arXiv subjects

Ion Nechita

Publications and source records attributed to Ion Nechita.

At least 19 recordsLinked to original sources

arXiv2026

Quantum Graph Theory by Example

Quantum graphs have been introduced by Duan, Severini, and Winter to describe the zero-error behaviour of quantum channels. Since then, quantum graph theory has become a field of study in its own right. A substantial source of difficulty in working with quantum graphs compared to classical graphs stems from the fact that they are no longer discrete objects. This makes it generally difficult to construct insightful, non-trivial examples. We present a collection of non-trivial quantum graphs that can be thought of in discrete terms, and that can be expressed in the diagrammatic formalism introduced by Musto, Reutter, and Verdon. The examples arise as the quantum graphs acted on by increasingly smaller classical matrix groups, and are parametrised by triples of matrices $(A, B, C)$. The parametrisation reveals a clean decomposition of quantum graph structure into classical and genuinely quantum components: $A$ and $C$ are described by a classical weighted graph called the strange graph, while $B$ provides a purely quantum contribution with no classical analogue. Based on this model, we give exact formulas or establish bounds for quantum graph parameters, such as the number of connected components, the chromatic number, the independence number, and the clique number. Our results provide the first large, parametric families of quantum graphs for which standard graph parameters can be computed analytically.

math.OA↗

arXiv2026

Towards Unconditional Uncloneable Encryption

Uncloneable encryption is a cryptographic primitive which encrypts a classical message into a quantum ciphertext, such that two quantum adversaries are limited in their capacity of being able to simultaneously decrypt, given the key and quantum side-information produced from the ciphertext. Since its initial proposal and scheme in the random oracle model by Broadbent and Lord [TQC 2020], uncloneable encryption has developed into an important primitive at the foundation of quantum uncloneability for cryptographic primitives. Despite sustained efforts, however, the question of unconditional uncloneable encryption (and in particular of the simplest case, called an uncloneable bit) has remained elusive. Here, we propose a candidate for the unconditional uncloneable bit problem, and provide strong evidence that the adversary's success probability in the related security game converges quadratically as ${1}/{2}+{1}/{(2\sqrt{K})}$, where $K$ represents the number of keys and ${1}/{2}$ is trivially achievable. We prove this bound's validity for $K$ ranging from $2$ to $7$ and demonstrate the validity up to $K = 17$ using computations based on the NPA hierarchy. We furthemore provide compelling heuristic evidence towards the general case. In addition, we prove an asymptotic upper bound of ${5}/{8}$ and give a numerical upper bound of $\sim 0.5980$, which to our knowledge is the best-known value in the unconditional model.

quant-ph↗

arXiv2026

Entanglement in the Dicke subspace

We provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors. This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. Using this framework, we construct explicit PPT entangled states in three or more qutrits, disproving a recent conjecture. We establish that PPT entanglement exists for all multipartite systems with local dimension d >= 3 and n >= 3 parties. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions. We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.

quant-ph↗

arXiv2026

Multi-entropy in random tensor networks

We study the evaluation of Rényi multi-entropies $S^{(q)}_n$ in Random Tensor Network (RTN) states in the large bond-dimension limit. For the case of Rényi index $n=2$ and arbitrary number of parties $q$, we prove that that multi-entropies are determined by minimal multiway cuts through the network. When the minimal multiway cut is degenerate, we characterize the full minimizer set via compatible families of minimal cuts and give a criterion for all minimizers to come from ordinary cut partitions. For $n=2$, this gives a natural generalization of the minimal cut description of bipartite entanglement to multipartite systems with arbitrarily many parties. For the case of integer $n>2$, we show that the minimal multiway cut conjecture is in general \emph{not true} by providing explicit counter examples for both the single random tensor and for the network built from isometric tilings. We discuss the implication for our results on the multipartite entanglement structures in RTN and holography.

hep-th↗

arXiv2025

Inclusion constants for free spectrahedra with applications to quantum incompatibility

Building on the matrix cube problem, inclusions of free spectrahedra have been used successfully to obtain relaxations of hard spectrahedral inclusion problems. The quality of such a relaxation is quantified by the inclusion constant associated with each free spectrahedron. While optimal values of inclusion constants were known in certain highly symmetric cases, no general method for computing them was available. In this work, we show that inclusion constants for Cartesian products of free simplices can be computed using methods from non-commutative polynomial optimization, together with a detailed analysis of the extreme points of the associated free spectrahedra. This analysis also yields new closed-form analytic expressions for these constants. As an application to quantum information theory, we prove new bounds on the amount of white noise that incompatible measurements can tolerate before they become compatible. In particular, we study the case of one dichotomic and one $k$-outcome measurement, as well as the case of four dichotomic qubit measurements.

