Conceptual Foundations and Foils for Quantum Information Processing - 2011

I will discuss what we know about creating randomness within physics. Although quantum theory prescribes completely random outcomes to particular processes, could it be that within a yet-to-be-discovered post-quantum theory these outcomes are predictable? We have recently shown that this is not possible, using a very natural assumption. In the present talk, I will discuss some recent progress towards relaxing this assumption, providing arguably the strongest evidence yet for truly random processes in our world.

The talk will focus primarily on recent work with Alexander Wilce in which we show that any locally tomographic composite of a qubit with any finite-dimensional homogeneous self-dual (equivalently Jordan-algebraic) system must be a standard finite-dimensional quantum (i.e. $C^*$-algebraic) system.  I may touch on work in progress with collaborators on composites of arbitrary homogeneous self-dual systems.  As motivation I will relate the properties of homogeneity and weak and strong self-duality to information processing phenomena, especially Schrooedingerian "steering" and teleportation (touching on earlier work with Wilce and Gaebler, as well as Barrett and Leifer).  If time permits I will explain the relation between some category-theoretic notions coming from the approach of Abramsky and Coecke and Selinger, notably compactness and dagger-compactness, to weak self-duality (work with Ross Duncan and Wilce).

In my talk I raise the question of the fundamental limits to the size of thermal machines - refrigerators, heat pumps and work producing engines - and I will present the smallest possible ones. I will also discuss the issue of a possible complementarity between size and efficiency and show that even the smallest machines could be maximally efficient. Finally I will present a new point of view over what is work and what do thermal machines actually do.

Three-partite quantum systems exhibit interesting features that are absent in bipartite ones. Several instances are classics by now: the GHZ argument, the W state, the UPB bound entangled states, Svetlichny inequalities... In this talk, I shall discuss some on-going research projects that we are pursuing in my group (in collaboration, or in friendly competition, with other groups) and that involve three-partite entanglement or non-locality: * Activation of non-locality in networks. * Device-independent assessment of the entangling power of a measurement. * Can one falsify all models of hidden communication with finite speed? * Information causality in the three-partite scenario. I shall conclude by a blind excursion into uncertainty relations and cryptography, which also shows 3>>2 albeit with a different meaning.

Non-contextuality is presented as an abstraction and at the same time generalisation of locality. Rather than in correlations, the underlying physical model leaves its signature in collections of expectation values, which are contrained by inequalities much like Bell's or Tsirelson's inequalities. These non-contextual inequalities reveal a deep connection to classic topics in graph theory, such as independence numbers, Lovasz numbers and other graph parameters. By considering the special case of bi-local experiments, we arrive at a semidefinite relaxation (and indeed a whole hierarchy of such relaxations) for the problem of determining the maximum quantum violation of a given Bell inequality.

The quantum mechanical state vector is a complicated object. In particular, the amount of data that must be given in order to specify the state vector (even approximately) increases exponentially with the number of quantum systems. Does this mean that the universe is, in some sense, exponentially complicated? I argue that the answer is yes, if the state vector is a one-to-one description of some part of physical reality. This is the case according to both the Everett and Bohm interpretations. But another possibility is that the state vector merely represents information about an underlying reality. In this case, the exponential complexity of the state vector is no more disturbing that that of a classical probability distribution: specifying a probability distribution over N variables also requires an amount of data that is exponential in N. This leaves the following question: does there exist an interpretation of quantum theory such that (i) the state vector merely represents information and (ii) the underlying reality is simple to describe (i.e., not exponential)? Adapting recent results in communication complexity, I will show that the answer is no. Just as any realist interpretation of quantum theory must be non-locally-causal (by Bell's theorem), any realist interpretation must describe an exponentially complicated reality.

Entanglement provides a coherent view of the physical origin of randomness and the growth and decay of correlations, even in macroscopic systems exhibiting few traditional quantum hallmarks. It helps explain why the future is more uncertain than the past, and how correlations can become macroscopic and classical by being redundantly replicated throughout a system's environment. The most private information, exemplified by a quantum eraser experiment, exists only transiently: after the experiment is over no record remains anywhere in the universe of what "happened". At the other extreme is information that has been so widely replicated as to be infeasible to conceal and unlikely to be forgotten. But such conspicuous information is exceptional: a comparison of entropy flows into and out of the Earth with estimates of the planet's storage capacity leads to the conclusion that most macroscopic classical information---for example the pattern of drops in last week's rainfall---is impermanent, eventually becoming nearly as ambiguous, from a terrestrial perspective, as the transient result of a quantum eraser experiment. Finally we discuss prerequisites for a system to accumulate and maintain in its present state, as our world does, a complex and redundant record of at least some features of its past. Not all dynamics and initial conditions lead to this behavior, and in those that do, the behavior itself tends to be temporary, with the system losing its memory, and even its classical character, as it relaxes to thermal equilibrium.

According to quantum theory, it is impossible to prepare the state of a system such that the outcome of any projective measurement on the system can be predicted with certainty. This limitation of predictive power, which is known as the uncertainty principle, is one of the main distinguishing properties of quantum theory when compared to classical theories. In this talk, I will discuss the implications of this principle to foundational questions. In particular, I will consider the hypothesis that the uncertainty principle, rather than (only) telling us something about reality, may be seen as a manifestation of the limitations of our (classical) methods used to describe reality.

We present “guess your neighbor input” (GYNI), a multipartite nonlocal task in which each player must guess the input received by his neighbor. We show that quantum correlations do not perform better than classical ones at this task, for any prior distribution of the inputs. There exist, however, input distributions for which general no-signalling correlations can outperform classical and quantum correlations. Some of the Bell inequalities associated to our construction correspond to facets of the local polytope. We then discuss implications of this game in connection with recent attempts of deriving quantum correlations from information based principles, such as non-trivial communication complexity, information causality and Gleason’s theorem. Our results show that truly multipartite concepts are necessary to obtain the set of quantum correlations for an arbitrary number of parties.

In 1964, John Bell proved that independent measurements on entangled quantum states lead to correlations that cannot be reproduced using local hidden variables. The core of his proof is that such distributions violate some logical constraints known as Bell inequalities. This remarkable result establishes the non-locality of quantum physics. Bell's approach is purely qualitative. This naturally leads to the question of quantifying quantum physics' non-locality. We will specifically consider two quantities introduced for this purpose. The first one is the maximum amount of Bell inequality violation, and the second one is the communication cost of simulating quantum distributions. In this talk, we prove that these two quantities are strongly related: the logarithm of the first is upper bounded by the second. We prove this theorem in the more general context of non-signalling distributions. This generalization gives us two clear benefits. First, the rich structure of the underlying affine space provides us with a very strong intuition. Secondly, non-signalling distributions capture traditional communication complexity of boolean functions. In that case, our theorem is equivalent to the factorization norm lower bound of Linial and Shraibman, for which we give an elementary proof.

We will explore generalisations of the Shannon and von Neumann entropy to other probabilistic theories, and their connection to the principle of information causality. We will also investigate the link between information causality and non-local games, leading to a new quantum bound on computing the inner product non-locally.