arXiv subjects

Guillaume Aubrun

Publications and source records attributed to Guillaume Aubrun.

At least 19 recordsLinked to original sources

arXiv2026

Factorization through Lorentz cones

A pair of proper cones $(\mathsf{C}_1,\mathsf{C}_2)$ is said to have the Lorentz factorization property (LFP) if every $(\mathsf{C}_1,\mathsf{C}_2)$-positive map factors through a direct sum of Lorentzian cones, i.e., cones over Euclidean balls. Clearly, $(\mathsf{C}_1,\mathsf{C}_2)$ has the LFP if either $\mathsf{C}_1$ or $\mathsf{C}_2$ is a direct sum of Lorentzian cones, and our main goal is to find other examples. We show that such examples cannot be found for pairs $(\mathsf{C}_1,\mathsf{C}_2)$ where $\mathsf{C}_1=\mathsf{C}_2$, or in the case where both $\mathsf{C}_1$ and $\mathsf{C}_2$ are polyhedral. We also focus on the case where $\mathsf{C}_1=\mathsf{C}_\square$ is the square-based cone in $\mathbf{R}^3$. Here, we show that $(\mathsf{C}_\square,\mathsf{C})$ has the LFP whenever $\mathsf{C}$ is a symmetric cone, i.e., a direct sum of Lorentz cones, cones of positive semidefinite matrices over the real numbers, complex numbers or quaternions, and the cone of $3\times 3$ positive semidefinite matrices over the octonions. We leave open the question whether there are more examples, but we show that this list cannot be extended by any strictly convex cone $\mathsf{C}$ or for a cone $\mathsf{C}$ with $\text{dim}(\mathsf{C})\leq 5$. Finally, we discuss an application to a problem in quantum information theory.

math.FA↗

arXiv2026

Sphere Packings in Higher Dimension (after Boaz Klartag)

Let $δ_n^L$ be the maximal density of a lattice sphere packing in the $n$-dimensional Euclidean space. We explain how Boaz Klartag proved the inequality $δ_n^L \geq c n^2 2^{-n}$ where $c>0$ is a universal constant. In higher dimension, even for non-lattice sphere packings, this new lower bound is a substantial improvement. Klartag's proof uses the probabilistic method in two different ways. The first, very standard, relies on the statistical properties of a uniformly chosen random lattice. The second, completely new, studies the stochastic evolution of an ellipsoid constrained to contain non nonzero lattice points in the interior.

math.MG↗

arXiv2025

Monogamy of entanglement between cones

A separable quantum state shared between parties $A$ and $B$ can be symmetrically extended to a quantum state shared between party $A$ and parties $B_1,\ldots ,B_k$ for every $k\in\mathbf{N}$. Quantum states that are not separable, i.e., entangled, do not have this property. This phenomenon is known as "monogamy of entanglement". We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones $\mathsf{C}_A$ and $\mathsf{C}_B$: The elements of the minimal tensor product $\mathsf{C}_A\otimes_{\min} \mathsf{C}_B$ are precisely the tensors that can be symmetrically extended to elements in the maximal tensor product $\mathsf{C}_A\otimes_{\max} \mathsf{C}^{\otimes_{\max} k}_B$ for every $k\in\mathbf{N}$. Equivalently, the minimal tensor product of two cones is the intersection of the nested sets of $k$-extendible tensors. It is a natural question when the minimal tensor product $\mathsf{C}_A\otimes_{\min} \mathsf{C}_B$ coincides with the set of $k$-extendible tensors for some finite $k$. We show that this is universally the case for every cone $\mathsf{C}_A$ if and only if $\mathsf{C}_B$ is a polyhedral cone with a base given by a product of simplices. Our proof makes use of a new characterization of products of simplices up to affine equivalence that we believe is of independent interest.

quant-ph↗

arXiv2025

A characterization of inner product spaces via norming vectors

A finite-dimensional normed space is an inner product space if and only if the set of norming vectors of any endomorphism is a linear subspace. This theorem was proved by Sain and Paul for real scalars. In this paper, we give a different proof which also extends to the case of complex scalars.

