Quantum Foundations

We begin with a fundamental approach to quantum mechanics based on the unitary representations of the group of diffeomorphisms of physical space (and correspondingly, self-adjoint representations of a local current algebra). From these, various classes of quantum configuration spaces arise naturally. One obtains in addition the usual exchange statistics for spatial dimension d >2, induced by representations of the symmetric group, while for d = 2, the approach led to an early prediction of intermediate or “anyonic” statistics induced by unitary representations of the braid group. After reviewing these ideas, which are based on joint work with R. Menikoff and D. H. Sharp at Los Alamos National Laboratory, we shall discuss briefly some analogous possibilities for infinite-dimensional configuration spaces, including anyonic statistics for extended objects in 3-dimensional space.

Ideal measurements are described in quantum mechanics textbooks by two postulates: the collapse of the wave packet and Born’s rule for the probabilities of outcomes. The quantum evolution of a system then has two components: a unitary (Hamiltonian) evolution in between measurements and non-unitary one when a measurement is performed. This situation was considered to be unsatisfactory by many people, including Einstein, Bohr, de Broglie, von Neumann and Wigner, but has remained unsolved to date. The quantum measurement problem, that is, understanding why a unique outcome is obtained in each individual run of an experiment, is tackled by solving a Hamiltonian model within standard quantum statistical mechanics. The model describes the measurement of the z-component of a spin through interaction with a magnetic memory. The latter apparatus is modeled by a Curie–Weiss magnet having N ≫ 1 spins weakly coupled to a phonon bath. The Hamiltonian evolution exhibits several time scales. The reduction, a rapid decay of the off-diagonal blocks of the system–apparatus density matrix, arises from the many degrees of freedom of the pointer (the magnetization). The registration occurs due to a phase transition from the initial metastable state to one of the final stable states triggered by the tested system. It yields a stationary state in which the apparatus and the system are correlated. Under proper conditions the process satisfies all the features of ideal measurements, including collapse and Born’s rule. As usual, irreversibility is ensured by the macroscopic size of the apparatus, in particular by the large value of N. Nothing else than the usual quantum statistical mechanics and Schro ̈dinger equation is needed, and the results support a specified version of the statistical interpretation. The solution of the quantum measurement problem requires a combination of the reduction and the registration, the properties of which arise from the irreversible dynamics.

I consider systems that consist of a few hot and a few cold two-level systems and define heat engines as unitaries that extract energy. These unitaries perform logical operations whose complexity depends on both the desired efficiency and the temperature quotient. I show cases where the optimal heat engine solves a hard computational task (e.g. an NP-hard problem) [2]. Heat engines can also drive refrigerators and use the temperature difference between two systems for cooling a third one. I argue that these triples of systems define a classification of thermodynamic resources [1]. All the above assumes that unitaries are implemented by an external controller. To get a thermodynamically reversible process, the joint process on system and controller must be reversible. Then, the implementation of the joint process requires a "meta-controller", and so on. To study thermodynamic limits without such an infinite sequence of controllers, I introduce the model of "physically universal cellular automata", in which the boundary between system and controller can be shifted (in analogy to the Heisenberg-cut for the quantum measurement problem). I show that this model raises a lot of fundamental questions [3]. Literature: [1] Janzing et al: Thermodynamic cost of reliability and low temperatures: Tightening Landauer's principle and the second law, J. Stat. Phys. 2000 [2] Janzing: On the computation power of molecular heat engines, J. Stat. Phys. 2006 [3] Janzing: Is there a physically universal cellular automaton or Hamiltonian? arXiv:1009.1720

Many fundamental results in quantum foundations and quantum information theory can be framed in terms of information-theoretic tasks that are provably (im)possible in quantum mechanics but not in classical mechanics. For example, Bell's theorem, the no-cloning and no-broadcasting theorems, quantum key distribution and quantum teleportation can all naturally be described in this way. More generally, quantum cryptography, quantum communication and quantum computing all rely on intrinsically quantum information-theoretic advantages. Much less attention has been paid to the information-theoretic power of relativistic quantum theory, although it appears to describe nature better than quantum mechanics. This talk describes some simple information-theoretic tasks that distinguish relativistic quantum theory from quantum mechanics and relativistic classical physics, and a general framework for defining tasks that includes all previously known (im)possibility theorems and raises many open questions. This suggests a new way of thinking about relativistic quantum theory, and a possible new approach to defining non-trivial relativistic quantum theories rigorously. I also describe some simple and surprisingly powerful applications of these ideas to cryptography, including a new secure scheme for simultaneously committing to and encrypting a prediction and ways of securely "tagging" an inaccessible object so as to guarantee its position.

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