arXiv subjects

Mihály Weiner

Publications and source records attributed to Mihály Weiner.

At least 19 recordsLinked to original sources

arXiv2026

Error bounds for composite quantum hypothesis testing and a new characterization of the weighted Kubo-Ando geometric means

The optimal error exponents of binary composite i.i.d.~state discrimination are trivially bounded by the worst-case pairwise exponents of discriminating individual elements of the sets representing the two hypotheses, and in the finite-dimensional classical case, these bounds in fact give exact single-copy expressions for the error exponents. In contrast, in the non-commutative case, the optimal exponents are only known to be expressible in terms of regularized divergences, resulting in formulas that, while conceptually relevant, are practically not very useful. In this paper, we develop further an approach initiated in [Mosonyi, Szilágyi, Weiner, IEEE Trans.~Inf.~Th.~68(2):1032--1067, 2022] to give improved single-copy bounds on the error exponents by comparing not only individual states from the two hypotheses, but also various unnormalized positive semi-definite operators associated to them. Here, we show a number of equivalent characterizations of such operators giving valid bounds, and show that in the commutative case, considering weighted geometric means of the states, and in the case of two states per hypothesis, considering weighted Kubo-Ando geometric means, are optimal for this approach. As a result, we give a new characterization of the weighted Kubo-Ando geometric means as the only $2$-variable operator geometric means that are block additive, tensor multiplicative, continuous on monotone decreasing sequences, and satisfy the arithmetic-geometric mean inequality. We also extend our results to composite quantum channel discrimination, and show an analogous optimality property of the weighted Kubo-Ando geometric means of two quantum channels, a notion that seems to be new. We extend this concept to defining the notion of superoperator perspective function and establish some of its basic properties, which may be of independent interest.

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arXiv2026

Classical, quantum, and general probabilistic state-discrimination profiles

For a collection of states indexed by $[n]={1,\ldots,n}$, let $F(H)$ denote the optimal unnormalized minimum discrimination error for each nonempty subfamily $H\subseteq[n]$. We call $F$ the discrimination profile and characterize exactly which profiles can arise in a general probabilistic theory (GPT). For fixed $n$, the GPT region $\mathfrak{G}_n$ is a bounded rational polytope, giving a hierarchy $\mathfrak{C}_n\subseteq\mathfrak{Q}_n\subseteq\mathfrak{G}_n$ of classical, quantum, and GPT profiles, analogous to the local--quantum--no-signalling hierarchy in Bell theory. For $n=3$ we determine the classical and GPT polytopes explicitly. Within $\mathfrak{G}_3$, the classical region is characterized by $β:=F(123)-F(12)-F(13)-F(23)\le0$, while the GPT maximum is $1$. For qubit triples, the maximal value is attained by the trine, $β_{\mathrm{tr}}=3\sqrt3/2-2$, and we prove the dimension-independent quantum bound $β_{\mathrm Q}\le2/3$, with a slight further improvement. Hence $0=β_{\mathrm C}<3\sqrt3/2-2\leβ_{\mathrm Q}\le2/3<1=β_{\mathrm{GPT}}$. We conjecture $β_{\mathrm Q}=β_{\mathrm{tr}}$ and prove this for arbitrary pure-state triples, together with further supporting evidence. For arbitrary $n$, every classical facet admitting a GPT violation already admits a quantum violation in dimension at most $3$. A four-state qubit example shows that quantum success profiles need not be submodular, unlike for $n=3$. Finally, we exhibit a four-dimensional quantum triple whose discrete profile is classical for all subfamilies but becomes nonclassical when unequal priors are allowed.

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arXiv2026

Schatten norms and determinants of linear combinations of matrix tensor powers via virtual representations

Let $$X_n=\sum_{i=1}^s t_i A_i^{\otimes n},$$ where $A_1,\ldots,A_s\in M_d(\mathbb C)$ and $t_1,\ldots,t_s\in\mathbb C$ are fixed, while $n$ grows. Direct computation of determinants or Schatten norms of $X_n$ is exponential in $n$. For a single tensor power these quantities are elementary, and even the determinant of a generic two-term combination admits a reduction to polynomially many scalar factors; however, no analogous elementary reduction is available for three or more terms. We give an exact representation-theoretic method which, for fixed $d$ and $s$, computes $\|X_n\|_p$, $0<p<\infty$, and determinants in polynomial time in $n$. Schur--Weyl duality yields a simultaneous block decomposition, while Jacobi--Trudi identities in the Grothendieck ring replace Schur modules by signed combinations of tensor products of symmetric powers. For $d=3$, each irreducible contribution reduces to the difference of two explicitly computable symmetric-power terms, leading to an open-source implementation. In a single-thread CPU benchmark, a genuine three-term $3\times3$ trace-norm problem with $n=18$ is evaluated in about $47$ seconds, whereas just storing the unreduced matrix would require approximately $2.4\times10^{18}$ bytes. Direct and reduced computations agree to relative error below $3.4\times10^{-15}$ throughout their common range $n\leq9$.

