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

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 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

Correlated Binomial Models and Correlation Structures

We discuss a general method to construct correlated binomial distributions by imposing several consistent relations on the joint probability function. We obtain self-consistency relations for the conditional correlations and conditional probabilities. The beta-binomial distribution is derived by a strong symmetric assumption on the conditional correlations. Our derivation clarifies the 'correlation' structure of the beta-binomial distribution. It is also possible to study the correlation structures of other probability distributions of exchangeable (homogeneous) correlated Bernoulli random variables. We study some distribution functions and discuss their behaviors in terms of their correlation structures.Comment: 12 pages, 7 figure

Curie-Weiss model of the quantum measurement process

A hamiltonian model is solved, which satisfies all requirements for a realistic ideal quantum measurement. The system S is a spin-\half, whose $z$-component is measured through coupling with an apparatus A=M+B, consisting of a magnet \RM formed by a set of $N\gg 1$ spins with quartic infinite-range Ising interactions, and a phonon bath \RB at temperature $T$. Initially A is in a metastable paramagnetic phase. The process involves several time-scales. Without being much affected, A first acts on S, whose state collapses in a very brief time. The mechanism differs from the usual decoherence. Soon after its irreversibility is achieved. Finally the field induced by S on M, which may take two opposite values with probabilities given by Born's rule, drives A into its up or down ferromagnetic phase. The overall final state involves the expected correlations between the result registered in M and the state of S. The measurement is thus accounted for by standard quantum statistical mechanics and its specific features arise from the macroscopic size of the apparatus.Comment: 5 pages Revte

Financial instability from local market measures

We study the emergence of instabilities in a stylized model of a financial market, when different market actors calculate prices according to different (local) market measures. We derive typical properties for ensembles of large random markets using techniques borrowed from statistical mechanics of disordered systems. We show that, depending on the number of financial instruments available and on the heterogeneity of local measures, the market moves from an arbitrage-free phase to an unstable one, where the complexity of the market - as measured by the diversity of financial instruments - increases, and arbitrage opportunities arise. A sharp transition separates the two phases. Focusing on two different classes of local measures inspired by real markets strategies, we are able to analytically compute the critical lines, corroborating our findings with numerical simulations.Comment: 17 pages, 4 figure

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

A quantum framework for likelihood ratios

The ability to calculate precise likelihood ratios is fundamental to many STEM areas, such as decision-making theory, biomedical science, and engineering. However, there is no assumption-free statistical methodology to achieve this. For instance, in the absence of data relating to covariate overlap, the widely used Bayes’ theorem either defaults to the marginal probability driven “naive Bayes’ classifier”, or requires the use of compensatory expectation-maximization techniques. Equally, the use of alternative statistical approaches, such as multivariate logistic regression, may be confounded by other axiomatic conditions, e.g., low levels of co-linearity. This article takes an information-theoretic approach in developing a new statistical formula for the calculation of likelihood ratios based on the principles of quantum entanglement. In doing so, it is argued that this quantum approach demonstrates: that the likelihood ratio is a real quality of statistical systems; that the naive Bayes’ classifier is a special case of a more general quantum mechanical expression; and that only a quantum mechanical approach can overcome the axiomatic limitations of classical statistics

Collective Autoionization in Multiply-Excited Systems: A novel ionization process observed in Helium Nanodroplets

Free electron lasers (FELs) offer the unprecedented capability to study reaction dynamics and image the structure of complex systems. When multiple photons are absorbed in complex systems, a plasma-like state is formed where many atoms are ionized on a femtosecond timescale. If multiphoton absorption is resonantly-enhanced, the system becomes electronically-excited prior to plasma formation, with subsequent decay paths which have been scarcely investigated to date. Here, we show using helium nanodroplets as an example that these systems can decay by a new type of process, named collective autoionization. In addition, we show that this process is surprisingly efficient, leading to ion abundances much greater than that of direct single-photon ionization. This novel collective ionization process is expected to be important in many other complex systems, e.g. macromolecules and nanoparticles, exposed to high intensity radiation fields

From Composite Indicators to Partial Orders: Evaluating Socio-Economic Phenomena Through Ordinal Data

In this paper we present a new methodology for the statistical evaluation of ordinal socio-economic phenomena, with the aim of overcoming the issues of the classical aggregative approach based on composite indicators. The proposed methodology employs a benchmark approach to evaluation and relies on partially ordered set (poset) theory, a branch of discrete mathematics providing tools for dealing with multidimensional systems of ordinal data. Using poset theory and the related Hasse diagram technique, evaluation scores can be computed without performing any variable aggregation into composite indicators. This way, ordinal scores need not be turned into numerical values, as often done in evaluation studies, inconsistently with the real nature of the phenomena at hand. We also face the problem of \u201cweighting\u201d evaluation dimensions, to account for their different relevance, and show how this can be handled in pure ordinal terms. A specific focus is devoted to the binary variable case, where the methodology can be specialized in a very effective way. Although the paper is mainly methodological, all of the basic concepts are illustrated through real examples pertaining to material deprivation