Talks

Sage is a collection of mature open source software for mathematics, and new code, all unified into one powerful and easy-to-use package. The mission statement of the Sage project is: "Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab." More information is available at www.sagemath.org. I will use the Sage notebook (a web interface) to demonstrate the use of Sage for a variety of mathematical problems and comment on its design and future direction.

We propose an operationally motivated definition of the physical equivalence of states in General Probabilistic Theories and consider the principle of the physical equivalence of pure states, which turns out to be equivalent to the symmetric structure of the state space. We further consider a principle of the decomposability with distinguishable pure states and give classification theorems of the state spaces for each principle, and derive the Bloch ball in 2 and 3 dimensional systems.

Usually, quantum theory (QT) is introduced by giving a list of abstract mathematical postulates, including the Hilbert space formalism and the Born rule. Even though the result is mathematically sound and in perfect agreement with experiment, there remains the question why this formalism is a natural choice, and how QT could possibly be modified in a consistent way. My talk is on recent work with Lluis Masanes, where we show that five simple operational axioms actually determine the formalism of QT uniquely. This is based to a large extent on Lucien Hardy's seminal work. We start with the framework of "general probabilistic theories", a simple, minimal mathematical description for outcome probabilities of measurements. Then, we use group theory and convex geometry to show that the state space of a bit must be a 3D (Bloch) ball, finally recovering the Hilbert space formalism. There will also be some speculation on how to find natural post-quantum theories by dropping one of the axioms.

We consider theories that satisfy: information causality, reversibility, local discriminability, all tight effects are measurable. A property of these theories is that binary systems (with two perfectly distinguishable states and no more) have state spaces with the shape of a unit ball (the Bloch ball) of arbitrary dimension. It turns out that for dimension different than three these systems cannot be entangled. Hence, the only theory with entanglement which satisfying the above assumptions is quantum theory.

Quantum Theory can be derived from six operational axioms. We introduce the operational and probabilistic language that is used to formulate the principles. After the basic notions of system, state, effect and transformation are reviewed, the principles are stated, and their immediate consequences and interpretations are analyzed. Finally, some key results that represent milestones of the derivation are discussed, with particular focus on their implications on information processing and their relation with the standard quantum formalism. The global picture of the presentation highlights quantum theory as a particular operational language emerging from a background of information processing theories, thanks to the purification postulate that singles out the strictly quantum features of information.

I provide a reformulation of finite dimensional quantum theory in the circuit framework in terms of mathematical axioms, and a reconstruction of quantum theory from operational postulates. The mathematical axioms for quantum theory are the following: [Axiom 1] Operations correspond to operators. [Axiom 2] Every complete set of positive operators corresponds to a complete set of operations. The following operational postulates are shown to be equivalent to these mathematical axioms: [P1] Definiteness. Associated with any given pure state is a unique maximal effect giving probability equal to one. This maximal effect does not give probability equal to one for any other pure state. [P2] Information locality. A maximal measurement on a composite system is effected if we perform maximal measurements on each of the components. [P3] Tomographic locality. The state of a composite system can be determined from the statistics collected by making measurements on the components. [P4] Compound permutatability. There exists a compound reversible transformation on any system effecting any given permutation of any given maximal set of distinguishable states for that system. [P5] Preparability. Filters are non-mixing and non-flattening. Hence, from these postulates we can reconstruct all the usual features of quantum theory: States are represented by positive operators, transformations by completely positive trace non-increasing maps, and effects by positive operators. The Born rule (i.e. the trace rule) for calculating probabilities also follows. See arXiv:1104.2066 for more details. These operational postulates are deeper than those I gave ten years ago in quant-ph/0101012.

Consider the two great physical theories of the twentieth century: relativity and quantum mechanics. Einstein derived relativity from very simple principles. By contrast, the foundation of quantum mechanics is built on a set of rather strange, disjointed and ad hoc axioms, reflecting at best the history that led to discovering this new world order. The purpose of this talk is to argue that a better foundation for quantum mechanics lies within the teachings of quantum information science. The basic postulate is that the truly fundamental laws of Nature concern information, not waves or particles. For example, it is known that quantum key distribution is possible but quantum bit commitment is not and that nature is nonlocal but not as nonlocal as is imposed by causality. But should these statements be considered as theorems or axioms? It's time to pause and reflect on what is really fundamental and what are merely consequences. Could information be the key?

The equilibration dynamics of a closed quantum system is encoded in the long-time distribution function of generic observables. In this paper we consider the Loschmidt echo generalized to finite temperature, and show that we can obtain an exact expression for its long-time distribution for a closed system described by a quantum XY chain following a sudden quench. In the thermodynamic limit the logarithm of the Loschmidt echo becomes normally distributed, whereas for small quenches in the opposite, quasi-critical regime, the distribution function acquires a universal double-peaked form indicating poor equilibration. These findings, obtained by a central limit theorem-type result, extend to completely general models in the small-quench regime.

It is often assumed that the first evidence for direct dark matter detection will come from experiments probing spin-independent interactions, because of higher sensitivities due to coherence effects. We explore the possibility of models that would be invisible in such experiments, but detectable via spin-dependent interactions. The existence of much larger (or only) spin-dependent tree-level interactions is not sufficient, due to potential spin-independent subdominant or loop-induced interactions. We find that most models with detectable spin-dependent interactions would also generate detectable spin-independent interactions. Models in which a light pseudoscalar acts as the mediator seem to uniquely evade this conclusion. We present a viable dark matter model generating such an interaction.

TeV-scale models of quantum gravity predict the formation of mini black holes at the Large Hadron Collider. If these black holes can be treated, at least for part of their evolution, as semi-classical objects, they will emit Hawking radiation. In this talk we review the modeling of this evaporation process, particularly for the case when the black hole is rotating. A detailed understanding of the Hawking radiation is necessary for accurate simulations of black hole events at the LHC.