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arXiv2025

Intrinsic volumes of the quantum state space and mutually unbiased bases

Previous studies on the geometrical properties of the state space of a finite-level quantum system have determined its volume and surface area. Building on this foundation, we derive explicit formulas for two additional intrinsic volume quantities. The question of whether a complete set of mutually unbiased bases exists in dimension $d$ can be equivalently framed as whether a specific convex polytope can be inscribed within the state space of a $d$-level quantum system. One motivation for our work was the hypothesis that a smaller intrinsic volume of the state space compared to the corresponding intrinsic volume of the mentioned polytope could rule out such an inscription. While our computations of these two intrinsic volumes do not lead to this conclusion, they nonetheless provide fundamental insights into the geometric structure of quantum state spaces. In particular, we show that these quantities can be used to rule out the existence of some unit-vector ``configurations'' (though not the one formed by the bases vectors of a complete set of mutually unbiased bases).

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arXiv2025

Triplets of Mutually Unbiased Bases

We initiate a systematic study of triplets of mutually unbiased bases (MUBs). We show that in $\mathbb{C}^d$ each MUB-triplet is characterized by a $d\times d\times d$ object that we call a Hadamard cube. We describe the basic properties of Hadamard cubes, and show how an MUB-triplet can be reconstructed from such a cube, up to unitary equivalence. We also present an algebraic identity which is conjectured to hold for all MUB-triplets in dimension 6. If true, it would imply the long-standing conjecture of Zauner that the maximum number of MUBs in dimension 6 is three.

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arXiv2025

Higher rank antipodality

Motivated by general probability theory, we say that the set $S$ in $\mathbb{R}^d$ is \emph{antipodal of rank $k$}, if for any $k+1$ elements $q_1,\ldots q_{k+1}\in S$, there is an affine map from $\mathrm{conv}(S)$ to the $k$-dimensional simplex $Δ_k$ that maps $q_1,\ldots q_{k+1}$ bijectively onto the $k+1$ vertices of $Δ_k$. For $k=1$, it coincides with the well-studied notion of (pairwise) antipodality introduced by Klee. We consider the following natural generalization of Klee's problem on antipodal sets: What is the maximum size of an antipodal set of rank $k$ in $\mathbb{R}^d$? We present a geometric characterization of antipodal sets of rank $k$ and adapting the argument of Danzer and Grünbaum originally developed for the $k=1$ case, we prove an upper bound which is exponential in the dimension. We show that this problem can be connected to a classical question in computer science on finding perfect hashes, and it provides a lower bound on the maximum size, which is also exponential in the dimension. By connecting rank-$k$ antipodality to $k$-neighborly polytopes, we obtain another upper bound when $k>d/2$.

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arXiv2023

Local energy bounds and strong locality in chiral CFT

A family of quantum fields is said to be strongly local if it generates a local net of von Neumann algebras. There are few methods of showing directly strong locality of a quantum field. Among them, linear energy bounds are the most widely used, yet a chiral conformal field of conformal weight $d>2$ cannot admit linear energy bounds. In this paper we give a new direct method to prove strong locality in two-dimensional conformal field theory. We prove that if a chiral conformal field satisfies an energy bound of degree $d-1$, then it also satisfies a certain local version of the energy bound, and this in turn implies strong locality. A central role in our proof is played by diffeomorphism symmetry. As a concrete application, we show that the vertex operator algebra given by a unitary vacuum representation of the $\mathcal{W}_3$-algebra is strongly local. For central charge $c > 2$, this yields a new conformal net. We further prove that these nets do not satisfy strong additivity, and hence are not completely rational.

math-ph↗

arXiv2023

Realization of an arbitrary structure of perfect distinguishability of states in general probability theory

Let $s_1,s_2,\ldots s_n$ be states of a general probability theory, and $\mathcal A$ be the set of all subsets of indices $H \subset [n]\equiv\{1,2,\ldots n\}$ such that the states $(s_j)_{j\in H}$ are jointly perfectly distinguishable. All subsets with a single element are of course in $\mathcal A$, and since smaller collections are easier to distinguish, if $H\in \mathcal A$ and $L \subset H$ then $L\in \mathcal A$; in other words, $\mathcal A$ is a so-called $\textit{independence system}$ on the set of indices $[n]$. In this paper it is shown that every independence system on $[n]$ can be realized in the above manner.

