Quantum Foundations

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.

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.

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.

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.

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.

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.

Much of the recent progress in understanding quantum theory has been achieved within an operational approach. Within this context quantum mechanics is viewed as a theory for making probabilistic predictions for measurement outcomes following specified preparations. However, thus far some of the essential elements of the theory – space, time and causal structure – elude such an operational formulation and are assumed to be fixed. Is it possible to extend the operational approach to quantum mechanics such that the notions of an underlying spacetime or causal structure are not assumed? What new phenomenology can follow from such an approach? We develop a framework for multipartite quantum correlations that does not presume these notions, but simply that experimenters in their local laboratories are free to perform arbitrary quantum operations. All known situations that respect definite causal order, including signalling and no-signalling correlations between space-like and time-like separated experiments, as well as probabilistic mixtures of these, can be expressed in this framework. Remarkably, we find quantum correlations which are neither causally ordered nor in a probabilistic mixture of definite causal orders. These correlations are shown to enable performing a communication task that is impossible if a fixed background time is assumed and the events are sufficiently localized in the time.

Over the last 10 years there has been an explosion of “operational reconstructions” of quantum theory. This is great stuff: For, through it, we come to see the myriad ways in which the quantum formalism can be chopped into primitives and, through clever toil, brought back together to form a smooth whole. An image of an IQ-Block puzzle comes to mind, http://www.prismenfernglas.de/iqblock_e.htm. There is no doubt that this is invaluable work, particularly for our understanding of the intricate connections between so many quantum information protocols. But to me, it seems to miss the mark for an ultimate understanding of quantum theory; I am left hungry. I still want to know what strange property of matter forces this formalism upon our information accounting. To play on something Einstein once wrote to Max Born, “The quantum reconstructions are certainly imposing. But an inner voice tells me that they are not yet the real thing. The reconstructions say a lot, but do not really bring us any closer to the secret of the 'old one’." In this talk, I hope to expand on these points and convey some sense of why I am fascinated with the problem of the symmetric informationally complete POVMs to an extent greater than axiomatic reconstructions.

This talk reviews recent and on-going work, much of it joint with Howard Barnum, on the origins of the Jordan-algebraic structure of finite-dimensional quantum theory. I begin by describing a simple recipe for constructing highly symmetrical probabilistic models, and discuss the ordered linear spaces generated by such models. I then consider the situation of a probabilistic theory consisting of a symmetric monoidal *-category of finite-dimensional such models: in this context, the state and effect cones are self-dual. Subject to a further ``steering" axiom, they are also homogenous, and hence, by the Koecher-Vinberg Theorem, representable as the cones of formally real Jordan algebras. Finally, if the theory contains a single system with the structure of a qubit, then (by a result of H. Hanche-Olsen), each model in the category is the self-adjoint part of a C*-algebra.

It is now exactly 75 years ago that John von Neumann denounced his own Hilbert space formalism: ``I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more.'' (sic) [1] His reason was that Hilbert space does not elucidate in any direct manner the key quantum behaviors. One year later, together with Birkhoff, they published "The logic of quantum mechanics". However, it is fair to say that this program was never successful nor does it have anything to do with logic. So what is logic? We will conceive logic in two manners: (1) Something which captures the mathematical content of language (cf `and', `or', `no', `if ... then' are captured by Boolean algebra); (2) something that can be encoded in a `machine' and enables it to reason. Recently we have proposed a new kind of `logic of quantum mechanics' [4]. It follows Schrodinger in that the behavior of compound quantum systems, described by the tensor product [2, again 75 years ago], that captures the quantum behaviors. Over the past couple of years we have played the following game: how much quantum phenomena can be derived from `composition + epsilon'. It turned out that epsilon can be taken to be `very little', surely not involving anything like continuum, fields, vector spaces, but merely a `two-dimensional space' of temporal composition (cf `and then') and compoundness (cf `while'), together with some very natural purely operational assertion. In a very short time, this radically different approach has produced a universal graphical language for quantum theory which helped to resolve some open problems. Most importantly, it paved the way to automate quantum reasoning [5,6], and also enables to model meaning for natural languages [7,8]. That is, we are truly taking `quantum logic' now! If time permits, we also discuss how this logical view has helped to solve concrete problems in quantum information. [1] M Redei (1997) Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead). Stud Hist Phil Mod Phys 27, 493-510. [2] G Birkhoff and J von Neumann (1936) The logic of quantum mechanics. Annals of Mathematics 37, 823843. [3] E Schroedinger, (1935) Discussion of probability relations between separated systems. Proc Camb Phil Soc 31, 555-563; (1936) 32, 446-451. [4] B Coecke (2010) Quantum picturalism. Contemporary Physics 51, 59-83. arXiv:0908.1787 [5] L Dixon, R Duncan, A Kissinger and A Merry. http://dream.inf.ed.ac.uk/projects/quantomatic/ [6] L Dixon and R Duncan (2009) Graphical reasoning in compact closed categories for quantum computation. Annals of Mathematics and Articial Intelligence 56, 2342. [7] B Coecke, M Sadrzadeh & S Clark (2010) Linguistic Analysis 36. Mathematical foundations for a compositional distributional model of meaning. arXiv:1003.4394 [8] New scientist (11 Dec 2011) Quantum links let computers read.