Quantum Foundations

The presumed irreversibility of quantum measurements (whatever they are) leads, in conventional approaches to quantum theory, to an asymmetry between state preparation and post-selection. Is it possible that a trajectory can be predicted from the former, yet not inferred from the latter? Especially in light of the exciting applications of non-unitary operations (i.e., postselection) in quantum information, it becomes timely to reconsider how much one can say about a post-selected subensemble. I will review the weak-measurement formalism of Aharonov, Vaidman et al., and discuss some applications and extensions. These will include a proposed experiment to study the duration of the tunneling process (a question controversial since the 1930s) and a recently completed experiment aiming to \'resolve\' Hardy\'s retrodiction paradox.

My favorite version of quantum mechanics is Bohmian mechanics, a theory about particle trajectories. What is so great about it is that it removes all the mystery from quantum mechanics. I will provide a Bohmian perspective on some issues about time, including time measurements (Why is there no time operator?), tunnelling times (How long did the particle stay inside the barrier?), and the problem of time in quantum gravity (How can it be that the wave function of the Wheeler-de Witt equation is time-independent?). I will particularly address the arrow of time, including the question whether quantum measurements are examples of fundamental time asymmetry (like PCT violations, as Roger Penrose has suggested) or merely of irreversibility (like thermodynamics), and the problem how to transfer Boltzmann\'s explanation of the arrow of time from classical to quantum mechanics.

A brief review of the Two State Vector Formalism (TSVF) will be presented. It will be argued that we need to consider also backwards evolving quantum state because information given by forwards evolving quantum states is not complete. Both past and future measurements are required for providing complete information about quantum systems. Peculiar properties of pre- and post-selected quantum systems which can be efficiently analyzed in the framework of the TSVF and which can be observed using weak measurements will be described. An example is a particle reaching a certain location without being on the path that leads to and from this location. An extension of the TSVF to multiple space-time points will be discussed.

Are time and space independently existing entities? Or is their existence secondary in that they are merely properties of other, more fundamental physical systems? The parameters of this enduring debate have shifted according to the physical theory in which it are set. In the 17th century, Newton\'s notions of Absolute Time and Space strongly favored the idea of independently existing times and spaces. Yet Leibniz famously plagued Newton by pointing to changes that Newton must suppose real even though they issued in no observable differences. Most recently, with the advent of general relativity and quantum theories of gravity that seek to incorporate it, the balance has shifted once again. Is the independence of space and time now finally revealed by the metric field of space and time absorbing the matter of the gravitational fields? Or has time and space has lost its independence from matter in so far as space and time have been absorbed into the matter of the gravitational field? The focus of my talk will be an intermediate episode of this debate that plays out in the context of special relativity. Lorentz noted that moving electrodynamical systems slow in time and shrink in space. The realist tradition explains this slowing and shrinking through the adaptation of matter fields to a real, independently existing Minkowski spacetime. A dissident constructive tradition has long felt that the reverse is the case. These spatio-temporal effects are best explained by the properties of matter theories, most notably, their Lorentz covariance. Harvey Brown has advocated a form of this latter constructivism in his Physical Relativity: Space- time Structure from a Dynamical Perspective. This debate between these two views has proven hard to resolve. That is largely because the notion of explanation is not sufficiently understood for us to adjudicate cleanly between competing claims of what explains what better. In my talk, I will review a new approach to the debate. Constructivists have tacitly assumed a technical result, that it is indeed possible to construct a Minkowski spacetime from Lorentz covariant matter theories. I will show that this is incorrect. This construction project can succeed only in so far as constructivists presume antecedently the basic tenets of the realist view of spacetime. Hence constructivism fails as an alternative to realism about spacetime.

I will examine a number of time-related issues arising in quantum theory, and in particular attempt to address the following basic questions from a quantum perspective: 1. What is a clock? 2. Why do uniformly moving clocks dilate? 3. What is the behaviour of accelerating clocks?

In 1898, Poincaré identified two fundamental issues in the theory of time: 1)What is the basis for saying that a second today is the same as a second tomorrow? 2) How can one define simultaneity at spatially separated points? Poincaré outlined the solution to the first problem { which amounts to a theory of duration { in his 1898 paper, and in 1905 he and Einstein simultaneously solved the second problem. Einstein\'s daring and elegant approach so gripped the imagination of theoreticians, especially after Minkowski\'s introduction of spacetime,that the definition of duration, and with it the theory of clocks, has received virtually no attention for over a century. This is a remarkable state of affairs and is a major cause of the conceptual confusion surrounding the problem of time in the canonical approach to the creation of a quantum theory of gravity. In my talk I shall develop Poincaré\'s outline into a potentially definitive theory of duration and clocks.

