1,987 research outputs found
Exchangeability and sets of desirable gambles
Sets of desirable gambles constitute a quite general type of uncertainty
model with an interesting geometrical interpretation. We give a general
discussion of such models and their rationality criteria. We study
exchangeability assessments for them, and prove counterparts of de Finetti's
finite and infinite representation theorems. We show that the finite
representation in terms of count vectors has a very nice geometrical
interpretation, and that the representation in terms of frequency vectors is
tied up with multivariate Bernstein (basis) polynomials. We also lay bare the
relationships between the representations of updated exchangeable models, and
discuss conservative inference (natural extension) under exchangeability and
the extension of exchangeable sequences.Comment: 40 page
Accept & Reject Statement-Based Uncertainty Models
We develop a framework for modelling and reasoning with uncertainty based on
accept and reject statements about gambles. It generalises the frameworks found
in the literature based on statements of acceptability, desirability, or
favourability and clarifies their relative position. Next to the
statement-based formulation, we also provide a translation in terms of
preference relations, discuss---as a bridge to existing frameworks---a number
of simplified variants, and show the relationship with prevision-based
uncertainty models. We furthermore provide an application to modelling symmetry
judgements.Comment: 35 pages, 17 figure
Infinite Divisibility in Euclidean Quantum Mechanics
In simple -- but selected -- quantum systems, the probability distribution
determined by the ground state wave function is infinitely divisible. Like all
simple quantum systems, the Euclidean temporal extension leads to a system that
involves a stochastic variable and which can be characterized by a probability
distribution on continuous paths. The restriction of the latter distribution to
sharp time expectations recovers the infinitely divisible behavior of the
ground state probability distribution, and the question is raised whether or
not the temporally extended probability distribution retains the property of
being infinitely divisible. A similar question extended to a quantum field
theory relates to whether or not such systems would have nontrivial scattering
behavior.Comment: 17 pages, no figure
A quantum de Finetti theorem in phase space representation
The quantum versions of de Finetti's theorem derived so far express the
convergence of n-partite symmetric states, i.e., states that are invariant
under permutations of their n parties, towards probabilistic mixtures of
independent and identically distributed (i.i.d.) states. Unfortunately, these
theorems only hold in finite-dimensional Hilbert spaces, and their direct
generalization to infinite-dimensional Hilbert spaces is known to fail. Here,
we address this problem by considering invariance under orthogonal
transformations in phase space instead of permutations in state space, which
leads to a new type of quantum de Finetti's theorem that is particularly
relevant to continuous-variable systems. Specifically, an n-mode bosonic state
that is invariant with respect to this continuous symmetry in phase space is
proven to converge towards a probabilistic mixture of i.i.d. Gaussian states
(actually, n identical thermal states).Comment: 5 page
A de Finetti representation for finite symmetric quantum states
Consider a symmetric quantum state on an n-fold product space, that is, the
state is invariant under permutations of the n subsystems. We show that,
conditioned on the outcomes of an informationally complete measurement applied
to a number of subsystems, the state in the remaining subsystems is close to
having product form. This immediately generalizes the so-called de Finetti
representation to the case of finite symmetric quantum states.Comment: 22 pages, LaTe
Additivity of the Renyi entropy of order 2 for positive-partial-transpose-inducing channels
We prove that the minimal Renyi entropy of order 2 (RE2) output of a
positive-partial-transpose(PPT)-inducing channel joint to an arbitrary other
channel is equal to the sum of the minimal RE2 output of the individual
channels. PPT-inducing channels are channels with a Choi matrix which is bound
entangled or separable. The techniques used can be easily recycled to prove
additivity for some non-PPT-inducing channels such as the depolarizing and
transpose depolarizing channels, though not all known additive channels. We
explicitly make the calculations for generalized Werner-Holevo channels as an
example of both the scope and limitations of our techniques.Comment: 4 page
Existence of equilibria in countable games: an algebraic approach
Although mixed extensions of finite games always admit equilibria, this is
not the case for countable games, the best-known example being Wald's
pick-the-larger-integer game. Several authors have provided conditions for the
existence of equilibria in infinite games. These conditions are typically of
topological nature and are rarely applicable to countable games. Here we
establish an existence result for the equilibrium of countable games when the
strategy sets are a countable group and the payoffs are functions of the group
operation. In order to obtain the existence of equilibria, finitely additive
mixed strategies have to be allowed. This creates a problem of selection of a
product measure of mixed strategies. We propose a family of such selections and
prove existence of an equilibrium that does not depend on the selection. As a
byproduct we show that if finitely additive mixed strategies are allowed, then
Wald's game admits an equilibrium. We also prove existence of equilibria for
nontrivial extensions of matching-pennies and rock-scissors-paper. Finally we
extend the main results to uncountable games
Imprecise Markov chains and their limit behaviour
When the initial and transition probabilities of a finite Markov chain in
discrete time are not well known, we should perform a sensitivity analysis.
This can be done by considering as basic uncertainty models the so-called
credal sets that these probabilities are known or believed to belong to, and by
allowing the probabilities to vary over such sets. This leads to the definition
of an imprecise Markov chain. We show that the time evolution of such a system
can be studied very efficiently using so-called lower and upper expectations,
which are equivalent mathematical representations of credal sets. We also study
how the inferred credal set about the state at time n evolves as n goes to
infinity: under quite unrestrictive conditions, it converges to a uniquely
invariant credal set, regardless of the credal set given for the initial state.
This leads to a non-trivial generalisation of the classical Perron-Frobenius
Theorem to imprecise Markov chains.Comment: v1: 28 pages, 8 figures; v2: 31 pages, 9 figures, major revision
after review: added, modified, and removed material (no results dropped,
results added), moved proofs to an appendi
Computable de Finetti measures
We prove a computable version of de Finetti's theorem on exchangeable
sequences of real random variables. As a consequence, exchangeable stochastic
processes expressed in probabilistic functional programming languages can be
automatically rewritten as procedures that do not modify non-local state. Along
the way, we prove that a distribution on the unit interval is computable if and
only if its moments are uniformly computable.Comment: 32 pages. Final journal version; expanded somewhat, with minor
corrections. To appear in Annals of Pure and Applied Logic. Extended abstract
appeared in Proceedings of CiE '09, LNCS 5635, pp. 218-23
A de Finetti representation theorem for infinite dimensional quantum systems and applications to quantum cryptography
According to the quantum de Finetti theorem, if the state of an N-partite
system is invariant under permutations of the subsystems then it can be
approximated by a state where almost all subsystems are identical copies of
each other, provided N is sufficiently large compared to the dimension of the
subsystems. The de Finetti theorem has various applications in physics and
information theory, where it is for instance used to prove the security of
quantum cryptographic schemes. Here, we extend de Finetti's theorem, showing
that the approximation also holds for infinite dimensional systems, as long as
the state satisfies certain experimentally verifiable conditions. This is
relevant for applications such as quantum key distribution (QKD), where it is
often hard - or even impossible - to bound the dimension of the information
carriers (which may be corrupted by an adversary). In particular, our result
can be applied to prove the security of QKD based on weak coherent states or
Gaussian states against general attacks.Comment: 11 pages, LaTe
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