Quantum Physics

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.

A quantum channel models a physical process in which noise is added to a quantum system via interaction with its environment. Protecting quantum systems from such noise can be viewed as an extension of the classical communication problem introduced by Shannon sixty years ago. A fundamental quantity of interest is the quantum capacity of a given channel, which measures the amount of quantum information which can be protected, in the limit of many transmissions over the channel. In this talk, I will show that certain pairs of channels, each with a capacity of zero, can have a strictly positive capacity when used together, implying that the quantum capacity does not completely characterize a channel\'s ability to transmit quantum information. As a corollary, I will show that a commonly used lower bound on the quantum capacity - the coherent information, or hashing bound - is an overly pessimistic benchmark against which to measure the performance of quantum error correction because the gap between this bound and the capacity can be arbitrarily large.

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.

I will report on efforts to implement a new method for simulating concatenated quantum error correction, where many levels of concatenation are simulated together explicitly. That is, the approach involves a Monte Carlo simulation of a noisy circuit involving many thousands of qubits, rather than tens of qubits previously. The new approach allows the threshold and resource usage of concatenated quantum error correction to be determined more accurately than before. Also, the approach makes it possible to better study the effects of circuit optimizations and message-passing algorithms [Poulin, PRA, 2006] on the performance of fault-tolerant concatenated quantum error correction. Such studies are necessary in order make a proper comparison with competing families of error-correction protocols, such as those involving surface codes. In the talk, a range of new numerical results will be presented.

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.

One simple way to think about physics is in terms of information. We gain information about physical systems by observing them, and with luck this data allows us to predict what they will do next. Quantum mechanics doesn't just change the rules about how physical objects behave - it changes the rules about how information behaves. In this talk we explore what quantum information is, and how strangely it differs from our intuitions. In particular we see how information about quantum particles can become entangled, leading to seemingly impossibly coordinated behaviour for separate objects, and to phenomena such as quantum teleportation.

One of the quintessential features of quantum information is its exclusivity, the inability of strong quantum correlations to be shared by many physical systems. Likewise, complementarity has a similar status in quantum mechanics as the sine qua non of quantum phenomena. We show that this is no coincidence, and that the central role of exclusivity in quantum information theory stems from the phenomenon of complementarity. We adopt an information-theoretic approach to complementarity, which leads to a new and simple definition of private states and new proofs of the achievable asymptotic rates of both secret key and entanglement distillation. From the latter follows a new proof of the direct part of the quantum noisy channel coding theorem.

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.

In deBroglie-Bohm theory the quantum state plays the role of a guiding agent. In this seminar we will explore if this is a universal feature shared by all hidden variable theories or merely a peculiar feature of deBroglie-Bohm theory. We present the bare bones of a model in which the quantum state represents a probability distribution and does not act as a guiding agent. The theory is also psi-epistemic according to Spekken\'s and Harrigan\'s definition. For simplicity we develop the model for a 1D discrete lattice but the generalization to higher dimensions is straightforward. The ontic state consists of a definite particle position and in addition possible non-local links between spatially separated lattice points. These non-local links comes in two types: directed links and non-directed links. Entanglement manifests itself through these links. Interestingly, this ontology seems to be the simplest possible and immediately suggested by the structure of quantum theory itself. For N lattice points there are N*3^(N(N-1)) ontic states growing exponentially with the Hilbert space dimension N as expected. We further require that the evolution of the probability distribution on the ontic state space is dictated by a master equation with non-negative transition rates. It is then easy to show that one can reproduce the Schroedinger equation if an only if there are positive solutions to a gigantic system of linear equations. This is a highly non-trivial problem and whether there exists such positive solutions or not is still not clear to me. Alternatively one can view this set of linear equations as constraints on the possible types of Hamiltonians. We end by speculating how one might incorporate gravity into this theory by requiring permutation invariance of the dynamical evolution law.