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

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

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

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

A Quantum-Bayesian Route to Quantum-State Space

In the quantum-Bayesian approach to quantum foundations, a quantum state is viewed as an expression of an agent's personalist Bayesian degrees of belief, or probabilities, concerning the results of measurements. These probabilities obey the usual probability rules as required by Dutch-book coherence, but quantum mechanics imposes additional constraints upon them. In this paper, we explore the question of deriving the structure of quantum-state space from a set of assumptions in the spirit of quantum Bayesianism. The starting point is the representation of quantum states induced by a symmetric informationally complete measurement or SIC. In this representation, the Born rule takes the form of a particularly simple modification of the law of total probability. We show how to derive key features of quantum-state space from (i) the requirement that the Born rule arises as a simple modification of the law of total probability and (ii) a limited number of additional assumptions of a strong Bayesian flavor.Comment: 7 pages, 1 figure, to appear in Foundations of Physics; this is a condensation of the argument in arXiv:0906.2187v1 [quant-ph], with special attention paid to making all assumptions explici

Finite size mean-field models

We characterize the two-site marginals of exchangeable states of a system of quantum spins in terms of a simple positivity condition. This result is used in two applications. We first show that the distance between two-site marginals of permutation invariant states on N spins and exchangeable states is of order 1/N. The second application relates the mean ground state energy of a mean-field model of composite spins interacting through a product pair interaction with the mean ground state energies of the components.Comment: 20 page

Don't know, can't know: Embracing deeper uncertainties when analysing risks

This article is available open access through the publisher’s website at the link below. Copyright @ 2011 The Royal Society.Numerous types of uncertainty arise when using formal models in the analysis of risks. Uncertainty is best seen as a relation, allowing a clear separation of the object, source and ‘owner’ of the uncertainty, and we argue that all expressions of uncertainty are constructed from judgements based on possibly inadequate assumptions, and are therefore contingent. We consider a five-level structure for assessing and communicating uncertainties, distinguishing three within-model levels—event, parameter and model uncertainty—and two extra-model levels concerning acknowledged and unknown inadequacies in the modelling process, including possible disagreements about the framing of the problem. We consider the forms of expression of uncertainty within the five levels, providing numerous examples of the way in which inadequacies in understanding are handled, and examining criticisms of the attempts taken by the Intergovernmental Panel on Climate Change to separate the likelihood of events from the confidence in the science. Expressing our confidence in the adequacy of the modelling process requires an assessment of the quality of the underlying evidence, and we draw on a scale that is widely used within evidence-based medicine. We conclude that the contingent nature of risk-modelling needs to be explicitly acknowledged in advice given to policy-makers, and that unconditional expressions of uncertainty remain an aspiration

Anomalous scaling due to correlations: Limit theorems and self-similar processes

We derive theorems which outline explicit mechanisms by which anomalous scaling for the probability density function of the sum of many correlated random variables asymptotically prevails. The results characterize general anomalous scaling forms, justify their universal character, and specify universality domains in the spaces of joint probability density functions of the summand variables. These density functions are assumed to be invariant under arbitrary permutations of their arguments. Examples from the theory of critical phenomena are discussed. The novel notion of stability implied by the limit theorems also allows us to define sequences of random variables whose sum satisfies anomalous scaling for any finite number of summands. If regarded as developing in time, the stochastic processes described by these variables are non-Markovian generalizations of Gaussian processes with uncorrelated increments, and provide, e.g., explicit realizations of a recently proposed model of index evolution in finance.Comment: Through text revision. 15 pages, 3 figure