quant-ph↗

arXiv2025

Entanglement in cyclic sign invariant quantum states

We introduce and study bipartite quantum states that are invariant under the local action of the cyclic sign group. Due to symmetry, these states are sparse and can be parameterized by a triple of vectors. Their 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. We study in detail the entanglement properties of this family of symmetric states, showing that it contains PPT entangled states. For states that are diagonal in the Dicke basis, deciding separability is equivalent to a circulant version of the complete positivity problem. In local dimension d <= 5, we completely characterize these sets and construct entanglement witnesses; some partial results are also obtained for d = 6, 7. We construct a new family of states for which the properties of PPT and separability can be characterized for all dimensions, generalizing results from the literature. Our results show that this class of symmetric states has a rich entanglement structure, even in the bosonic subspace.

quant-ph↗

arXiv2025

Positive maps and extendibility hierarchies from copositive matrices

This work introduces and systematically studies a new convex cone of PCOP (pairwise copositive). We establish that this cone is dual to the cone of PCP (pairwise completely positive) and, critically, provides a complete characterization for the positivity of the broad class of covariant maps. We provide a way to lift matrices from the cone of COP to PCOP, thereby creating a powerful bridge between the theory of copositive forms and the positive maps. We develop an analogous framework for decomposable maps, introducing the cone PDEC. As a primary application of this framework, we define a novel family of linear maps $Φ_t^G$ parameterized by a graph $G$ and a real parameter $t$. We derive exact thresholds on $t$ that determine when these maps are positive or decomposable, linking these properties to fundamental graph-theoretic parameters. This construction yields vast new families of positive indecomposable maps, for which we provide explicit examples derived from infinite classes of graphs, most notably rank 3 strongly regular graphs such as Paley graphs. On the dual side, we investigate the entanglement properties of large classes of (symmetric) states. We prove that the SOS hierarchies used in polynomial optimization to approximate the cone of copositive matrices correspond precisely to dual cones of witnesses for different levels of the PPT bosonic extendibility hierarchy}-. In the setting of the DPS hierarchy for separability, we construct a large family of optimal entanglement witnesses that are not certifiable by any level of the PPT bosonic extendibility hierarchy, answering a long standing open question from [DPS04]. Leveraging the duality, we also provide an explicit construction of (mixture of) bipartite Dicke states that are simultaneously entangled and $K_r$-PPT bosonic extendible for any desired hierarchy level $r \geq 2$ and local dimension $n \geq 5$.

quant-ph↗

arXiv2025

Estimating the entanglement of random multipartite quantum states

Genuine multipartite entanglement of a given multipartite pure quantum state can be quantified through its geometric measure of entanglement, which, up to logarithms, is simply the maximum overlap of the corresponding unit tensor with product unit tensors, a quantity that is also known as the injective norm of the tensor. Our general goal in this work is to estimate this injective norm of randomly sampled tensors. To this end, we study and compare various algorithms, based either on the widely used alternating least squares method or on a novel normalized gradient descent approach, and suited to either symmetrized or non-symmetrized random tensors. We first benchmark their respective performances on the case of symmetrized real Gaussian tensors, whose asymptotic average injective norm is known analytically. Having established that our proposed normalized gradient descent algorithm generally performs best, we then use it to obtain numerical estimates for the average injective norm of complex Gaussian tensors (i.e., up to normalization, uniformly distributed multipartite pure quantum states), with or without permutation-invariance. We also estimate the average injective norm of random matrix product states constructed from Gaussian local tensors, with or without translation-invariance. All these results constitute the first numerical estimates on the amount of genuinely multipartite entanglement typically present in various models of random multipartite pure states. Finally, motivated by our numerical results, we posit two conjectures on the injective norms of random Gaussian tensors (real and complex) and Gaussian MPS in the asymptotic limit of the physical dimension.