math.FA↗

arXiv2024

Limit formulas for norms of tensor power operators

Given an operator $ϕ:X\rightarrow Y$ between Banach spaces, we consider its tensor powers $ϕ^{\otimes k}$ as operators from the $k$-fold injective tensor product of $X$ to the $k$-fold projective tensor product of $Y$. We show that after taking the $k$th root, the operator norm of $ϕ^{\otimes k}$ converges to the $2$-dominated norm $γ^*_2(ϕ)$, one of the standard operator ideal norms.

math.FA↗

arXiv2024

Optimal constants in concentration inequalities on the sphere and in the Gauss space

We show several variants of concentration inequalities on the sphere stated as subgaussian estimates with optimal constants. For a Lipschitz function, we give one-sided and two-sided bounds for deviation from the median as well as from the mean. For example, we show that if $μ$ is the normalized surface measure on $S^{n-1}$ with $n\geq 3$, $f : S^{n-1} \to \mathbb{R}$ is $1$-Lipschitz, $M$ is the median of $f$, and $t >0$, then $μ\big(f \geq M +t\big) \leq \frac 12 e^{-nt^2/2}$. If $M$ is the mean of $f$, we have a two-sided bound $μ\big(|f - M| \geq t\big) \leq e^{-nt^2/2}$. Consequently, if $γ$ is the standard Gaussian measure on $\mathbb{R}^n$ and $f : \mathbb{R}^{n} \to \mathbb{R}$ (again, $1$-Lipschitz, with the mean equal to $M$), then $γ\big(|f - M| \geq t\big) \leq e^{-t^2/2}$. These bounds are slightly better and arguably more elegant than those available elsewhere in the literature.

math.PR↗

arXiv2024

Completely Bounded Norms of $k$-positive Maps

Given an operator system $\mathcal{S}$, we define the parameters $r_k(\mathcal{S})$ (resp. $d_k(\mathcal{S})$) defined as the maximal value of the completely bounded norm of a unital $k$-positive map from an arbitrary operator system into $\mathcal{S}$ (resp. from $\mathcal{S}$ into an arbitrary operator system). In the case of the matrix algebras $M_n$, for $1 \leq k \leq n$, we compute the exact value $r_k(M_n) = \frac{2n-k}{k}$ and show upper and lower bounds on the parameters $d_k(M_n)$. Moreover, when $\mathcal{S}$ is a finite-dimensional operator system, adapting recent results of Passer and the 4th author, we show that the sequence $(r_k( \mathcal{S}))$ tends to $1$ if and only if $\mathcal{S}$ is exact and that the sequence $(d_k(\mathcal{S}))$ tends to $1$ if and only if $\mathcal{S}$ has the lifting property.

math.OA↗

arXiv2023

Maximal exponent of the Lorentz cones

We show that the maximal exponent (i.e., the minimum number of iterations required for a primitive map to become strictly positive) of the n-dimensional Lorentz cone is equal to n. As a byproduct, we show that the optimal exponent in the quantum Wielandt inequality for qubit channels is equal to 3.

math.MG↗

arXiv2021

Annihilating Entanglement Between Cones

Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The entanglement annihilating maps transform the $k$-fold maximal tensor product of a cone $C_1$ into the $k$-fold minimal tensor product of a cone $C_2$, and the pair $(C_1,C_2)$ is called resilient if all entanglement annihilating maps are entanglement breaking. Our main result is that $(C_1,C_2)$ is resilient if either $C_1$ or $C_2$ is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.

quant-ph↗

arXiv2021

Asymptotic Tensor Powers of Banach Spaces

We study the asymptotic behaviour of large tensor powers of normed spaces and of operators between them. We define the tensor radius of a finite-dimensional normed space $X$ as the limit of the sequence $A_k^{1/k}$, where $A_k$ is the equivalence constant between the projective and injective norms on $X^{\otimes k}$. We show that Euclidean spaces are characterized by the property that their tensor radius equals their dimension. Moreover, we compute the tensor radius for spaces with enough symmetries, such as the spaces $\ell_p^n$. We also define the tensor radius of an operator $T$ as the limit of the sequence $B_k^{1/k}$, where $B_k$ is the injective-to-projective norm of $T^{\otimes k}$. We show that the tensor radius of an operator whose domain or range is Euclidean is equal to its nuclear norm, and give some evidence that this property might characterize Euclidean spaces.

