or regularized over large tensor powers, often called "asymptotic" tensor parameters, play a central role in several areas including algebraic complexity theory (constructing fast matrix multiplication algorithms), quantum information (entanglement cost and distillable entanglement), and additive combinatorics (bounds on cap sets, sunflower-free sets, etc.). Examples are the asymptotic tensor rank, asymptotic slice rank and asymptotic subrank. Recent works (Costa--Dalai, Blatter--Draisma--Rupniewski, Christandl--Gesmundo--Zuiddam) have investigated notions of discreteness (no accumulation points) or "gaps" in the values of such tensor parameters. We prove that (1) over any finite set of coefficients in any field, the asymptotic rank, the asymptotic subrank and the asymptotic slice rank have no accumulation points, and (2) over the complex numbers, the asymptotic slice rank has no accumulation points.

Léo Le Nestour - Explicit Renormalization Group Flow from Perturbed Matrix Product Operators Using Random Walks

We explore the renormalization group (RG) flow starting from perturbed renormalization fixed points (RFPs) in the context of matrix product density operators (MPOs). Building on the framework outlined in arXiv:2204.05940, we introduce an explicit and local coarse-graining quantum channel for various MPOs to systematically track their renormalization. By applying small perturbations to these RFPs, we follow the resulting RG flow to study which new RFPs can be reached. We mainly detail the structure in the case of the quantum double model corresponding to group algebras as well as the dual case, in order to further understand the emerging phase diagram of matrix product density operators that are renormalization fixed points.

Kieran McShane - Threshold result for the p_n-PPT criterion

Deciding whether a given bipartite quantum state is entangled or separable (or even just close to separable) is known to be a computationally hard task.

Several much more easily checkable necessary conditions for separability do exist though, among the most famous and widely used ones is the positivity of partial transpose (PPT) criterion. Non-PPT states are necessarily entangled  and this is the simplest test to detect entanglement. The model that we study is that of a bipartite quantum system ℂ^d⊗ℂ^d,  coupled with some environment ℂ^s, the whole system thus being ℂ^d⊗ℂ^d⊗ℂ^s, with the assumption that the dimension of the environment scales as s=s_d~ cd², where c>0 is a constant parameter. 

We would like to know whether there is a threshold on the parameter c at which the typical behaviour of the state ρ switches from typically entangled to typically separable. Guillaume Aubrun proved that a threshold occurs at c=4 result for the PPT criterion. The state ρ is typically non-PPT if c<4 and typically PPT if c>4. Similar threshold results apply to the weaker p_n-PPT criteria, with a threshold at c=1 for lowest order p₃-PPT criterion.

Lubashan Pathirana - Random Dynamical Systems Approach to Study (non-IID) Repeated Interaction Dynamics

A discretization of the dynamics of an open quantum system is the repeated interaction or collision model, where one assumes that the system interacts with IID copies of the environment or an IID collection of smaller units (ancillas). However, using methods in random dynamical systems (RDS) one may study repeated interaction dynamics for the non-IID case. In the context of repeated-interactions, one assumes that the interactions are modeled by a random stationary sequence of quantum operations-valued maps, $(\phi_n)_{n\in\mathbb N}$, defined on the measure-preserving system $(\Omega,\mathcal F,Pr,\theta)$ where $\phi_n = \phi_1\circ\theta^{n-1}$. Therefore, the discrete parameter dynamic propagators are then given by a \emph{random} sequence, $(\Phi^{(n)})_{n\in\mathbb N}$, where $\Phi^{(n)}_\omega = \phi_n^\omega\circ\ldots\phi_1^\omega$, and they satisfy a cocycle property. We study the asymptotic behavior of $\Phi^{(n)}$ using RDS tools and obtain limiting results for both the discrete-parameter case and the continuous time dynamics. Furthermore, a class of random matrix products states (MPS) obtained by random CP-maps are introduced. These random MPS are distributionally translation invariant (a generalization of the translation invariant MPS with periodic boundary conditions) and certain probabilistic correlation inequalities are presented for local observables in the thermodynamic limit. By assuming that $\phi_1$ is given by a (random) perfect Kraus measurement we may also study the trajectory of an initial state $\rho$ under the measurement sequence $(\phi_n)_{n\in\mathbb N}$. This setting gives a time inhomogeneous Markov chain with random transition kernels (a.k.a. Markov chain in a random environment). We study ergodic-type results for these quantum trajectories and their asymptotic purification which is characterized by (lack of the) existence of a random multi-valued function called random dark subspace.

