De Finetti Theorem Research Articles

SINGLETON CONDITIONS AND QUANTUM DE FINETTI'S THEOREM

We investigate the validity of quantum De Finetti's Theorem for algebraic random processes satisfying the singleton condition. We first prove a conditional version of the theorem, then we give some natural conditions which guarantee the full validity of De Finetti's Theorem.

Quantum Cryptography with Finite Resources: Unconditional Security Bound for Discrete-Variable Protocols with One-Way Postprocessing

We derive a bound for the security of quantum key distribution with finite resources under one-way postprocessing, based on a definition of security that is composable and has an operational meaning. While our proof relies on the assumption of collective attacks, unconditional security follows immediately for standard protocols such as Bennett-Brassard 1984 and six-states protocol. For single-qubit implementations of such protocols, we find that the secret key rate becomes positive when at least N approximately 10(5) signals are exchanged and processed. For any other discrete-variable protocol, unconditional security can be obtained using the exponential de Finetti theorem, but the additional overhead leads to very pessimistic estimates.

Comparison Results for Exchangeable Credit Risk Portfolios

This paper is dedicated to the risk analysis of credit portfolios. Assuming that default indicators form an exchangeable sequence of Bernoulli random variables and as a consequence of de Finetti's theorem, default indicators are Binomial mixtures. We can characterize the supermodular order between two exchangeable Bernoulli random vectors in terms of the convex ordering of their corresponding mixture distributions. Thus we can proceed to some comparisons between stoploss premiums, CDO tranche premiums and convex risk measures on aggregate losses. This methodology provides a unified analysis of dependence for a number of CDO pricing models based on factor copulas, multivariate Poisson and structural approaches.

SECURITY OF QUANTUM KEY DISTRIBUTION

Quantum Information Theory is an area of physics which studies both fundamental and applied issues in quantum mechanics from an information-theoretical viewpoint. The underlying techniques are, however, often restricted to the analysis of systems which satisfy a certain independence condition. For example, it is assumed that an experiment can be repeated independently many times or that a large physical system consists of many virtually independent parts. Unfortunately, such assumptions are not always justified. This is particularly the case for practical applications — e.g. in quantum cryptography — where parts of a system might have an arbitrary and unknown behavior. We propose an approach which allows us to study general physical systems for which the above mentioned independence condition does not necessarily hold. It is based on an extension of various information-theoretical notions. For example, we introduce new uncertainty measures, called smooth min- and max-entropy, which are generalizations of the von Neumann entropy. Furthermore, we develop a quantum version of de Finetti's representation theorem, as described below. Consider a physical system consisting of n parts. These might, for instance, be the outcomes of n runs of a physical experiment. Moreover, we assume that the joint state of this n-partite system can be extended to an (n + k)-partite state which is symmetric under permutations of its parts (for some k ≫ 1). The de Finetti representation theorem then says that the original n-partite state is, in a certain sense, close to a mixture of product states. Independence thus follows (approximatively) from a symmetry condition. This symmetry condition can easily be met in many natural situations. For example, it holds for the joint state of n parts, which are chosen at random from an arbitrary (n + k)-partite system. As an application of these techniques, we prove the security of quantum key distribution (QKD), i.e. secret key agreement by communication over a quantum channel. In particular, we show that, in order to analyze QKD protocols, it is generally sufficient to consider so-called collective attacks, where the adversary is restricted to applying the same operation to each particle sent over the quantum channel separately. The proof is generic and thus applies to known protocols such as BB84 and B92 (where better bounds on the secret-key rate and on the the maximum tolerated noise level of the quantum channel are obtained) as well as to continuous variable schemes (where no full security proof has been known). Furthermore, the security holds with respect to a strong so-called universally composable definition. This implies that the keys generated by a QKD protocol can safely be used in any application, e.g. for one-time pad encryption — which, remarkably, is not the case for most standard definitions.