math.FA↗

arXiv2021

Principal angles between random subspaces and polynomials in two free projections

We use the geometric concept of principal angles between subspaces to compute the noncommutative distribution of an expression involving two free projections. For example, this allows to simplify a formula by Fevrier-Mastnak-Nica-Szpojankowski about the free Bernoulli anticommutator. We also derive economically an explicit formula for the free additive convolution of Bernoulli distributions. As a byproduct, we observe the remarkable fact that the principal angles between random half-dimensional subspaces are asymptotically distributed according to the uniform measure.

math.PR↗

arXiv2021

Entanglement and superposition are equivalent concepts in any physical theory

We prove that any two general probabilistic theories (GPTs) are entangleable, in the sense that their composite exhibits either entangled states or entangled measurements, if and only if they are both non-classical, meaning that neither of the state spaces is a simplex. This establishes the universal equivalence of the (local) superposition principle and the existence of global entanglement, valid in a fully theory-independent way. As an application of our techniques, we show that all non-classical GPTs exhibit a strong form of incompatibility of states and measurements, and use this to construct a version of the BB84 protocol that works in any non-classical GPT.

quant-ph↗

arXiv2021

Entangleability of cones

We solve a long-standing conjecture by Barker, proving that the minimal and maximal tensor products of two finite-dimensional proper cones coincide if and only if one of the two cones is generated by a linearly independent set. Here, given two proper cones $C_1$, $C_2$, their minimal tensor product is the cone generated by products of the form $x_1 \otimes x_2$, where $x_1 \in C_1$ and $x_2 \in C_2$, while their maximal tensor product is the set of tensors that are positive under all product functionals $f_1 \otimes f_2$, where $f_1$ is positive on $C_1$ and $f_2$ is positive on $C_2$. Our proof techniques involve a mix of convex geometry, elementary algebraic topology, and computations inspired by quantum information theory. Our motivation comes from the foundations of physics: as an application, we show that any two non-classical systems modelled by general probabilistic theories can be entangled.

math.FA↗

arXiv2019

Universal entangleability of non-classical theories

Inspired by its fundamental importance in quantum mechanics, we define and study the notion of entanglement for abstract physical theories, investigating its profound connection with the concept of superposition. We adopt the formalism of general probabilistic theories (GPTs), encompassing all physical models whose predictive power obeys minimal requirements. Examples include classical theories, which do not exhibit superposition and whose state space has the shape of a simplex, quantum mechanics, as well as more exotic models such as Popescu-Rohrlich boxes. We call two GPTs entangleable if their composite admits either entangled states or entangled measurements, and conjecture that any two non-classical theories are in fact entangleable. We present substantial evidence towards this conjecture by proving it (1) for the simplest case of 3-dimensional theories; (2) when the local state spaces are discrete, which covers foundationally relevant cases; (3) when one of the local theories is quantum mechanics. Furthermore, (4) we envision the existence of a quantitative relation between local non-classicality and global entangleability, explicitly describing it in the geometrically natural case where the local state spaces are centrally symmetric.