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arXiv2022

On entanglement assistance to a noiseless classical channel

For a classical channel, neither the Shannon capacity, nor the sum of conditional probabilities corresponding to the cases of successful transmission can be increased by the use of shared entanglement, or, more generally, a non-signaling resource. Yet, perhaps somewhat counterintuitively, entanglement assistance can help and actually elevate the chances of success even in a one-way communicational task that is to be completed by a single-shot use of a noiseless classical channel. To quantify the help that a non-signaling resource provides to a noiseless classical channel, one might ask how many extra letters should be added to the alphabet of the channel in order to perform equally well without the specified non-signaling resource. As was observed by Cubitt, Leung, Matthews, and Winter, there is no upper bound on the number of extra letters required for substituting the assistance of a general non-signaling resource to a noiseless one-bit classical channel. In contrast, here we prove that if this resource is a bipartite quantum system in a maximally entangled state, then an extra classical bit always suffices as a replacement.

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arXiv2021

A rigidity property of complete systems of mutually unbiased bases

Suppose that for some unit vectors $b_1,\ldots b_n$ in $\mathbb C^d$ we have that for any $j\neq k$ $b_j$ is either orthogonal to $b_k$ or $|\langle b_j,b_k\rangle|^2 = 1/d$ (i.e. $b_j$ and $b_k$ are unbiased). We prove that if $n=d(d+1)$, then these vectors necessarily form a complete system of mutually unbiased bases, that is, they can be arranged into $d+1$ orthonormal bases, all being mutually unbiased with respect to each other.

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arXiv2021

On the error exponents of binary state discrimination with composite hypotheses

The trade-off between the two types of errors in binary state discrimination may be quantified in the asymptotics by various error exponents. In the case of simple i.i.d. hypotheses, each of these exponents is equal to a divergence (pseudo-distance) of the two states. In the case of composite hypotheses, represented by sets of states $R,S$, one always has the inequality $\mathrm{e}(R\|S)\le \mathrm{E}(R\|S)$, where $\mathrm{e}$ is the exponent, $\mathrm{E}$ is the corresponding divergence, and the question is whether equality holds. The relation between the composite exponents and the worst pairwise exponents may be influenced by a number of factors: the type of exponents considered; whether the problem is classical or quantum; the cardinality and the geometric properties of the sets representing the hypotheses; and, on top of the above, possibly whether the underlying Hilbert space is finite- or infinite-dimensional. Our main contribution in this paper is clarifying this landscape considerably: We exhibit explicit examples for hitherto unstudied cases where the above inequality fails to hold with equality, while we also prove equality for various general classes of state discrimination problems. In particular, we show that equality may fail for any of the error exponents even in the classical case, if the system is allowed to be infinite-dimensional, and the alternative hypothesis contains countably infinitely many states. Moreover, we show that in the quantum case strict inequality is the generic behavior in the sense that, starting from any pair of non-commuting density operators of any dimension, and for any of the exponents, it is possible to construct an example with a simple null-hypothesis and an alternative hypothesis consisting of only two states, such that strict inequality holds for the given exponent.

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arXiv2020

Unitary representations of the $\mathcal{W}_3$-algebra with $c\geq 2$

We prove unitarity of the vacuum representation of the $\mathcal{W}_3$-algebra for all values of the central charge $c\geq 2$. We do it by modifying the free field realization of Fateev and Zamolodchikov resulting in a representation which, by a nontrivial argument, can be shown to be unitary on a certain invariant subspace, although it is not unitary on the full space of the two currents needed for the construction. These vacuum representations give rise to simple unitary vertex operator algebras. We also construct explicitly unitary representations for many positive lowest weight values. Taking into account the known form of the Kac determinants, we then completely clarify the question of unitarity of the irreducible lowest weight representations of the $\mathcal{W}_3$-algebra in the $2\leq c\leq 98$ region.

