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The Minimal Modal Interpretation of Quantum Theory
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David Kagan University of Massachusetts Dartmouth
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Jacob Barandes Harvard University
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Non local weak measurements
Aharon Brodutch Institute for Quantum Computing (IQC)
Space-Time Circuit-to-Hamiltonian construction and Its Applications
Barbara Terhal Delft University of Technology
Inferring causal structure: a quantum advantage
Katja Ried Universität Innsbruck
Physics, Logic and Mathematics of Time
Louis Kauffman University of Illinois at Chicago
Quantum thermalization and many-body Anderson localization
David Huse Princeton University
Categories of Convex Sets and C*-Algebras
Robert Furber Radboud Universiteit Nijmegen
A non-commuting stabilizer formalism
Xiaotong Ni Max Planck Institute for Gravitational Physics - Albert Einstein Institute (AEI)
Beyond local causality: causation and correlation after Bell
Matthew Pusey University of York
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Disturbance in weak measurements and the difference between quantum and classical weak values
The role of measurement induced disturbance in weak measurements is of central importance for the interpretation of the weak value. Uncontrolled disturbance can interfere with the postselection process and make the weak value dependent on the details of the measurement process. Here we develop the concept of a generalized weak measurement for classical and quantum mechanics. The two cases appear remarkably similar, but we point out some important differences. A priori it is not clear what the correct notion of disturbance should be in the context of weak measurements. We consider three different notions and get three different results: (1) For a `strong' definition of disturbance, we find that weak measurements are disturbing. (2) For a weaker definition we find that a general class of weak measurements are non-disturbing, but that one gets weak values which depend on the measurement process. (3) Finally, with respect to an operational definition of the `degree of disturbance', we find that the AAV weak measurements are the least disturbing, but that the disturbance is still non-zero. -
The Minimal Modal Interpretation of Quantum Theory
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David Kagan University of Massachusetts Dartmouth
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Jacob Barandes Harvard University
A persistent mystery of quantum theory is whether it admits an interpretation that is realist, self-consistent, model-independent, and unextravagant in the sense of featuring neither multiple worlds nor pilot waves. In this talk, I will present a new interpretation of quantum theory -- called the minimal modal interpretation (MMI) -- that aims to meet these conditions while also hewing closely to the basic structure of the theory in its widely accepted form. The MMI asserts that quantum systems -- whether closed or open -- have actual states that evolve along kinematical trajectories through their state spaces, and that those trajectories are governed by specific (if approximate) dynamical rules determined by a general new class of conditional probabilities, and in a manner that differs significantly from the de Broglie-Bohm formulation. The MMI is axiomatically parsimonious, leaves the usual dynamical content of quantum theory essentially intact, and includes only metaphysical entities that are either already a standard part of quantum theory or that have counterparts in classical physics. I will also address a number of important issues and implicit assumptions in the foundations community that I believe merit reconsideration and re-evaluation going forward. -
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Non local weak measurements
Aharon Brodutch Institute for Quantum Computing (IQC)
Weak measurement is increasingly acknowledged as an important theoretical and experimental tool. Weak values- the results of weak measurements- are often used to understand seemingly paradoxical quantum behavior. Until now however, it was not known how to perform a weak non-local measurement of a general operator. Such a procedure is necessary if we are to take the associated `weak values' seriously as a physical quantity. We propose a novel scheme for performing non-local weak measurement which is based on the principle of quantum erasure. This method can be used for a large class of observables including those related to Hardy's paradox. -
Space-Time Circuit-to-Hamiltonian construction and Its Applications
Barbara Terhal Delft University of Technology
The circuit-to-Hamiltonian construction translates a dynamics (a quantum circuit and its output) into statics (the groundstate of a circuit Hamiltonian) by explicitly defining a quantum register for a clock. The standard Feynman-Kitaev construction uses one global clock for all qubits while we consider a different construction in which a clock is assigned to each point in space where a qubit of the quantum circuit resides. We show how one can apply this construction to one-dimensional quantum circuits for which the circuit Hamiltonian realizes the dynamics of a vibrating string. We discuss how the construction can be used (1) in quantum complexity theory to obtain new and stronger results in QMA and (2) how one can realize, based on this construction, universal quantum adiabatic computation and a universal quantum walk using a 2D interacting particle Hamiltonian. See http://arxiv.org/abs/1311.6101 -