Reassessing Accuracy Rates of Median Decisions

We show how Bruno de Finetti's fundamental theorem of prevision has computable applications in statistical problems that involve only partial information. Specifically, we assess accuracy rates for median decision procedures used in the radiological diagnosis of asbestosis. Conditional exchangeability of individual radiologists' diagnoses is recognized as more appropriate than independence which is commonly presumed. The FTP yields coherent bounds on probabilities of interest when available information is insufficient to determine a complete distribution. Further assertions that are natural to the problem motivate a partial ordering of conditional probabilities, extending the computation from a linear to a quadratic programming problem.

Computational complexity of the quantum separability problem

Ever since entanglement was identified as a computational and cryptographic resource, researchers have sought efficient ways to tell whether a given density matrix represents an unentangled, or \emph{separable}, state. This paper gives the first systematic and comprehensive treatment of this (bipartite) quantum separability problem, focusing on its deterministic (as opposed to randomized) computational complexity. First, I review the one-sided tests for separability, paying particular attention to the semidefinite programming methods. Then, I discuss various ways of formulating the quantum separability problem, from exact to approximate formulations, the latter of which are the paper's main focus. I then give a thorough treatment of the problem's relationship with NP, NP-completeness, and co-NP. I also discuss extensions of Gurvits' NP-hardness result to strong NP-hardness of certain related problems. A major open question is whether the NP-contained formulation (QSEP) of the quantum separability problem is Karp-NP-complete; QSEP may be the first natural example of a problem that is Turing-NP-complete but not Karp-NP-complete. Finally, I survey all the proposed (deterministic) algorithms for the quantum separability problem, including the bounded search for symmetric extensions (via semidefinite programming), based on the recent quantum de Finetti theorem \cite{DPS02,DPS04,qphCKMR06}; and the entanglement-witness search (via interior-point algorithms and global optimization) \cite{ITCE04,IT06}. These two algorithms have the lowest complexity, with the latter being the best under advice of asymptotically optimal point-coverings of the sphere.

One-and-a-Half Quantum de Finetti Theorems

We prove a new kind of quantum de Finetti theorem for representations of the unitary group U(d). Consider a pure state that lies in the irreducible representation U_{mu+nu} for Young diagrams mu and nu. U_{mu+nu} is contained in the tensor product of U_mu and U_nu; let xi be the state obtained by tracing out U_nu. We show that xi is close to a convex combination of states Uv, where U is in U(d) and v is the highest weight vector in U_mu. When U_{mu+nu} is the symmetric representation, this yields the conventional quantum de Finetti theorem for symmetric states, and our method of proof gives near-optimal bounds for the approximation of xi by a convex combination of product states. For the class of symmetric Werner states, we give a second de Finetti-style theorem (our 'half' theorem); the de Finetti-approximation in this case takes a particularly simple form, involving only product states with a fixed spectrum. Our proof uses purely group theoretic methods, and makes a link with the shifted Schur functions. It also provides some useful examples, and gives some insight into the structure of the set of convex combinations of product states.

De Finetti theorem and Borel states in [0, 1]-valued algebraic logic

In this paper de Finetti’s (no-Dutch-Book) criterion for coherent probability assignments is extended to large classes of logics and their algebras. Given a set A of “events” and a closed set W ⊆ [ 0 , 1 ] A of “possible worlds” we show that a map s : A → [ 0 , 1 ] satisfies de Finetti’s criterion if, and only if, it has the form s ( a ) = ∫ W V ( a ) d μ ( V ) for some probability measure μ on W . Our results are applicable to all logics whose connectives are continuous operations on [ 0 , 1 ] , notably (i) every [ 0 , 1 ] -valued logic with finitely many truth-values, (ii) every logic whose conjunction is a continuous t-norm, and whose negation is ¬ x = 1 - x , possibly also equipped with its t-conorm and with some continuous implication, (iii) any extension of Łukasiewicz logic with constants or with a product-like connective. We also extend de Finetti’s criterion to the noncommutative underlying logic of GMV-algebras.

