De Finetti Research Articles

Learning properties of quantum states without the IID assumption

We develop a framework for learning properties of quantum states beyond the assumption of independent and identically distributed (i.i.d.) input states. We prove that, given any learning problem (under reasonable assumptions), an algorithm designed for i.i.d. input states can be adapted to handle input states of any nature, albeit at the expense of a polynomial increase in training data size (aka sample complexity). Importantly, this polynomial increase in sample complexity can be substantially improved to polylogarithmic if the learning algorithm in question only requires non-adaptive, single-copy measurements. Among other applications, this allows us to generalize the classical shadow framework to the non-i.i.d. setting while only incurring a comparatively small loss in sample efficiency. We leverage permutation invariance and randomized single-copy measurements to derive a new quantum de Finetti theorem that mainly addresses measurement outcome statistics and, in turn, scales much more favorably in Hilbert space dimension.

De Finetti’s theorem and related results for infinite weighted exchangeable sequences

De Finetti’s theorem and related results for infinite weighted exchangeable sequences

Postselection Technique for Optical Quantum Key Distribution with Improved de Finetti Reductions

The postselection technique is an important proof technique for proving the security of quantum key distribution protocols against coherent attacks via the uplift of any security proof against independent identically distributed collective attacks. In this work, we go through multiple steps to rigorously apply the postselection technique to optical quantum key distribution protocols. First, we place the postselection technique on a rigorous mathematical foundation by fixing a technical flaw in the original postselection paper. Second, we extend the applicability of the postselection technique to prepare-and-measure protocols by using a de Finetti reduction with a fixed marginal. Third, we show how the postselection technique can be used for decoy-state protocols by tagging the source. Finally, we extend the applicability of the postselection technique to realistic optical setups by developing a new variant of the flag-state squasher. We also improve existing de Finetti reductions, which reduce the effect of using the postselection technique on the key rate. These improvements can be more generally applied to other quantum information processing tasks. As an example to demonstrate the applicability of our work, we apply our results to the time-bin-encoded three-state protocol. We observe that the postselection technique performs better than all other known proof techniques against coherent attacks. Published by the American Physical Society 2024

A connection between tests for absolute convergence of infinite series, or how to be fair

The Ratio Test and the Root Test for absolute convergence/divergence of series of numbers ∑ n = 0 ∞ a n are frequently discussed and proved independently in Calculus courses. The Root Test is stronger (verifies convergence for more series) than the Ratio Test. This relation inspires introduction of some intermediate strength tests (stronger than the Ratio Test and weaker than the Root Test) that we call Power Mean Tests (they, in particular, include the Arithmetic Mean Test). We show the connection between the Root, the Power Mean and the Ratio Tests. We also note that all these tests are related to the test that we formulate and call Generalised f-mean Test (or Kolmogorov–Nagumo–de Finetti mean Test). We provide an example of an infinite series, where the Arithmetic Mean test is the test that should be used for convergence verification, because the Root and the Ratio Tests are not easy to apply. We conclude this work with the statement emphasising why it is ‘fair’ to include the summarising Corollary 3.1 in our Calculus course.

On trivalent logics, probabilistic weak deduction theorems, and a general import-export principle

In this paper we first recall some results for conditional events, compound conditionals, conditional random quantities, p-consistency, and p-entailment. We discuss the equivalence between conditional bets and bets on conditionals, and review de Finetti's trivalent analysis of conditionals. But we go beyond de Finetti's early trivalent logical analysis and his later ideas, aiming to take his proposals to a higher level. We examine two recent articles that explore trivalent logics for conditionals and their definitions of logical validity and compare them with the approach to compound conditionals introduced by Gilio and Sanfilippo within the framework of conditional random quantities. As we use the notion of p-entailment, the full deduction theorem does not hold. We prove a Probabilistic Weak Deduction Theorem for conditional events. After that we study some variants of it, with further results, and we present several examples. Moreover, we illustrate how to derive new inference rules related to selected Aristotelian syllogisms. We focus on iterated conditionals and the invalidity of the Import-Export principle in the light of our Probabilistic Weak Deduction Theorem. We use the inference from a disjunction, A or B, to the conditional, if not-A then B, as an example to show the invalidity of this principle. We introduce a General Import-Export principle by examining examples and counterexamples. In particular, when considering the inference rules of System P, we find that a General Import-Export principle is satisfied, even if the assumptions of the Probabilistic Weak Deduction Theorem do not hold. We also deepen further aspects related to p-entailment and p-consistency. Finally, we briefly discuss some related work relevant to AI.

