Scientific Series

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.

The speculation that Dark Energy can be explained by the backreaction of present inhomogeneities on the evolution of the background cosmology has been increasingly debated in the recent literature. We demonstrate quantitively that the backreaction of linear perturbations on the Friedmann equations is small but is nevertheless non-vanishing. This indicates the need for an improved averaging procedure capable of averaging tensor quantities in a generally covariant way. We present an averaging process which decomposes the metric into Vielbeins selected employing a variational principle, and parallel-transports them to a single point at which they can be averaged. The functionality of the process is discussed in specific 2-d examples, and its application to 3-surfaces and metric recovery in cosmology is outlined.

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.

At the end of inflation, dynamical instability can rapidly deposit the energy of homogeneous cold inflation into excitations of other fields. This process (known as preheating) essentially starts the hot big bang as we know it. I will present simulations of several preheating models using a new numerical solver DEFROST I developed. The results trace the evolution of the fields, which quickly become very inhomogeneous as the instability kicks in. Surprisingly, there appears to be a certain universality across preheating models with different decay channels. After initial transient, the field density distributions quickly become stationary and lognormal to high precision. I will discuss possible connection of this observation to scalar field turbulence.

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.

We discuss two methods to encode one qubit into six physical qubits. Each of our two examples corrects an arbitrary single-qubit error. Our first example is a degenerate six-qubit quantum error-correcting code. We explicitly provide the stabilizer generators, encoding circuits, codewords, logical Pauli operators, and logical CNOT operator for this code. We also show how to convert this code into a non-trivial subsystem code that saturates the subsystem Singleton bound. We then prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a non-degenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender and the receiver and corrects an arbitrary single-qubit error. The code we obtain is globally equivalent to the Steane seven-qubit code and thus corrects an arbitrary error on the receiver\'s half of the ebit as well. We prove that this code is the smallest code with a CSS structure that uses only one ebit and corrects an arbitrary single-qubit error on the sender\'s side. We discuss the advantages and disadvantages for each of the two codes.

We investigate the strengths and weaknesses of the Spekkens toy model for quantum states. We axiomatize the Spekkens toy model into a set of five axioms, regarding valid states, transformations, measurements and composition of systems. We present two relaxations of the Spekkens toy model, giving rise to two variant toy theories. By relaxing the axiom regarding valid transformations a group of toy operations is obtained that is equivalent to the projective extended Clifford Group for one and two qubits. However, the physical state of affairs resulting from this relaxation is undesirable, violating the desideratum that single toy bit operations must compose under the tensor product. The second variant toy theory is obtained by relaxing the axioms regarding valid states and measurements, resulting in a toy model that exhibits the Kochen-Specker property. Like the previous toy model, the relaxation renders the toy model physically undesirable. Therefore, we claim that the Spekkens toy model is optimal; altering its axioms does not yield a better epistemic description of quantum theory. This work is a collaboration with Gilad Gour, Aidan Roy and Barry C. Sanders.

Set theory provides foundations of mathematics in the sense that all the mathematical notions like numbers, functions, relations, structures are defined in the axiomatic set theory called ZFC. Quantum set theory naturally extends ZFC to quantum logic. Hence, we can expect that quantum set theory provides mathematics based on quantum logic. In this talk, I will show a useful application of quantum set theory to quantum mechanics based on the fact that the real numbers constructed in quantum set theory exactly corresponds to the quantum observables. The standard formulation of quantum mechanics answers the question as to in what state an observable A has the value in an interval I. However, the question is not answered as to in what state two observables A and B have the same value. The notion of equality between the values of observables will play many important roles in foundations of quantum mechanics. The notion of measurement of an observable relies on the condition that the observable to be measured and the meter after the measurement should have the same value. We can define the notion of quantum disturbance through the condition whether the values of the given observable before and after the process is the same. It is shown that all the observational propositions on a quantum system corresponds to some propositions in quantum set theory and the equality relation naturally provides the proposition that two observables have the same value. It has been broadly accepted that we cannot speak of the values of quantum observables without assuming a hidden variable theory. However, quantum set theory enables us to do so without assuming hidden variables but alternatively under the consist use of quantum logic, which is more or less considered as logic of the superposition principle. [1] M. Ozawa, Transfer principle in quantum set theory, J. Symbolic Logic 72, 625-648 (2007), online preprint: http://arxiv.org/abs/math.LO/0604349. [2] M. Ozawa, Quantum perfect correlations, Ann. Phys. (N.Y.) 321, 744--769 (2006), online preprint: LANL quant-ph/0501081.

I will comment on the prevailing atmosphere and attitudes that provoked the CJS theorem, aspects of the theorem itself, some features of the aftermath following the theorem and, finally, a critique of the relevance of the theorem based on my own research on position operators in Lorentz covariant quantum theory.