Quantum Foundations

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 of 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 present a new formulation of quantum mechanics for closed systems like the universe using an extension of familiar probability theory that incorporates negative probabilities. Probabilities must be positive for alternative histories that are the basis of settleable bets. However, quantum mechanics describes alternative histories are not the basis for settleable bets as in the two-slit experiment. These alternatives can be assigned extended probabilities that are sometimes negative. We will compare this with the decoherent (consistent) histories formulation of quantum theory. The prospects for using this formulation as a starting point for testable alternatives to quantum theory or further generalizations of it will be briefly discussed.

The nature of antimatter is examined in the context of algebraic quantum field theory. It is shown that the notion of antimatter is more general than that of antiparticles. Properly speaking, then, antimatter is not matter made up of antiparticles --- rather, antiparticles are particles made up of antimatter. We go on to discuss whether the notion of antimatter is itself completely general in quantum field theory. Does the matter-antimatter distinction apply to all field theoretic systems? The answer depends on which of several possible criteria we should impose on the space of physical states.

Recently rediscovered results in the theory of partial differential equations show that for free fields, the properties of the field in an arbitrarily small volume of space, traced through eternity, determine completely the field everywhere at all times. Over finite times, the field is determined in the entire region spanned by the intersection of the future null cone of the earliest event and the past null cone of the latest event. Thus this paradigm of classical field theory exhibits a fascinating form of nonlocality. I'll discuss this result and what it tells us about the possibility of constructing a classical, nonlocal theory which accommodates all the phenomena we observe.

Symmetric monoidal categories provide a convenient and enlightening framework within which to compare and contrast physical theories on a common mathematical footing. In this talk we consider two theories: stabiliser qubit quantum mechanics and the toy bit theory proposed by Rob Spekkens. Expressed in the categorical framework the two theories look very similar mathematically, reflecting their common physical features. There are differences though: in particular a finite Abelian group emerges naturally in the categorical framework, and this group is different in each case ($Z_4$ for the stabiliser theory and $Z_2 \times Z_2$ for the toy bit theory). It turns out that this mathematical difference corresponds directly with a key physical difference between the theories: the stabiliser theory cannot be modelled by local hidden variables, while the toy bit theory can. This analysis can be extended to other Abelian groups yielding a group-theoretic criterion for determining the possibility of local hidden variable interpretations for other physical theories.

Quantum mechanics does not allow us to measure all possible combinations of observables on one system. Even in the simplest case of two observables we know, that measuring one of the observables changes the system in such way, that the other measurement will not give us desired precise information about the state of the system. Prominent examples of such observables are measurement of position and momentum of a particle, or measuring spin along two orthogonal directions. However, once we accept the possibility of imprecise measurements, we can consider to perform such measurement within one experiment. This is the basis of the notion of coexistence. I will present the basics of coexistence by showing how to perform the spin measurement in two directions, while considering the imprecision of the measurement described by POVMs. We can also go a little further and consider coexistence of instruments, i.e. measurements, where on the output besides classical information we are also left out with quantum post-measurement state.

Nonlocality is arguably one of the most remarkable features of quantum mechanics. On the other hand nature seems to forbid other no-signaling correlations that cannot be generated by quantum systems. Usual approaches to explain this limitation is based on information theoretic properties of the correlations without any reference to physical theories they might emerge from. However, as shown in [PRL 104, 140401 (2010)], it is the structure of local quantum systems that determines the bipartite correlations possible in quantum mechanics. We investigate this connection further by introducing toy systems with regular polygons as local state spaces. This allows us to study the transition between bipartite classical, no-signaling and quantum correlations by modifying only the local state space. It turns out that the strength of nonlocality of the maximally entangled state depends crucially on a simple geometric property of the local state space, known as strong self-duality. We prove that the limitation of nonlocal correlations is a general result valid for the maximally entangled state in any model with strongly self-dual local state spaces, since such correlations must satisfy the principle of macroscopic locality. This implies notably that Tsirelson’s bound for correlations of the maximally entangled state in quantum mechanics can be regarded as a consequence of strong self-duality of local quantum systems. Finally, our results also show that there exist models which are locally almost identical to quantum mechanics, but can nevertheless generate maximally nonlocal correlations.

A picture can be used to represent an experiment. In this talk we will consider such pictures and show how to turn them into pictures representing calculations (in the style of Penrose's diagrammatic tensor notation). In particular, we will consider circuits described probabilistically. A circuit represents an experiment where we act on various systems with boxes, these boxes being connected by the passage of systems between them. We will make two assumptions concerning such circuits. These two assumptions allow us to set up the duotensor framework (a duotensor is like a tensor except that each position is associated with two possible bases). We will see that quantum theory can be formulated in this framework. Each of the usual objects of quantum theory (states, measurements, transformations) are special cases of duotensors. The framework is motivated by the objective of providing a formulation of quantum theory which is local in the sense that, in doing a calculation pertaining to a particular region of spacetime, we need only use mathematical objects that pertain to this same region. This is, I argue, a prerequisite in a theory of quantum gravity. Reference for this talk: http://arxiv.org/abs/1005.5164

I will review some recent advances on the line of deriving quantum field theory from pure quantum information processing. The general idea is that there is only Quantum Theory (without quantization rules), and the whole Physics---including space-time and relativity---is emergent from the processing. And, since Quantum Theory itself is made with purely informational principles, the whole Physics must be reformulated in information-theoretical terms. Here's the TOC of the talk: a) Very short review of the informational axiomatization of Quantum Theory; b) How space-time and relativistic covariance emerge from the quantum computation; c) Special relativity without space: other ideas; d) Dirac equation derived as information flow (without the need of Lorentz covariance); e) Information-theoretical meaning of inertial mass and Planck constant; f) Observable consequences (at the Planck scale?); h) What about Gravity? Three alternatives as a start for a brainstorming.

The second law of thermodynamics tells that physics imposes a fundamental constraint on the efficiency of all thermal machines. Here I will address the question of whether size imposes further constraints upon thermal machines, namely whether there is a minimum size below which no machine can run, and whether when they are small if they can still be efficient? I will present a simple model which shows that there is no size limitation and no limit on the efficiency of thermal machine and that this leads to a unified view of small refrigerators, pumps and engines.

Nonlocality is the most striking feature of quantum mechanics. It might even be considered its defining feature and understanding it may be the most important step towards understanding the whole theory. Yet for a long time it was impossible to pinpoint the reason behind the exact amount of nonlocality allowed by quantum mechanics expressed by Tsirelson bound. Recently information causality has been shown to be the principle from which this bound can be derived. However the whole set of nonlocal correlations and nonlocal information processing protocols that quantum mechanics allows is not specified by the Tsirelson bound. It remains an open question whether this whole zoo of nonlocality can be derived from information causality. In this talk I present the fields where information causality is applied together with most recent results or lack of such.