Pareto’s legacy in the Italian tradition: the case of mathematical economics

Pareto’s personality had a great influence on the evolution of economic theory in the last century at least in the Italian context. At the beginning of the twentieth century a small but combative circle of followers aimed to defend and spread the new Paretian approach based on the idea of general economic equilibrium. Paretian scholars were well recognizable in the Italian context for their contributions to mathematical and statistical aspects. The purpose of this contribution is to outline the essential features of the Italian Paretian school in the field of pure economics or as we would say today of mathematical economics. This favorable position of the Italian tradition in the field of mathematical economics was basically due to two circumstances. The first was represented by the legacy of Pareto and his school while the second was to be found in the fact that among the students of Pareto there was a pure mathematician of great value Luigi Amoroso. We see then that an important voice even if poorly valued was that of Bruno de Finetti who published a remarkable series of articles on mathematical economics in the second half of the 1930s. After World War II the Paretian tradition in this field weakened considerably and effectively disappeared internationally.

De Finetti’s control problem with a concave bound on the control rate

AbstractWe consider De Finetti’s control problem for absolutely continuous strategies with control rates bounded by a concave function and prove that a generalized mean-reverting strategy is optimal in a Brownian model. In order to solve this problem, we need to deal with a nonlinear Ornstein–Uhlenbeck process. Despite the level of generality of the bound imposed on the rate, an explicit expression for the value function is obtained up to the evaluation of two functions. This optimal control problem has, as special cases, those solved in Jeanblanc-Picqué and Shiryaev (1995) and Renaud and Simard (2021) when the control rate is bounded by a constant and a linear function, respectively.

A probabilistic analysis of selected notions of iterated conditioning under coherence

It is well known that basic conditionals satisfy some desirable basic logical and probabilistic properties, such as the compound probability theorem. However checking the validity of these becomes trickier when we switch to compound and iterated conditionals. Herein we consider de Finetti's notion of conditional both in terms of a three-valued object and as a conditional random quantity in the betting framework. We begin by recalling the notions of conjunction and disjunction among conditionals in selected trivalent logics. Then we analyze the notions of iterated conditioning in the frameworks of the specific three-valued logics introduced by Cooper-Calabrese, by de Finetti, and by Farrel. By computing some probability propagation rules we show that the compound probability theorem and other important properties are not always preserved by these formulations. Then, for each trivalent logic we introduce an iterated conditional as a suitable random quantity which satisfies the compound prevision theorem as well as some other desirable properties. We also check the validity of two generalized versions of Bayes' Rule for iterated conditionals. We study the p-validity of generalized versions of Modus Ponens and two-premise centering for iterated conditionals. Finally, we observe that all the basic properties are satisfied within the framework of iterated conditioning followed in recent papers by Gilio and Sanfilippo in the setting of conditional random quantities.

Quantum de Finetti Theorems as Categorical Limits, and Limits of State Spaces of C*-algebras

Quantum de Finetti Theorems as Categorical Limits, and Limits of State Spaces of C*-algebras

Short Communication: The Birth of (a Robust) Arbitrage Theory in de Finetti’s Early Contributions

Short Communication: The Birth of (a Robust) Arbitrage Theory in de Finetti’s Early Contributions

On Complex Matrix-Variate Dirichlet Averages and Its Applications in Various Sub-Domains.

This paper is about Dirichlet averages in the matrix-variate case or averages of functions over the Dirichlet measure in the complex domain. The classical power mean contains the harmonic mean, arithmetic mean and geometric mean (Hardy, Littlewood and Polya), which is generalized to the y-mean by de Finetti and hypergeometric mean by Carlson; see the references herein. Carlson's hypergeometric mean averages a scalar function over a real scalar variable type-1 Dirichlet measure, which is known in the current literature as the Dirichlet average of that function. The idea is examined when there is a type-1 or type-2 Dirichlet density in the complex domain. Averages of several functions are computed in such Dirichlet densities in the complex domain. Dirichlet measures are defined when the matrices are Hermitian positive definite. Some applications are also discussed.

Uniform probability in cosmology

Problems with uniform probabilities on an infinite support show up in contemporary cosmology. This paper focuses on the context of inflation theory, where it complicates the assignment of a probability measure over pocket universes. The measure problem in cosmology, whereby it seems impossible to pick out a uniquely well-motivated measure, is associated with a paradox that occurs in standard probability theory and crucially involves uniformity on an infinite sample space. This problem has been discussed by physicists, albeit without reference to earlier work on this topic. The aim of this article is both to introduce philosophers of probability to these recent discussions in cosmology and to familiarize physicists and philosophers working on cosmology with relevant foundational work by Kolmogorov, de Finetti, Jaynes, and other probabilists. As such, the main goal is not to solve the measure problem, but to clarify the exact origin of some of the current obstacles. The analysis of the assumptions going into the paradox indicates that there exist multiple ways of dealing consistently with uniform probabilities on infinite sample spaces. Taking a pluralist stance towards the mathematical methods used in cosmology shows there is some room for progress with assigning probabilities in cosmological theories.