Cumulants in noncommutative probability theory IV. Noncrossing cumulants: De Finetti's theorem and [formula omitted]-inequalities

De Finetti's theorem states that any exchangeable sequence of classical random variables is conditionally i.i.d. with respect to some σ-algebra. In this paper we prove a “free” noncommutative analog of this theorem, namely we show that any noncrossing exchangeability system with a faithful state which satisfies a so called weak singleton condition can be embedded into an free product with amalgamation over a certain subalgebra such that the interchangeable algebras remain interchangeable with respect to the operator-valued expectation. Vanishing of crossing cumulants can be verified by checking a certain weak freeness condition and the weak singleton condition is satisfied e.g. when the state is tracial. The proof follows the classical proof of De Finetti's theorem, the main technical tool being a noncommutative L p -inequality for i.i.d. sums of centered noncommutative random variables in noncrossing exchangeability systems.

Exchangeable measures for subshifts

Let Ω be a Borel subset of S N where S is countable. A measure is called exchangeable on Ω, if it is supported on Ω and is invariant under every Borel automorphism of Ω which permutes at most finitely many coordinates. De-Finetti's theorem characterizes these measures when Ω = S N . We apply the ergodic theory of equivalence relations to study the case Ω ≠ S N , and obtain versions of this theorem when Ω is a countable state Markov shift, and when Ω is the collection of beta expansions of real numbers in [ 0 , 1 ] (a non-Markovian constraint).

Examples of Bosonic de Finetti States over Finite Dimensional Hilbert Spaces

According to the Quantum de Finetti Theorem, locally normal infinite particle states with Bose-Einstein symmetry can be represented as mixtures of infinite tensor powers of vector states. This note presents examples of infinite-particle states with Bose-Einstein symmetry that arise as limits of Gibbs ensembles on finite dimensional spaces, and displays their de Finetti representations. We consider Gibbs ensembles for systems of bosons in a finite dimensional setting and discover limits as the number of particles tends to infinity, provided the temperature is scaled in proportion to particle number.

Enumerative Induction and Lawlikeness

The paper is based on ranking theory, a theory of degrees of disbelief (and hence belief). On this basis, it explains enumerative induction, the confirmation of a law by its positive instances, which may, indeed, take various schemes. It gives a ranking theoretic explication of a possible law or a nomological hypothesis. It proves, then, that such schemes of enumerative induction uniquely correspond to mixtures of such nomological hypotheses. Thus, it shows both that de Finetti's probabilistic representation theorems may be transformed into an account of confirmation of possible laws and that enumerative induction is equivalent to such an account. The paper concludes with some remarks about the apriority of lawfulness, or the uniformity of nature.

Measurements of Entanglement and a Quantum de Finetti Theorem

The problem of defining a natural measure of entanglement of mixed states on tensor products is considered from the point of view of a quantum de Finetti theorem for Bosons.

Central Limit Theorems for Exchangeable Random Variables When Limits Are Scale Mixtures of Normals

Central limit theorems for exchangeable random variables are studied when limits are scale mixtures of normals. First, necessary and sufficient conditions are given under the asymptotic tail probability condition for the mixands: $$nP^\omega \left\{ {\left| {\xi _1 } \right| >\varepsilon b_n } \right\}\xrightarrow{P}0.$$ Second, when the weak limits have a particular form, i.e., the mixing measure comes directly from de Finetti's Theorem, necessary and sufficient conditions are given. Finally, some applications are discussed.

Unknown quantum states: The quantum de Finetti representation

We present an elementary proof of the quantum de Finetti representation theorem, a quantum analog of de Finetti’s classical theorem on exchangeable probability assignments. This contrasts with the original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanced mathematics and does not share the same potential for generalization. The classical de Finetti theorem provides an operational definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be degrees of belief instead of objective states of nature. The quantum de Finetti theorem, in a closely analogous fashion, deals with exchangeable density-operator assignments and provides an operational definition of the concept of an “unknown quantum state” in quantum-state tomography. This result is especially important for information-based interpretations of quantum mechanics, where quantum states, like probabilities, are taken to be states of knowledge rather than states of nature. We further demonstrate that the theorem fails for real Hilbert spaces and discuss the significance of this point.