Inferring causal structure: a quantum advantage
Katja Ried Universität Innsbruck
A fundamental question in trying to understand the world -- be it classical or quantum -- is why things happen. We seek a causal account of events, and merely noting correlations between them does not provide a satisfactory answer. Classical statistics provides a better alternative: the framework of causal models proved itself a powerful tool for studying causal relations in a range of disciplines. We aim to adapt this formalism to allow for quantum variables and in the process discover a new perspective on how causality is different in the quantum world. Causal inference is a central task in the context of causal models: given observed statistics over a set of variables, one aims to infer how they are causally related. Yet in the seemingly simple case of just two classical variables, this is impossible (unless one makes additional assumptions). I will show how the analogous task for quantum variables can be solved. This quantum advantage is reminiscent of the advantages that quantum mechanics offers in computing and communication, and may lead to similarly rich insights. Our scheme is corroborated by data obtained in collaboration with Kevin Resch's experimental group. Time permitting, I will also address other applications of the quantum causal models. arXiv:1406.5036 
Local quanta, unitary inequivalence, and vacuum entanglement
In this work we develop a formalism for describing localised quanta for a real-valued Klein-Gordon field in a one-dimensional box [0, R]. We quantise the field using non-stationary local modes which, at some arbitrarily chosen initial time, are completely localised within the left or the right side of the box. In this concrete set-up we directly face the problems inherent to a notion of local field excitations, usually thought of as elementary particles. Specifically, by computing the Bogoliubov coefficients relating local and standard (global) quantizations, we show that the local quantisation yields a Fock space F^L which is unitarily inequivalent to the standard one F^G. In spite of this, we find that the local creators and annihilators remain well defined in the global Fock space F^G, and so do the local number operators associated to the left and right partitions of the box. We end up with a useful mathematical toolbox to analyse and characterise local features of quantum states in F^G . Specifically, an analysis of the global vacuum state |0_G> ∈ F^G in terms of local number operators shows, as expected, the existence of entanglement between the left and right regions of the box. The local vacuum |0_L> ∈ F^L , on the contrary, has a very different character. It is neither cyclic nor separating and displays no entanglement. Further analysis shows that the global vacuum also exhibits a distribution of local excitations reminiscent, in some respects, of a thermal bath. We discuss how the mathematical tools developed herein may open new ways for the analysis of fundamental problems in local quantum field theory.-
Physics, Logic and Mathematics of Time
Louis Kauffman University of Illinois at Chicago
Consider discrete physics with a minimal time step taken to be
tau. A time series of positions q,q',q'', ... has two classical
observables: position (q) and velocity (q'-q)/tau. They do not commute,
for observing position does not force the clock to tick, but observing
velocity does force the clock to tick. Thus if VQ denotes first observe
position, then observe velocity and QV denotes first observe velocity,
then observe position, we have
VQ: (q'-q)q/tau
QV: q'(q'-q)/tau
(since after one tick the position has moved from q to q').
Thus [Q,V]= QV - VQ = (q'-q)^2/tau. If we consider the equation
[Q,V] = k (a constant), then k = (q'-q))^2/tau and this is recognizably
the diffusion constant that arises in a process of Brownian motion.
Thus, starting with the simplest assumptions for discrete physics, we are
lead to recognizable physics. We take this point of view and follow it
in both physical and mathematical directions. A first mathematical
direction is to see how i, the square root of negative unity, is related
to the simplest time series: ..., -1,+1,-1,+1,... and making the
above analysis of time series more algebraic leads to the following
interpetation for i. Let e=[-1,+1] and e'=[+1,-1] denote, as ordered
pairs, two phase-shifted versions of the alternating series above.
Define an operator b such that eb = be' and b^2 = 1. Regard b as a time
shifting operator. The operator b shifts the alternating series by one
half its period. Regard e' = -e and ee' = [-1.-1] = -1 (combining term by
term). Then let i = eb. We have ii = (eb)(eb) = ebeb = ee'bb = -1. Thus ii = -1
through the definition of i as eb, a temporally sensitive entity that
shifts it phase in the course of interacting with (a copy of) itself.