De Finetti Theorems for Quantum Conditional Probability Distributions with Symmetry

The aim of device-independent quantum key distribution (DIQKD) is to study protocols that allow the generation of a secret shared key between two parties under minimal assumptions on the devices that produce the key. These devices are merely modeled as black boxes and mathematically described as conditional probability distributions. A major obstacle in the analysis of DIQKD protocols is the huge space of possible black box behaviors. De Finetti theorems can help to overcome this problem by reducing the analysis to black boxes that have an iid structure. Here we show two new de Finetti theorems that relate conditional probability distributions in the quantum set to de Finetti distributions (convex combinations of iid distributions) that are themselves in the quantum set. We also show how one of these de Finetti theorems can be used to enforce some restrictions onto the attacker of a DIQKD protocol. Finally we observe that some desirable strengthenings of this restriction, for instance to collective attacks only, are not straightforwardly possible.

Understanding Ramsey-de Finetti Probabilities and the St. Petersburg Paradox

We provide simple interpretations of probability with a view towards its potential for real world (i.e., practical) applications. Our discussion is along the lines initiated by David Ramsey and Bruno de Finetti, though we do not provide a theory of probability but merely use their axioms and results to interpret mathematical probability. Subsequently, we provide a critique as well as a resolution of the well-known St. Petersburg paradox that is consistent with expected utility theory.

Stable dividends under linear-quadratic optimisation

The optimisation criterion for dividends from a risky business is most often formalised in terms of the expected present value of future dividends. That criterion disregards a potential, explicit demand for the stability of dividends. In particular, within actuarial risk theory, the maximisation of future dividends has been studied as the so-called de Finetti problem. However, there the optimal strategies typically become so-called barrier strategies. These are far from stable, and suboptimal affine dividend strategies have recently received attention. In contrast, in the class of linear-quadratic problems, the demand for stability is explicitly stressed. These have often been studied in diffusion models different from the actuarial risk models. We bridge the gap between these thinking patterns by deriving optimal affine dividend strategies under a linear-quadratic criterion for an additive process. We characterise the value function by the Hamilton-Jacobi-Bellman equation, solve it, and compare the objective and the optimal controls to the classical objective of maximising the expected present value of future dividends. Thereby we provide a framework within which stability of dividends from a risky business, e.g. in classical risk theory, is explicitly demanded and obtained.

Probability propagation rules for Aristotelian syllogisms

We present a coherence-based probability semantics and probability propagation rules for (categorical) Aristotelian syllogisms. For framing the Aristotelian syllogisms as probabilistic inferences, we interpret basic syllogistic sentence types A, E, I, O by suitable precise and imprecise conditional probability assessments. Then, we define validity of probabilistic inferences and probabilistic notions of the existential import which is required, for the validity of the syllogisms. Based on a generalization of de Finetti's fundamental theorem to conditional probability, we investigate the coherent probability propagation rules of argument forms of the syllogistic Figures I, II, and III, respectively. These results allow to show, for all three figures, that each traditionally valid syllogism is also valid in our coherence-based probability semantics. Moreover, we interpret the basic syllogistic sentence types by suitable defaults and negated defaults. Thereby, we build a bridge from our probability semantics of Aristotelian syllogisms to nonmonotonic reasoning. Then we show that reductio by conversion does not work while reductio ad impossibile can be applied in our approach. Finally, we show how the proposed probability propagation rules can be used to analyze syllogisms involving generalized quantifiers (like Most).

The inflation hierarchy and the polarization hierarchy are complete for the quantum bilocal scenario

It is a fundamental but difficult problem to characterize the set of correlations that can be obtained by performing measurements on quantum mechanical systems. The problem is particularly challenging when the preparation procedure for quantum states is assumed to comply with a given causal structure. Recently, a first completeness result for this quantum causal compatibility problem has been given based on the so-called quantum inflation technique. However, completeness was achieved by imposing additional technical constraints, such as an upper bound on the Schmidt rank of the observables. Here, we show that these complications are unnecessary in the quantum bilocal scenario, a much-studied abstract model of entanglement swapping experiments. We prove that the quantum inflation hierarchy is complete for the bilocal scenario in the commuting observable model of locality. We also give a bilocal version of an observation by Tsirelson, namely, in finite dimensions, the commuting observable model and the tensor product model of locality coincide. These results answer questions recently posed by Renou and Xu [arXiv:2210.09065v2 (2022)]. Finally, we point out that our techniques can be interpreted more generally as giving rise to a semidefinite programming hierarchy that is complete for the problem of optimizing polynomial functions in the states of operator algebras defined by generators and relations. The completeness of this polarization hierarchy follows from a quantum de Finetti theorem for states on maximal C*-tensor products.

