De Finetti Theorem 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.

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

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.

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.

Some Arithmetic Properties of Pólya's Urn

Following Hales (2018), the evolution of Pólya's urn may be interpreted as a walk, a Pólya walk, on the integer lattice $\mathbb{N}^2$. We study the visibility properties of Pólya's walk or, equivalently, the divisibility properties of the composition of the urn. In particular, we are interested in the asymptotic average time that a Pólya walk is visible from the origin, or, alternatively, in the asymptotic proportion of draws so that the resulting composition of the urn is coprime. Via de Finetti's exchangeability theorem, Pólya's walk appears as a mixture of standard random walks. This paper is a follow-up of Cilleruelo-Fernández-Fernández (2019), where similar questions were studied for standard random walks.

Markovianity and the Thompson monoid F+

We introduce a new distributional invariance principle, called ‘partial spreadability’, which emerges from the representation theory of the Thompson monoid F+ in noncommutative probability spaces. We show that a partially spreadable sequence of noncommutative random variables is adapted to a local Markov filtration. Conversely we show that a large class of noncommutative stationary Markov sequences provides representations of the Thompson monoid F+. In the particular case of a classical probability space, we arrive at a de Finetti theorem for stationary Markov sequences with values in a standard Borel space.

De Finetti-type theorems on quasi-local algebras and infinite Fermi tensor products

Local actions of [Formula: see text], the group of finite permutations on [Formula: see text], on quasi-local algebras are defined and proved to be [Formula: see text]-abelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann algebra. Infinite graded tensor products of [Formula: see text]-algebras, which include the CAR algebra, are then addressed as particular examples of quasi-local algebras acted upon [Formula: see text] in a natural way. Extreme invariant states are characterized as infinite products of a single even state, and a de Finetti theorem is established. Finally, infinite products of factorial even states are shown to be factorial by applying a twisted version of the tensor product commutation theorem, which is also derived here.

De Finetti Theorems for the Unitary Dual Group

We prove several de Finetti theorems for the unitary dual group, also called the Brown algebra. Firstly, we provide a finite de Finetti theorem characterizing $R$-diagonal elements with an identical distribution. This is surprising, since it applies to finite sequences in contrast to the de Finetti theorems for classical and quantum groups; also, it does not involve any known independence notion. Secondly, considering infinite sequences in $W^*$-probability spaces, our characterization boils down to operator-valued free centered circular elements, as in the case of the unitary quantum group $U_n^+$. Thirdly, the above de Finetti theorems build on dual group actions, the natural action when viewing the Brown algebra as a dual group. However, we may also equip the Brown algebra with a bialgebra action, which is closer to the quantum group setting in a way. But then, we obtain a no-go de Finetti theorem: invariance under the bialgebra action of the Brown algebra yields zero sequences, in $W^*$-probability spaces. On the other hand, if we drop the assumption of faithful states in $W^*$-probability spaces, we obtain a non-trivial half a de Finetti theorem similar to the case of the dual group action.

Symmetric states for C∗-Fermi systems

After introducing the infinite Fermi [Formula: see text]-tensor product of a single [Formula: see text]-graded [Formula: see text]-algebra as an inductive limit, we systematically study the structure of the so-called symmetric states, that is those which are invariant under the group consisting of all finite permutations of a countable set. Among the obtained results, we mention the extension of De Finetti theorem which asserts that a symmetric state is a “mixture” of product states, each of which is a product of a single even state. This result induces a canonical morphism of the simplexes made of the symmetric even states on the usual infinite [Formula: see text]-tensor product and the symmetric states on the infinite Fermi [Formula: see text]-tensor product. We then extend the so-called Klein transformation to the infinite Fermi [Formula: see text]-tensor product, available when the parity automorphism is inner. In such a situation, we investigate further properties of product states, the last being the extremal symmetric states on such an infinite Fermi [Formula: see text]-tensor product [Formula: see text]-algebra. This paper is complemented with a finite dimensional illustrative example for which the Klein transformation is not implementable, and then the Fermi tensor product might not generate a usual tensor product. Therefore, in general, the study of the symmetric states on the Fermi algebra cannot be easily reduced to that of the corresponding symmetric states on the usual infinite tensor product, even if both share many common properties.

De Finetti's coherence and exchangeability in infinitary logic

We continue the investigation towards a logic-based approach to statistics within the infinitary conservative extension of Łukasiewicz logic IRL and prove versions of de Finetti's theorems on coherence and exchangeability. In particular we will prove a coherence criterion for a subclass of the variety of σ-complete Riesz MV-algebras in the conditional and unconditional case, and discuss de Finetti's exchangeability in a special case.