Carlos Pérez-Sánchez - Loop equations for noncommutative geometries on a graph

Since in Connes' noncommutative setting geometric (alongside of physical) data are encoded in a Dirac operator, the partition function that averages over noncommutative geometries on a fixed graph amounts to integrating over an ensemble of Dirac operators. Using elementary quiver representation theory, in this poster we:

- associate a Dirac operator to a quiver representation (in a category that emerges in noncommutative geometry);

- derive the constraints that the set of Wilson loops satisfy (generalised Makeenko-Migdal equations);

- explore the consequences of the positivity of a certain matrix of Wilson loops ('bootstrap').

In the special case that our graph is a rectangular lattice and our physical action quartic, we get lattice gauge field theory, hence the terminology. Unsurprisingly, our ensembles (but this for any graph) boil down to integrating noncommutative polynomials against a product Haar measure on unitary groups. Sources: Classical aspects were addressed in [2401.03705], and the loop equations in [2409.03705].

Anas Rahman - A Multiscale Cavity Method for Sublinear-rank Symmetric Matrix Factorization

We consider a statistical model for symmetric matrix factorization with additive Gaussian noise in the high-dimensional regime where the rank M of the signal matrix to infer scales with its size N as M = o(N^1/10). Allowing for an N-dependent rank offers new challenges and requires new methods. Working in the Bayes-optimal setting, we show that whenever the signal has i.i.d. entries the limiting mutual information between signal and data is given by a variational formula involving a rank-one replica symmetric potential. In other words, from the information-theoretic perspective, the case of a (slowly) growing rank is the same as when M = 1 (namely, the standard spiked Wigner model). The proof is primarily based on a novel multiscale cavity method allowing for growing rank along with some information-theoretic identities on worst noise for the Gaussian vector channel. We believe that the cavity method developed here will play a role in the analysis of a broader class of inference and spin models where the degrees of freedom are large arrays instead of vectors. arXiv:2403.07189.

Saswato Sen - Field theories on Bethe lattice

We study Euclidean quantum field theories on the Bethe lattice or the d - regular homogenous trees, which describe the local limit of large random regular graphs. We use walk representations to derive the Green functions, local density of states, and the spectral dimension for free scalar and Fermionic theories on the Bethe lattice. We also comment on the renormalization of field theories on the Bethe lattice (and arbitrary graphs) and show the emergent 3D behavior of the Bethe lattice in the continuum limit.

Mirte van der Eyden - Tensor products of convex cones    

Does there exist a self-dual tensor product of convex cones? We generalise the framework of operator systems to general conic systems by promoting the underlying structure of cones of positive semidefinite matrices to far more general structures. This allows us to prove the existence of a self-dual tensor product for finite dimensional proper cones and to reveal connections between the many examples that arise as siblings of operator systems.

Lu Wei - Cumulant Structures of Entanglement Entropy

We will present new methods to, in principle, obtain all cumulants of von Neumann entropy over different models of random states. The new methods uncover the structures of cumulants in terms of lower-order joint cumulants involving families of ancillary linear statistics. Importantly, the new methods avoid the task of simplifying nested summations when using existing methods in the literature that becomes prohibitively tedious as the order of cumulant increases. This poster is based on an ongoing joint work with Youyi Huang.