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arXiv2019

Rigidity and a common framework for mutually unbiased bases and k-nets

Many deep, mysterious connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called $k$-nets (and in particular, between complete collections of MUBs and finite affine - or equivalently: finite projective - planes). Here we introduce the notion of a $k$-net over an algebra $\mathfrak{A}$ and thus provide a common framework for both objects. In the commutative case, we recover (classical) $k$-nets, while choosing $\mathfrak{A} := M_d(\mathbb C)$ leads to collections of MUBs. A common framework allows one to find shared properties and proofs that "inherently work" for both objects. As a first example, we derive a certain rigidity property which was previously shown to hold for $k$-nets that can be completed to affine planes using a completely different, combinatorial argument. For $k$-nets that cannot be completed and for MUBs, this result is new, and, in particular, it implies that the only vectors unbiased to all but $k \leq \sqrt{d}$ bases of a complete collection of MUBs in $\mathbb C^d$ are the elements of the remaining $k$ bases (up to phase factors). In general, this is false when $k$ is just the next integer after $\sqrt{d}$; we present an example of this in every prime-square dimension, demonstrating that the derived bound is tight. As an application of the rigidity result, we prove that if a large enough collection of MUBs constructed from a certain type of group representation (e.g. a construction relying on discrete Weyl operators or generalized Pauli matrices) can be extended to a complete system, then in fact every basis of the completion must come from the same representation. In turn, we use this to show that certain large systems of MUBs cannot be completed.

math-ph↗

arXiv2018

From vertex operator algebras to conformal nets and back

We consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. We present a general procedure which associates to every strongly local vertex operator algebra V a conformal net A_V acting on the Hilbert space completion of V and prove that the isomorphism class of A_V does not depend on the choice of the scalar product on V. We show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex operator algebra V, the map W\mapsto A_W gives a one-to-one correspondence between the unitary subalgebras W of V and the covariant subnets of A_V. Many known examples of vertex operator algebras such as the unitary Virasoro vertex operator algebras, the unitary affine Lie algebras vertex operator algebras, the known c=1 unitary vertex operator algebras, the moonshine vertex operator algebra, together with their coset and orbifold subalgebras, turn out to be strongly local. We give various applications of our results. In particular we show that the even shorter Moonshine vertex operator algebra is strongly local and that the automorphism group of the corresponding conformal net is the Baby Monster group. We prove that a construction of Fredenhagen and Jörss gives back the strongly local vertex operator algebra V from the conformal net A_V and give conditions on a conformal net A implying that A= A_V for some strongly local vertex operator algebra V.

math.OA↗

arXiv2018

Conformal covariance and the split property

We show that for a conformal local net of observables on the circle, the split property is automatic. Both full conformal covariance (i.e. diffeomorphism covariance) and the circle-setting play essential roles in this fact, while by previously constructed examples it was already known that even on the circle, Möbius covariance does not imply the split property. On the other hand, here we also provide an example of a local conformal net living on the two-dimensional Minkowski space, which - although being diffeomorphism covariant - does not have the split property.

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arXiv2017

Character tables and the problem of existence of finite projective planes

Recently, the authors of the present work (together with M. N. Kolountzakis) introduced a new version of the non-commutative Delsarte scheme and applied it to the problem of mutually unbiased bases. Here we use this method to investigate the existence of a finite projective plane of a given order d. In particular, a short new proof is obtained for the nonexistence of a projective plane of order 6. For higher orders like 10 and 12, the method is non decisive but could turn out to give important supplementary informations.

math.CO↗

arXiv2017

Second order equation of motion for electromagnetic radiation back-reaction

We take the viewpoint that the physically acceptable solutions of the Lorentz--Dirac equation for radiation back-reaction are actually determined by a second order equation of motion, the self-force being given as a function of spacetime location and velocity. We propose three different methods to obtain this self-force function. For two example systems, we determine the second order equation of motion exactly in the nonrelativistic regime via each of these three methods, the three methods leading to the same result. We reveal that, for both systems considered, back-reaction induces a damping proportional to velocity and, in addition, it decreases the effect of the external force.

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arXiv2016

An application of positive definite functions to the problem of MUBs

We present a new approach to the problem of mutually unbiased bases (MUBs), based on positive definite functions on the unitary group. The method provides a new proof of the fact that there are at most $d+1$ MUBs in ${\mathbb C}^d$. It may also lead to a proof of non-existence of complete systems of MUBs in dimension 6 via a conjectured algebraic identity.

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