Bayesian Representation of Stochastic Processes under Learning: de Finetti Revisited

A probability distribution governing the evolution of a stochastic process has infinitely many Bayesian representations of the form μ = ∫ Θ μ θ dλ(θ). Among these, a natural representation is one whose components (μ θ 's) are learnable (one can approximate μ θ by conditioning μ on observation of the process) and sufficient for prediction (μ θ 's predictions are not aided by conditioning on observation of the process). We show the existence and uniqueness of such a representation under a suitable asymptotic mixing condition on the process. This representation can be obtained by conditioning on the tail-field of the process, and any learnable representation that is sufficient for prediction is asymptotically like the tail-field representation. This result is related to the celebrated de Finetti theorem, but with exchangeability weakened to an asymptotic mixing condition, and with his conclusion of a decomposition into i.i.d. component distributions weakened to components that are learnable and sufficient for prediction.

Some properties of reduced density operators

Some questions concerning reduced density operators are reviewed. A new proof of the condition for Fermionic N-representability of one-particle density operators is given which is based on the functional analytic method of the dual cone. The application of a quantum analog of de Finetti's theorem to the interpretation of quantum theory is described. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 595–599, 1999

Ramsey Theory and Sequences of Random Variables

We consider probability spaces which contain a family {EA[ratio ]A⊆{1, 2, …, n}, [mid ]A[mid ]=k} of events indexed by the k-element subsets of {1, 2, …, n}. A pair (A, B) of k-element subsets of {1, 2, …, n} is called a shift pair if the largest k−1 elements of A coincide with the smallest k−1 elements of B. For a shift pair (A, B), Pr[AB¯] is the probability that event EA is true and EB is false. We investigate how large the minimum value of Pr[AB¯], taken over all shift pairs, can be. As n→∞, this value converges to a number λk, with ½−1/2k+2[les ]λk[les ] ½−1/4k+2. We show that λk is a strictly increasing function of k, with λ1=¼ and λ2=1/3.For k=1, our results have the following natural interpretation. If a fair coin is tossed repeatedly, and event Ei is true when the ith toss is heads, then for all i and j with i<j, Pr[EiĒj]=¼. Furthermore, as we show in this paper, for any ε>0, there is an n such that for any sequence E1, E2, …, En of events in an arbitrary probability space, there are indices i<j with Pr[EiĒj]<¼+ε. The results and techniques we develop in this research, together with further applications of Ramsey theory, are then used to show that the supremum of fractional dimensions of interval orders is exactly 4, answering a question of Brightwell and Scheinerman.Generalizing the ¼+ε result to random variables X1, X2, …, Xn with values in an m-element set, we obtain a finite version of de Finetti's theorem without the exchangeability hypothesis: for any fixed m, k and ε, every sufficiently long sequence of such random variables has a length-k subsequence at variation distance less than ε from an i.i.d. mix.

Effects of observed correlation structure among tumor types on experimentwise false positive rate in animal carcinogenicity studies.

Probabilities P1, P2, ..., Pn of n events E1, E2, ..., En can impose constraints on the probability of a further event E. deFinetti's fundamental theorem of probability characterizes the interval of coherent probability of E. Lad, Dickey, and Rahman (5) extended deFinetti's theorem in a few directions. In this work we will show some applications of these results to the calculation of experimentwise false positive error rate in multiple hypotheses testing procedure.

A Continuous Version of De Finetti's Theorem

A continuous version of De Finetti's theorem is proved in which the role of the homogeneous product states is played by the independent increment stationary processes on the real line. The proof is based on a conditional, finite De Finetti's theorem (i.e., a result involving only a finite number of random variables and exchangeable conditional expectations rather than exchangeable probabilities). Our technique of proof improves and simplifies a result of Freedman and includes a generalization of the quantum De Finetti's theorem as well as some more recent variants of it. The last section of the paper is an attempt to answer a question of Diaconis and Freedman.