Convex geometry of finite exchangeable laws and de Finetti style representation with universal correlated corrections

We present a novel analogue for finite exchangeable sequences of the de Finetti, Hewitt and Savage theorem and investigate its implications for multi-marginal optimal transport (MMOT) and Bayesian statistics. If \((Z_1, \ldots ,Z_N)\) is a finitely exchangeable sequence of N random variables taking values in some Polish space X, we show that the law \(\mu _k\) of the first k components has a representation of the form $$\begin{aligned} \mu _k=\int _{\mathcal{P}_{\frac{1}{N}}(X)} F_{N,k}(\lambda ) \, \text{ d } \alpha (\lambda ) \end{aligned}$$for some probability measure \(\alpha \) on the set of \(\frac{1}{N}\)-quantized probability measures on X and certain universal polynomials \(F_{N,k}\). The latter consist of a leading term \(N^{k-1}\! /\prod _{j=1}^{k-1}(N\! -\! j) \lambda ^{\otimes k}\) and a finite, exponentially decaying series of correlated corrections of order \(N^{-j}\) (\(j=1, \ldots ,k\)). The \(F_{N,k}(\lambda )\) are precisely the extremal such laws, expressed via an explicit polynomial formula in terms of their one-point marginals \(\lambda \). Applications include novel approximations of MMOT via polynomial convexification and the identification of the remainder which is estimated in the celebrated error bound of Diaconis and Freedman (Ann Probab 8(4):745–764, 1980) between finite and infinite exchangeable laws.

De Finetti’s Theorem in Categorical Probability

We present a novel proof of de Finetti's Theorem characterizing permutation-invariant probability measures of infinite sequences of variables, so-called exchangeable measures. The proof is phrased in the language of Markov categories, which provide an abstract categorical framework for probability and information flow. The diagrammatic and abstract nature of the arguments makes the proof intuitive and easy to follow. We also show how the usual measure-theoretic version of de Finetti's Theorem for standard Borel spaces is an instance of this result.

Particle Representation for the Solution of the Filtering Problem. Application to the Error Expansion of Filtering Discretizations

We introduce a weighted particle representation for the solution of the filtering problem based on a suitably chosen variation of the classical de Finetti theorem. This representation has important theoretical and numerical applications. In this paper, we explore some of its theoretical consequences. The first is to deduce the equations satisfied by the solution of the filtering problem in three different frameworks: the signal independent Brownian measurement noise model, the spatial observations with additive white noise model and the cluster detection model in spatial point processes. Secondly we use the representation to show that a suitably chosen filtering discretisation converges to the filtering solution. Thirdly we study the leading error coefficient for the discretisation. We show that it satisfies a stochastic partial differential equation by exploiting the weighted particle representation for both the approximation and the limiting filtering solution.

Schur\u2013Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations

Schur–Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the t-fold tensor powers U^{otimes t} of all unitaries Uin U(d) is spanned by the permutations of the t tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries. The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of fault-tolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications:We resolve an open problem in quantum property testing by showing that “stabilizerness” is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group.We find that tensor powers of stabilizer states have an increased symmetry group. Conversely, we provide corresponding de Finetti theorems, showing that the reductions of arbitrary states with this symmetry are well-approximated by mixtures of stabilizer tensor powers (in some cases, exponentially well).We show that the distance of a pure state to the set of stabilizers can be lower-bounded in terms of the sum-negativity of its Wigner function. This gives a new quantitative meaning to the sum-negativity (and the related mana) – a measure relevant to fault-tolerant quantum computation. The result constitutes a robust generalization of the discrete Hudson theorem.We show that complex projective designs of arbitrary order can be obtained from a finite number (independent of the number of qudits) of Clifford orbits. To prove this result, we give explicit formulas for arbitrary moments of random stabilizer states.

Semidefinite programming hierarchies for constrained bilinear optimization

We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form {mathrm {Tr}}big [H(Dotimes E)big ], maximized with respect to semidefinite constraints on D and E. Applied to the problem of approximate error correction in quantum information theory, this gives hierarchies of efficiently computable outer bounds on the success probability of approximate quantum error correction codes in any dimension. The first level of our hierarchies corresponds to a previously studied relaxation (Leung and Matthews in IEEE Trans Inf Theory 61(8):4486, 2015) and positive partial transpose constraints can be added to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our sum-of-squares hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo’s asymptotic de Finetti theorem for quantum channels. Finally, our proof methods answer a question of Brandão and Harrow (Proceedings of the forty-fourth annual ACM symposium on theory of computing, STOC’12, p 307, 2012) by improving the approximation factor of de Finetti theorems with no symmetry from O(d^{k/2}) to {mathrm {poly}}(d,k), where d denotes local dimension and k the number of copies.

Finitely additive mixtures of probability measures

Let D be a linear space of real bounded functions and P:D→R a coherent functional. Also, let Q be a collection of coherent functionals on D. Under mild conditions, there is a finitely additive probability Π on the power set of Q such that P(f)=∫QQ(f)Π(dQ) for each f∈D. This fact has various consequences and such consequences are investigated in this paper. Three types of results are provided: (i) Existence of common extensions satisfying certain properties, (ii) Finitely additive mixtures of extreme points, (iii) Countably additive mixtures. Among other things, we obtain new versions of Kolmogorov's consistency theorem and de Finetti's representation theorem.

Harmonic analysis of symmetric random graphs

This note attempts to understand graph limits as defined by Lovasz and Szegedy (2006)} in terms of harmonic analysis on semigroups. This is done by representing probability distributions of random exchangeable graphs as mixtures of characters on the semigroup of unlabeled graphs with node-disjoint union, thereby providing an alternative derivation of de Finetti's theorem for random exchangeable graphs.