Quantum Foundations

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.

Landauer's erasure principle states that there is an inherent work cost associated with all irreversible operations, like the erasure of the data stored in a system. The necessary work is determined by our uncertainty: the more we know about the system, the less it costs to erase it. Here, we analyse erasure in a general setting where our information about that system can be quantum mechanical. In this scenario, our uncertainty, measured by a conditional entropy, may become negative. We establish a general relation between quantum conditional entropies and a physical quantity, the work cost of erasure. As a consequence, we obtain a thermodynamic interpretation of negative entropies: they quantify the work that can be gained by a quantum observer erasing a system. (arXiv: 1009.1630)

I revisit an example of stronger-than-quantum correlations that was discovered by Ernst Specker in 1960. The example was introduced as a parable wherein an over-protective seer sets a simple prediction task to his daughter's suitors. The challenge cannot be met because the seer asks the suitors for a noncontextual assignment of values but measures a system for which the statistics are inconsistent with such an assignment. I will show how by generalizing these sorts of correlations, one is led naturally to some well-known proofs of nonlocality and contextuality, and to some new ones. Specker's parable involves a kind of complementarity that does not arise in quantum theory - three measurements that can be implemented jointly pairwise but not triplewise -- and therefore prompts the question of what sorts of foundational principles might rule out this kind of complementarity. This is joint work with Howard Wiseman and Yeong-Cherng Liang.

In this talk we quickly review the basics of the modal "toy model" of quantum theory described by Schumacher in his September 22 colloquium at PI. We then consider how the theory addresses more general open systems. Because the modal theory has a more primitive mathematical structure than actual quantum mechanics, it lacks density operators, positive operator measurements, and completely positive maps. As we will show, however, modal quantum theory has an elegant description of the states, effects and operations of open modal systems -- a description with close analogies to actual quantum mechanics.

The question of the existence of gravitational stress-energy in general relativity has exercised investigators in the field since the very inception of the theory. Folklore has it that no adequate definition of a localized gravitational stress-energetic quantity can be given. Most arguments to that effect invoke one version or another of the Principle of Equivalence. I argue that not only are such arguments of necessity vague and hand-waving but, worse, are beside the point and do not address the heart of the issue. Based on an analysis of what it may mean for one tensor to depend in the proper way on another, I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. Along the way, I prove a result of some interest in own right about the structure of the associated jet bundles of the bundle of Lorentz metrics over spacetime.

The uncertainty principle bounds the uncertainties about the outcomes of two incompatible measurements, such as position and momentum, on a particle. It implies that one cannot predict the outcomes for both possible choices of measurement to arbitrary precision, even if information about the preparation of the particle is available in a classical memory. However, if the particle is prepared entangled with a quantum memory, it is possible to predict the outcomes for both measurement choices precisely. I will explain a recent extension of the uncertainty principle to incorporate this case. The new relation gives a lower bound on the uncertainties, which depends on the amount of entanglement between the particle and the quantum memory. If time permits, I will also outline a couple of applications.