A Convergent Inflation Hierarchy for Quantum Causal Structures

A causal structure is a description of the functional dependencies between random variables. A distribution is compatible with a given causal structure if it can be realized by a process respecting these dependencies. Deciding whether a distribution is compatible with a structure is a practically and fundamentally relevant, yet very difficult problem. Only recently has a general class of algorithms been proposed: These so-called inflation techniques associate to any causal structure a hierarchy of increasingly strict compatibility tests, where each test can be formulated as a computationally efficient convex optimization problem. Remarkably, it has been shown that in the classical case, this hierarchy is complete in the sense that each non-compatible distribution will be detected at some level of the hierarchy. An inflation hierarchy has also been formulated for causal structures that allow for the observed classical random variables to arise from measurements on quantum states—however, no proof of completeness of this quantum inflation hierarchy has been supplied. In this paper, we construct a first version of the quantum inflation hierarchy that is provably convergent. It takes an additional parameter, r, which can be interpreted as an upper bound on the Schmidt rank of the observables involved. For each r, it provides a family of increasingly strict and ultimately complete compatibility tests for correlations that are compatible with a given causal structure under this Schmidt rank constraint. From a technical point of view, convergence proofs are built on de Finetti theorems, which show that certain symmetries (which can be imposed in convex optimization problems) imply independence of random variables (which is not directly a convex constraint). A main technical ingredient to our proof is a Quantum de Finetti Theorem that holds for general tensor products of C^*-algebras, generalizing previous work that was restricted to minimal tensor products.

A refinement of Reznick's Positivstellensatz with applications to quantum information theory

In his solution of Hilbert's 17th problem Artin showed that any positive definite polynomial in several variables can be written as the quotient of two sums of squares. Later Reznick showed that the denominator in Artin's result can always be chosen as an N-th power of the squared norm of the variables and gave explicit bounds on N. By using concepts from quantum information theory (such as partial traces, optimal cloning maps, and an identity due to Chiribella) we give simpler proofs and minor improvements of both real and complex versions of this result. Moreover, we discuss constructions of Hilbert identities using Gaussian integrals and we review an elementary method to construct complex spherical designs. Finally, we apply our results to give improved bounds for exponential quantum de Finetti theorems in the real and in the complex setting.

Conversations with Gábor J. Székely

Gábor J. Székely was born in Budapest, Hungary on February 4, 1947. He graduated from Eötvös Loránd University (ELTE) with an M.S. degree in 1970, and a Ph.D. degree in 1971. He received his Candidate Degree from the Hungarian Academy of Sciences in 1976, and the Doctor of Science Degree (D. Sc.) from the Hungarian Academy of Sciences in 1986. Székely joined the Department of Probability Theory of ELTE in 1970. In 1989, he became the founding chair of the Department of Stochastics of the Budapest Institute of Technology (Technical University of Budapest). In 1995, he moved to the United States as a tenured full professor at Bowling Green State University (BGSU) in Bowling Green, Ohio. Before that, in 1990–1991, he was the first Lukacs Distinguished Professor at BGSU. Székely had several visiting positions, for example, at the University of Amsterdam in 1977 and at Yale University in 1989. Between 2006 and 2022, he served as a Program Director in the Statistics Program of the Division of Mathematical Sciences at the U.S. National Science Foundation. Székely has about 250 publications, including 6 books in several languages. In 1988, he received the Rollo Davidson Prize from Cambridge University, jointly with Imre Z. Ruzsa for their work on algebraic probability theory. In 2010, Székely became an Elected Fellow of the Institute of Mathematical Statistics mostly for his works dealing with physics concepts in statistics like energy statistics and distance correlation. He had the fortune to know and work with world-class mathematicians and statisticians like (in chronological order of their first meetings): P. Erdős, A. Rényi, Y. Linnik, B. de Finetti, A. N. Kolmogorov, H. Robbins, G. Pólya, L. Shepp, G. Wahba, C. R. Rao, B. Efron, P. Bickel and E. Seneta.