quant-ph↗

arXiv2025

Factorization of multimeters: a unified view on nonclassical quantum phenomena

Quantum theory exhibits various nonclassical features, such as measurement incompatibility, contextuality, steering, and Bell nonlocality, which distinguish it from classical physics. These phenomena are often studied separately, but they possess deep interconnections. This work introduces a unified mathematical framework based on commuting diagrams that unifies them. By representing collections of measurements (multimeters) as maps to the set of column-stochastic matrices, we show that measurement compatibility and simulability correspond to specific factorizations of these maps through intermediate systems. We apply this framework to put forward connections between different nonclassical notions and provide factorization-based characterizations for steering assemblages and Bell correlations, including a new perspective on the CHSH inequality witnessing measurement incompatibility. We also investigate the symmetric $n$-extensions of multimeters and no-signaling behaviors and connect these extensions to a notion of $n$-wise compatibility and to the existence of $n$-wise LHV models, respectively. Furthermore, we investigate robustness to noise of nonlocal features by examining factorization conditions for maps involving noisy state spaces, providing geometric criteria for when noisy multimeters can be simulated by simpler measurement settings.

quant-ph↗

arXiv2025

Random measurements are almost maximally incompatible

In this work, we investigate the incompatibility of random quantum measurements. Most previous work has focused on characterizing the maximal amount of white noise that any fixed number of incompatible measurements with a fixed number of outcomes in a fixed dimension can tolerate before becoming compatible. This can be used to quantify the maximal amount of incompatibility available in such systems. The present article investigates the incompatibility of several classes of random measurements, i.e., the generic amount of incompatibility available. In particular, we show that for an appropriate choice of parameters, both random dichotomic projective measurements and random basis measurements are close to being maximally incompatible. We use the technique of incompatibility witnesses to certify incompatibility and combine it with tools from random matrices and free probability.

quant-ph↗

arXiv2025

Canonical partial ordering from min-cuts and quantum entanglement in random tensor networks

The \emph{max-flow min-cut theorem} has been recently used in the theory of random tensor networks in quantum information theory, where it is helpful for computing the behavior of important physical quantities, such as the entanglement entropy. In this paper, we extend the max-flow min-cut theorem to a relation among different \emph{partial orders} on the set of vertices of a network and introduce a new partial order for the vertices based on the \emph{min-cut structure} of the network. We apply the extended max-flow min-cut theorem to random tensor networks and find that the \emph{finite correction} to the entanglement Rényi entropy arising from the degeneracy of the min-cuts is given by the number of \emph{order morphisms} from the min-cut partial order to the partial order induced by non-crossing partitions on the symmetric group. Moreover, we show that the number of order morphisms corresponds to moments of a graph-dependent measure which generalizes the free Bessel law in some special cases in free probability theory.

math-ph↗

arXiv2025

Tensor free probability theory: asymptotic tensor freeness and central limit theorem

Voiculescu's notion of asymptotic free independence applies to a wide range of random matrices, including those that are independent and unitarily invariant. In this work, we generalize this notion by considering random matrices with a tensor product structure that are invariant under the action of local unitary matrices. Assuming the existence of the \emph{tensor distribution} limit described by tuples of permutations, we show that an independent family of local unitary invariant random matrices satisfies asymptotically a novel form of freeness, which we term \emph{tensor freeness}. It can be defined via the vanishing of mixed \emph{tensor free cumulants}, allowing the joint tensor distribution of tensor free elements to be described in terms of that of individual elements. We present several applications of these results in the context of random matrices with a tensor product structure, such as partial transpositions of (local) unitarily invariant random matrices and tensor embeddings of random matrices. Furthermore, we propose a tensor free version of the central limit theorem, which extends and recovers several previous results for tensor products of free variables.

math.OA↗

arXiv2025

On the simulation of quantum multimeters

In the quest for robust and universal quantum devices, the notion of simulation plays a crucial role, both from a theoretical and from an applied perspective. In this work, we go beyond the simulation of quantum channels and quantum measurements, studying what it means to simulate a collection of measurements, which we call a multimeter. To this end, we first explicitly characterize the completely positive transformations between multimeters. However, not all of these transformations correspond to valid simulations, as otherwise we could create any resource from nothing. For example, the set of transformations includes maps that always prepare the same multimeter regardless of the input, which we call trash-and-prepare. From the perspective of an experimenter with a given multimeter as part of a complicated setup, having to discard the multimeter and using a different one instead is undesirable. We give a new definition of multimeter simulations as transformations that are triviality-preserving, i.e., when given a multimeter consisting of trivial measurements they can only produce another trivial multimeter. In the absence of a quantum ancilla, we then characterize the transformations that are triviality-preserving and the transformations that are trash-and-prepare. Finally, we use these characterizations to compare our new definition of multimeter simulation to three existing ones: classical simulations, compression of multimeters, and compatibility-preserving simulations.