According to the second law of thermodynamics the entropy of a system cannot decrease by adiabatic state transformations. In quantum mechanics, the \'degree of entanglement\' of a state cannot increase under state transformations of a certain kind (local operations assisted by classical communication) In this talk I will explore the significance of the analogy between these two statements.

After using the complex Hilbert space formalism for quantum theory for so long, it is very easy to begin to take for granted features like projection operators and the projection postulate, the algebra of observables, symmetric transition probabilities, linear evolution, etc.... Over the past 50 years there have been many attempts to gain a better understanding of this formalism by reconstructing it from different kinds of (sometimes) physically motivated assumptions. By looking at how the above features are motivated and used in different reconstructions, it becomes clear just how special and restrictive many of them are. The question is then what a theory which does not have some of these features looks like. Another interesting question is whether there are any reasons to be suspicious of postulating them in reconstructions or when trying to generalize or apply the quantum formalism to untested situations.

Theoretical and experimental results on the Quantum Injected Optical Parametric Amplification (QI-OPA) of optical qubits in the high gain regime (g > 6) are reported. The entanglement of the related Schroedinger Cat-State (SCS) is demonstrated as well as the establishment of Phase-Covariant quantum cloning for a Macrostate consisting of about 106 particles. In addition, the violation of the CHSH inequality is has been realized experimentally. According to the original 1935 definition of the SCS, the overall apparatus establishes for the first time the nonlocal correlations between a microcopic spin (qubit) and a high J angular momentum i.e. a macroscopic multiparticle system close to the classical limit. Applications to Quantum Information will be discussed.

The fundamentally different localization concepts of QT, i.e. the Born-(Newton-Wigner) localization of (relativistic) QM as compared with the causal localization (modular localization) of QFT, lead to significant differences in the nature of local observables and affiliated states. This in turn results in a rather sharp distinction between a tensor-factorization and information-theoretic entanglement in QM on the one hand, and a more radical \"thermal entanglement” responsible for an area law for localization entropy. These surprising differences can be traced back to the very different nature of the localized operator algebras in QFT: they are all isomorphic (independent of the localization region) to one abstract \"monad\" (borrowing terminology from Leibniz) and the full reality of QFT (including its symmetries) is contained in the positioning of a finite rather small number (2 for chiral theories, 6 for d=1+3,...) within a joint Hilbert space. It is an important open question to what extend such positional characterizations (where the individual monads are void of any physical properties which reside fully in their relative placements) can be generalized to CST or QG.

Domains were introduced in computer science in the late 1960\'s by Dana Scott to provide a semantics for the lambda calculus (the lambda calculus is the basic prototype for a functional programming language i.e. ML). The study of domains with measurements was initiated in the speaker\'s thesis: a domain provides a qualitative view of information expressed in part by an \'information order\' and a measurement on a domain expresses a quantitative view of information with respect to the underlying qualitative aspect. The theory of domains and measurements was initially introduced to provide a first order model of computation, one in which a computation is viewed as a process that evolves in a space of informatic objects, where processes have informatic rates of change determined by the manner in which they manipulate information. There is a domain of binary channels with capacity as a measurement. There is a domain of finite probability distributions with entropy as a measurement. There is a domain of quantum mixed states with entropy as a measurement. There is a domain of spacetime intervals with global time as a measurement. In this setting, similarities between QM and GR emerge, but also some important differences. In a domain, if we write x <= y, then it means that x carries information about y, while x << y is a stronger relation that means x carries *essential* information about y. In GR, the domain theoretic relation << can be proven to be timelike causality. It possesses stronger mathematical properties than << does in QM. However, by an application of the maximum entropy principle, we can restrict the mixed states in consideration and this difference is removed: the domains of events and mixed states are both globally hyperbolic -- where globally hyperbolic is a purely order theoretic idea that just happens to coincide with the usual notion in the case of GR. Along the way, we will see domain theoretic ways of distinguishing between the Newtonian and relativistic notions of time, how to reconstruct the topology and geometry of spacetime in a purely order theoretic manner beginning from only a countable set, see that the Holevo capacity of a unital qubit channel is determined by the largest value of its informatic derivative and have reason to wonder if distance can be defined as the amount of information (capacity) that can be transmitted between two points.