quant-ph↗

arXiv2018

Universal gaps for XOR games from estimates on tensor norm ratios

We define and study XOR games in the framework of general probabilistic theories, which encompasses all physical models whose predictive power obeys minimal requirements. The bias of an XOR game under local or global strategies is shown to be given by a certain injective or projective tensor norm, respectively. The intrinsic (i.e.\ model-independent) advantage of global over local strategies is thus connected to a universal function $r(n,m)$ called 'projective-injective ratio'. This is defined as the minimal constant $ρ$ such that $\|\cdot\|_{X\otimes_πY}\leqρ\,\|\cdot\|_{X\otimes_\varepsilon Y}$ holds for all Banach spaces of dimensions $\dim X=n$ and $\dim Y=m$, where $X\otimes_πY$ and $X \otimes_\varepsilon Y$ are the projective and injective tensor products. By requiring that $X=Y$, one obtains a symmetrised version of the above ratio, denoted by $r_s(n)$. We prove that $r(n,m)\geq 19/18$ for all $n,m\geq 2$, implying that injective and projective tensor products are never isometric. We then study the asymptotic behaviour of $r(n,m)$ and $r_s(n)$, showing that, up to log factors: $r_s(n)$ is of the order $\sqrt{n}$ (which is sharp); $r(n,n)$ is at least of the order $n^{1/6}$; and $r(n,m)$ grows at least as $\min\{n,m\}^{1/8}$. These results constitute our main contribution to the theory of tensor norms. In our proof, a crucial role is played by an '$\ell_1$/$\ell_2$/$\ell_{\infty}$ trichotomy theorem' based on ideas by Pisier, Rudelson, Szarek, and Tomczak-Jaegermann. The main operational consequence we draw is that there is a universal gap between local and global strategies in general XOR games, and that this grows as a power of the minimal local dimension. In the quantum case, we are able to determine this gap up to universal constants. As a corollary, we obtain an improved bound on the scaling of the maximal quantum data hiding efficiency against local measurements.

quant-ph↗

arXiv2017

Dvoretzky's Theorem and the Complexity of Entanglement Detection

The well-known Horodecki criterion asserts that a state $ρ$ on $\mathbf{C}^d \otimes \mathbf{C}^d$ is entangled if and only if there exists a positive map $Φ: \mathsf{M}_d \to \mathsf{M}_d$ such that the operator $(Φ\otimes \mathrm{Id})(ρ)$ is not positive semi-definite. We show that the number of such maps needed to detect all the robustly entangled states (i.e., states $ρ$ which remain entangled even in the presence of substantial randomizing noise) exceeds $\exp(c d^3 / \log d)$. The proof is based on the 1977 inequality of Figiel--Lindenstrauss--Milman, which ultimately relies on Dvoretzky's theorem about almost spherical sections of convex bodies. We interpret that inequality as a statement about approximability of convex bodies by polytopes with few vertices or with few faces and apply it to the study of fine properties of the set of quantum states and that of separable states. Our results can be thought of as geometrical manifestations of the complexity of entanglement detection.

quant-ph↗

arXiv2015

Two proofs of Størmer's theorem

The structure of the set of positivity-preserving maps between matrix algebras is notoriously difficult to describe. The notable exceptions are the results by Størmer and Woronowicz from 1960s and 1970s settling the low dimensional cases. By duality, these results are equivalent to the Peres-Horodecki positive partial transpose criterion being able to unambiguously establish whether a state in a 2 x 2 or 2 x 3 quantum system is entangled or separable. However, even in these low dimensional cases, the existing arguments (known to the authors) were based on long and seemingly ad hoc computations. We present a simple proof, based on Brouwer's fixed point theorem, for the 2 x 2 case (Størmer's theorem). For completeness, we also include another argument (following the classical outline, but highly streamlined) based on a characterization of extreme self-maps of the Lorentz cone and on a link - noticed by R. Hildebrand - to the S-lemma, a well-known fact from control theory and quadratic/semi-definite programming.

math.FA↗

arXiv2015

Catalysis in the trace class and weak trace class ideals

Given operators $A,B$ in some ideal $\mathcal{I}$ in the algebra $\mathcal{L}(H)$ of all bounded operators on a separable Hilbert space $H$, can we give conditions guaranteeing the existence of a trace-class operator $C$ such that $B \otimes C$ is submajorized (in the sense of Hardy--Littlewood) by $A \otimes C$ ? In the case when $\mathcal{I} = \mathcal{L}_1$, a necessary and almost sufficient condition is that the inequalities ${\rm Tr} (B^p) \leq {\rm Tr} (A^p)$ hold for every $p \in [1,\infty]$. We show that the analogous statement fails for $\mathcal{I} = \mathcal{L}_{1,\infty}$ by connecting it with the study of Dixmier traces.

math.FA↗