Negative Quasi-Probability as a Resource for Quantum Computation

A central problem in quantum information is to determine the minimal physical resources that are required for quantum computational speedup and, in particular, for fault-tolerant quantum computation. We establish a remarkable connection between the potential for quantum speed-up and the onset of negative values in a distinguished quasi-probability representation, a discrete analog of the Wigner function for quantum systems of odd dimension. This connection allows us to resolve an open question on the existence of bound states for magic-state distillation: we prove that there exist mixed states outside the convex hull of stabilizer states that cannot be distilled to non-stabilizer target states using stabilizer operations. We also provide an efficient simulation protocol for Clifford circuits that extends to a large class of mixed states, including bound universal states.Comment: 15 pages v4: This is a major revision. In particular, we have added a new section detailing an explicit extension of the Gottesman-Knill simulation protocol to deal with positively represented states and measurement (even when these are non-stabilizer). This paper also includes significant elaboration on the two main results of the previous versio

Quasi-probability representations of quantum theory with applications to quantum information science

This article comprises a review of both the quasi-probability representations of infinite-dimensional quantum theory (including the Wigner function) and the more recently defined quasi-probability representations of finite-dimensional quantum theory. We focus on both the characteristics and applications of these representations with an emphasis toward quantum information theory. We discuss the recently proposed unification of the set of possible quasi-probability representations via frame theory and then discuss the practical relevance of negativity in such representations as a criteria for quantumness.Comment: v3: typos fixed, references adde

Robust Online Hamiltonian Learning

In this work we combine two distinct machine learning methodologies, sequential Monte Carlo and Bayesian experimental design, and apply them to the problem of inferring the dynamical parameters of a quantum system. We design the algorithm with practicality in mind by including parameters that control trade-offs between the requirements on computational and experimental resources. The algorithm can be implemented online (during experimental data collection), avoiding the need for storage and post-processing. Most importantly, our algorithm is capable of learning Hamiltonian parameters even when the parameters change from experiment-to-experiment, and also when additional noise processes are present and unknown. The algorithm also numerically estimates the Cramer-Rao lower bound, certifying its own performance.Comment: 24 pages, 12 figures; to appear in New Journal of Physic

Quantum Fourier transform, Heisenberg groups and quasiprobability distributions

This paper aims to explore the inherent connection among Heisenberg groups, quantum Fourier transform and (quasiprobability) distribution functions. Distribution functions for continuous and finite quantum systems are examined first as a semiclassical approach to quantum probability distribution. This leads to studying certain functionals of a pair of "conjugate" observables, connected via the quantum Fourier transform. The Heisenberg groups emerge naturally from this study and we take a rapid look at their representations. The quantum Fourier transform appears as the intertwining operator of two equivalent representation arising out of an automorphism of the group. Distribution functions correspond to certain distinguished sets in the group algebra. The marginal properties of a particular class of distribution functions (Wigner distributions) arise from a class of automorphisms of the group algebra of the Heisenberg group. We then study the reconstruction of Wigner function from the marginal distributions via inverse Radon transform giving explicit formulas. We consider applications of our approach to quantum information processing and quantum process tomography.Comment: 39 page

Structural and functional aspects of social support as predictors of mental and physical health trajectories: Whitehall II cohort study

BACKGROUND: Social support is associated with better health. However, only a limited number of studies have examined the association of social support with health from the adult life course perspective and whether this association is bidirectional. METHODS: Participants (n=6797; 30% women; age range from 40 to 77 years) who were followed from 1989 (phase 2) to 2006 (phase 8) were selected from the ongoing Whitehall II Study. Structural and functional social support was measured at follow-up phases 2, 5 and 7. Mental and physical health was measured at five consecutive follow-up phases (3–8). RESULTS: Social support predicted better mental health, and certain functional aspects of social support, such as higher practical support and higher levels of negative aspects in social relationships, predicted poorer physical health. The association between negative aspects of close relationships and physical health was found to strengthen over the adult life course. In women, the association between marital status and mental health weakened until the age of approximately 60 years. Better mental and physical health was associated with higher future social support. CONCLUSIONS: The strength of the association between social support and health may vary over the adult life course. The association with health seems to be bidirectional

Job insecurity and risk of coronary heart disease : Mediation analyses of health behaviors, sleep problems, physiological and psychological factors