By going to i as a discrete dynamical system, we can come back to the
general features of discrete dynamical systems and look in a new way at
the role of i in quantum mechanics. Note that the i we have constructed is
already part of a simple Clifford algebra generated by e and b with
ee = bb = 1 and eb + be = 0. We will discuss other mathematical physical
structures such as the Schrodinger equation, the Dirac equation and the
relationship of a simple logical operator (generalizing negation) with
Majorana Fermions. -
Quantum thermalization and many-body Anderson localization
David Huse Princeton University
Progress in physics and quantum information science motivates much recent study of the behavior of strongly-interacting many-body quantum systems fully isolated from their environment, and thus undergoing unitary time evolution. What does it mean for such a system to go to thermal equilibrium? I will explain the Eigenstate Thermalization Hypothesis (ETH), which posits that each individual exact eigenstate of the system's Hamiltonian is at thermal equilibrium, and which appears to be true for most (but not all) quantum many-body systems. Prominent among the systems that do not obey this hypothesis are quantum systems that are many-body Anderson localized and thus do not constitute a reservoir that can thermalize itself. When the ETH is true, one can do standard statistical mechanics using the `single-eigenstate ensembles', which are the limit of the microcanonical ensemble where the `energy window' contains only a single many-body quantum state. These eigenstate ensembles are more powerful than the traditional statistical mechanical ensembles, in that they can also "see" the quantum phase transition in to the localized phase, as well as a rich new world of phases and phase transitions within the localized phase. -
Categories of Convex Sets and C*-Algebras
Robert Furber Radboud Universiteit Nijmegen
The start of the talk will be an outline how the ordinary notions of quantum theory translate into the category of C*-algebras, where there are several possible choices of morphisms. The second half will relate this to a category of convex sets used as state spaces. Alfsen and Shultz have characterized the convex sets arising from state spaces C*-algebras and this result can be applied to get a categorical equivalence between C*-algebras and state spaces of C*-algebras which is a generalization of the equivalence between the Schroedinger and Heisenberg pictures. -
A non-commuting stabilizer formalism
Xiaotong Ni Max Planck Institute for Gravitational Physics - Albert Einstein Institute (AEI)
We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group {\alpha I, X,S}, where \alpha=e^{i\pi/4} and S=diag(1,i). We provide techniques to efficiently compute various properties, related to e.g. bipartite entanglement, expectation values of local observables, preparation by means of quantum circuits, parent Hamiltonians etc. We also highlight significant differences compared to the Pauli stabilizer formalism. In particular we give examples of states in our formalism which cannot arise in the Pauli stabilizer formalism, such as topological models that supports non-abelian anyons. This is a joint work with O. Buerschaper and M. van den Nest. -
Beyond local causality: causation and correlation after Bell
Matthew Pusey University of York
There is now a remarkable mathematical theory of causation. But applying this theory to a Bell scenario implies the Bell inequalities, which are violated in experiment. We alleviate this tension by translating the basic definitions of the theory into the framework of generalised probabilistic theories. We find that a surprising number of results carry over: the d-separation criterion for conditional independence (the no-signalling principle on steroids), and even certain quantitative limits on correlations. Finally, we begin a classification of the causal structures, such as the Bell scenarios, that are "interesting" from this perspective. Joint work with Joe Henson and Raymond Lal. 
Incompatibility of observables in quantum theory and other probabilistic theories
We introduce a new way of quantifying the degrees of incompatibility of two observables in a probabilistic physical theory and, based on this, a global measure of the degree of incompatibility inherent in such theories. This opens up a flexible way of comparing probabilistic theories with respect to the nonclassical feature of incompatibility. We show that quantum theory contains observables that are as incompatible as any probabilistic physical theory can have. In particular, we prove that two of the most common pairs of complementary observables (position and momentum; number and phase) are maximally incompatible. However, if one adopts a more refined measure of the degree of incompatibility, for instance, by restricting the comparison to binary observables, it turns out that there are probabilistic theories whose inherent degree of incompatibility is greater than that of quantum theory. Finally, we analyze the noise tolerance of the incompatibility of a pair of observables in a CHSH-Bell experiment.