quant-ph↗

arXiv2024

Monogamy of highly symmetric states

We investigate the extent to which two particles can be maximally entangled when they are also similarly entangled with other particles on a complete graph, focusing on Werner, isotropic, and Brauer states. To address this, we formulate and solve optimization problems that draw on concepts from many-body physics, computational complexity, and quantum cryptography. We approach the problem by formalizing it as a semi-definite program (SDP), which we solve analytically using tools from representation theory. Notably, we determine the exact maximum values for the projection onto the maximally entangled state and the antisymmetric Werner state, thereby resolving long-standing open problems in the field of quantum extendibility. Our results are achieved by leveraging SDP duality, the representation theory of symmetric, unitary and orthogonal groups, and the Brauer algebra.

quant-ph↗

arXiv2024

Random covariant quantum channels

The group symmetries inherent in quantum channels often make them tractable and applicable to various problems in quantum information theory. In this paper, we introduce natural probability distributions for covariant quantum channels. Specifically, this is achieved through the application of ``twirling operations'' on random quantum channels derived from the Stinespring representation that use Haar-distributed random isometries. We explore various types of group symmetries, including unitary and orthogonal covariance, hyperoctahedral covariance, diagonal orthogonal covariance (DOC), and analyze their properties related to quantum entanglement based on the model parameters. In particular, we discuss the threshold phenomenon for positive partial transpose and entanglement breaking properties, comparing thresholds among different classes of random covariant channels. Finally, we contribute to the PPT$^2$ conjecture by showing that the composition between two random DOC channels is generically entanglement breaking.

quant-ph↗

arXiv2024

A Max-Flow approach to Random Tensor Networks

We study the entanglement entropy of a random tensor network (RTN) using tools from free probability theory. Random tensor networks are simple toy models that help the understanding of the entanglement behavior of a boundary region in the ADS/CFT context. One can think of random tensor networks are specific probabilistic models for tensors having some particular geometry dictated by a graph (or network) structure. We first introduce our model of RTN, obtained by contracting maximally entangled states (corresponding to the edges of the graph) on the tensor product of Gaussian tensors (corresponding to the vertices of the graph). We study the entanglement spectrum of the resulting random spectrum along a given bipartition of the local Hilbert spaces. We provide the limiting eigenvalue distribution of the reduced density operator of the RTN state, in the limit of large local dimension. The limit value is described via a maximum flow optimization problem in a new graph corresponding to the geometry of the RTN and the given bipartition. In the case of series-parallel graphs, we provide an explicit formula for the limiting eigenvalue distribution using classical and free multiplicative convolutions. We discuss the physical implications of our results, allowing us to go beyond the semiclassical regime without any cut assumption, specifically in terms of finite corrections to the average entanglement entropy of the RTN.

quant-ph↗

arXiv2024

On random classical marginal problems with applications to quantum information theory

In this paper, we study random instances of the classical marginal problem. We encode the problem in a graph, where the vertices have assigned fixed binary probability distributions, and edges have assigned random bivariate distributions having the incident vertex distributions as marginals. We provide estimates on the probability that a joint distribution on the graph exists, having the bivariate edge distributions as marginals. Our study is motivated by Fine's theorem in quantum mechanics. We study in great detail the graphs corresponding to CHSH and Bell-Wigner scenarios providing rations of volumes between the local and non-signaling polytopes.

quant-ph↗

arXiv2024

Algebra of Nonlocal Boxes and the Collapse of Communication Complexity

Communication complexity quantifies how difficult it is for two distant computers to evaluate a function f(X,Y), where the strings X and Y are distributed to the first and second computer respectively, under the constraint of exchanging as few bits as possible. Surprisingly, some nonlocal boxes, which are resources shared by the two computers, are so powerful that they allow to collapse communication complexity, in the sense that any Boolean function f can be correctly estimated with the exchange of only one bit of communication. The Popescu-Rohrlich (PR) box is an example of such a collapsing resource, but a comprehensive description of the set of collapsing nonlocal boxes remains elusive. In this work, we carry out an algebraic study of the structure of wirings connecting nonlocal boxes, thus defining the notion of the "product of boxes" $P\boxtimes Q$, and we show related associativity and commutativity results. This gives rise to the notion of the "orbit of a box", unveiling surprising geometrical properties about the alignment and parallelism of distilled boxes. The power of this new framework is that it allows us to prove previously-reported numerical observations concerning the best way to wire consecutive boxes, and to numerically and analytically recover recently-identified noisy PR boxes that collapse communication complexity for different types of noise models.

quant-ph↗