Job insecurity has been linked to increased risk of coronary heart disease (CHD), but underlying mechanisms remain uncertain. Our aim was to assess the extent to which this association is mediated through life style, physiological, or psychological factors. A total of 3917 men and women free from CHD provided data on job insecurity in the Whitehall II cohort study in 1997-1999. The association between job insecurity and CHD was decomposed into a direct and indirect effect mediated through unhealthy behaviors (smoking, high alcohol consumption, physical inactivity), sleep disturbances, 'allostatic load', or psychological distress. The counterfactual analyses on psychological distress indicated a marginally significant association between job insecurity and incident CHD (hazard ratio (HR) 1.32; 95 % confidence interval (CI) 1.00-1.75). This association was decomposed into a direct (HR 1.22, 95 %CI 0.92-1.63) and indirect association (1.08, 95 %CI 1.01-1.15), suggesting that about 30 % of the total relationship was mediated by psychological distress. No mediation was indicated via health behaviors, sleep disturbances, or allostatic load, although job insecurity was related to disturbed sleep and C-reactive protein, which, in turn were associated with CHD. In conclusion, our results suggest that psychological distress may play a role in the relation between job insecurity and CHD.Peer reviewe

Framed Hilbert space: hanging the quasi-probability pictures of quantum theory

Building on earlier work, we further develop a formalism based on the mathematical theory of frames that defines a set of possible phase-space or quasi-probability representations of finite-dimensional quantum systems. We prove that an alternate approach to defining a set of quasi-probability representations, based on a more natural generalization of a classical representation, is equivalent to our earlier approach based on frames, and therefore is also subject to our no-go theorem for a non-negative representation. Furthermore, we clarify the relationship between the contextuality of quantum theory and the necessity of negativity in quasi-probability representations and discuss their relevance as criteria for non-classicality. We also provide a comprehensive overview of known quasi-probability representations and their expression within the frame formalism.Comment: 46 pages, 1 table, contains a review of finite dimensional quasi-probability function

Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations

Several finite dimensional quasi-probability representations of quantum states have been proposed to study various problems in quantum information theory and quantum foundations. These representations are often defined only on restricted dimensions and their physical significance in contexts such as drawing quantum-classical comparisons is limited by the non-uniqueness of the particular representation. Here we show how the mathematical theory of frames provides a unified formalism which accommodates all known quasi-probability representations of finite dimensional quantum systems. Moreover, we show that any quasi-probability representation satisfying two reasonable properties is equivalent to a frame representation and then prove that any such representation of quantum mechanics must exhibit either negativity or a deformed probability calculus.Comment: 13 pages, published versio

The Alignment Between 3-D Data and Articulated Shapes with Bending Surfaces

International audienceIn this paper we address the problem of aligning 3-D data with articulated shapes. This problem resides at the core of many motion tracking methods with applications in human motion capture, action recognition, medical-image analysis, etc. We describe an articulated and bending surface representation well suited for this task as well as a method which aligns (or registers) such a surface to 3-D data. Articulated objects, e.g., humans and animals, are covered with clothes and skin which may be seen as textured surfaces. These surfaces are both articulated and deformable and one realistic way to model them is to assume that they bend in the neighborhood of the shape's joints. We will introduce a surface-bending model as a function of the articulated-motion parameters. This combined articulated-motion and surface-bending model better predicts the observed phenomena in the data and therefore is well suited for surface registration. Given a set of sparse 3-D data (gathered with a stereo camera pair) and a textured, articulated, and bending surface, we describe a register-and-fit method that proceeds as follows. First, the data-to-surface registration problem is formalized as a classifier and is carried out using an EM algorithm. Second, the data-to-surface fitting problem is carried out by minimizing the distance from the registered data points to the surface over the joint variables. In order to illustrate the method we applied it to the problem of hand tracking. A hand model with 27 degrees of freedom is successfully registered and fitted to a sequence of 3-D data points gathered with a stereo camera pair

Smallest disentangling state spaces for general entangled bipartite quantum states

PACS numbers: 03.67.-a, 03.65.-w, 03.65.Ta, 03.65.Ud.Entangled quantum states can be given a separable decomposition if we relax the restriction that the local operators be quantum states. Motivated by the construction of classical simulations and local hidden variable models, we construct `smallest' local sets of operators that achieve this. In other words, given an arbitrary bipartite quantum state we construct convex sets of local operators that allow for a separable decomposition, but that cannot be made smaller while continuing to do so. We then consider two further variants of the problem where the local state spaces are required to contain the local quantum states, and obtain solutions for a variety of cases including a region of pure states around the maximally entangled state. The methods involve calculating certain forms of cross norm. Two of the variants of the problem have a strong relationship to theorems on ensemble decompositions of positive operators, and our results thereby give those theorems an added interpretation. The results generalise those obtained in our previous work on this topic [New J. Phys. 17, 093047 (2015)].EP/K022512/1/Engineering and Physical Sciences Research Counci