Session J29: Focus Session: Quantum Information for Quantum Foundations - Informational Principles and Fundamental Structures

Information Causality as a physical principle

It is known that the physical principle of ``no-signaling'' alone does not single out quantum correlations, and that the post-quantum no-signaling correlations share many of the features that are supposed to define quantum physics (intrinsic randomness, no-cloning, violation of Bell's inequalities...). This talk focuses on the principle of Information Causality (IC), which generalizes no-signaling and has been proved to come close to singling out quantum correlations. I shall review the successes of IC and also the difficulties that the subsequent research is meeting. In particular, I shall emphasize how a generalization of the initial bipartite scenario to a multipartite one is a most urgent necessary step.   

Tensor network states for quantum foundations

Penrose developed a graphical language to reason about networks of connected tensors and applied these techniques to quantum theory. Here we present the development of a framework and tool set based on Penrose tensor networks that enables one to address certain questions of a foundational nature in quantum theory. A quantum tensor theory is one in which a fixed collection of tensors with clearly defined composition laws defines a physical theory. We show that each element of a universal collection of such tensors gives rise to a physical operation, allowed by the rules of quantum mechanics. Although each of these (possibly) atemporal operations is indeed physical, certain sequences of them could represent processes that violate the rules of quantum theory. The question is to determine when this is the case and we arrive at some perplexing conclusions.   

On the Feynman Problem

Foundations of Quantum Field Theory can be connected to foundations of Quantum Theory if we can derive the former in terms of the latter with two additional postulates of locality and topological homogeneity of interactions between quantum systems, in the hypothesis that a quantum field is ultimately made of a numerable set of quantum systems that are unitarily interacting. But, in order to do that we need to be able to simulate quantumm field by a quantum computer. In the paper ``Simulating Physics with Computers'' (Int. J. Th. Phys, 21 467 (1982)) Richard Feynman raised the problem whether it is possible to simulate Fermi fields by a quantum computer---in short if we can ``qubit-ize'' a Fermi field, keeping the field interaction local on qubits. In this talk I will show how this problem is solved for any space-dimension, upon introducing a new general Jordan-Wigner map between field and qubits, and by adding witnessing auxiliary qubits. I will also derive the unique vacuum for the Fermi fields and for the auxiliary fields. The solution of the Feynman problem allows us to simulate quantum fields by a quantum cellular automata, also providing a kind of Planck-scale version of quantum field theory. Computer simulations will be projected at the end of the talk   

Origin of Feynman's Rules of Quantum Theory and Complementarity

Complex numbers are an intrinsic part of the mathematical formalism of quantum theory, and are perhaps its most mysterious feature. If one considers how to combine experimental arrangements to generate new experimental arrangements, a set of five simple symmetries involving two binary operators naturally arises. Recently, I have shown that these symmetries, together with the probabilistic nature of measurement outcomes and the principle of complementarity (formalized in a novel way as the Principle of Information Duality) naturally lead to Feynman's rules of quantum theory (including their complex nature) [1]. In this paper, I present recent development of this work showing that the assumption of complementarity can be dropped, and, instead, a fundamental theorem from number theory---Frobenius' theorem---can be applied to show that the only formalism compatible with the experimental symmetries are real and complex quantum theory. I shall conclude with a discussion of physical principles which can be used to rule out real quantum theory. \\[4pt] [1] Origin of Complex Quantum Amplitudes and Feynman's Rules, P. Goyal, K. Knuth, J. Skilling, Phys. Rev. A 81, 022109 (2010)   

Partially Ordered Sets of Quantum Measurements and the Dirac Equation

Events can be ordered according to whether one event influences another. This results in a partially ordered set (poset) of events often referred to as a causal set. In this framework, an observer can be represented by a chain of events. Quantification of events and pairs of events, referred to as intervals, can be performed by projecting them onto an observer chain, or even a pair of observer chains, which in specific situations leads to a Minkowski metric replete with Lorentz transformations (Bahreyni \& Knuth, 2011. APS B21.00007). In this work, we unify this picture with the Process Calculus, which coincides with the Feynman rules of quantum mechanics (Goyal, Knuth, Skilling, 2010, arXiv:0907.0909; Goyal \& Knuth, Symmetry 2011, 3(2), 171), by considering quantum measurements to be events. This is performed by quantifying pairs of events, which represent transitions, with a pair of numbers, or a quantum amplitude. In the 1+1D case this results in the Feynman checkerboard model of the Dirac equation (Feynman \& Hibbs, 1965). We further demonstrate that in the case of 3+1 dimensions, we recover Bialnycki-Birula's (1994, Phys. Rev. D, 49(12), 6920) body-centered cubic cellular automata model of the Dirac equation studied more recently by Earle (2011, arXiv:1102.1200v1).   

An Observer-Based Foundation of Geometry

The fact that some events influence other events enables one to define a partially ordered set (poset) of events, often referred to as a causal set. A chain of events, called observer chain, can be quantified by labeling its events numerically. Other events in a poset may be quantified with respect to an observer chain/chains by projecting them onto the chain, resulting in a pair of numbers. Similarly, pairs of events, called intervals, can be quantified with four numbers. Under certain conditions, this leads to the Minkowski metric, Lorentz transformations and the mathematics of special relativity (Bahreyni {\&} Knuth, APS March Meeting 2011). We exploit the same techniques to demonstrate that geometric concepts can be \textit{derived} from order-theoretic concepts. We show how chains in a poset can be used to define points and line segments. Subsequent quantification results in the Pythagorean Theorem and the inner product as well as other geometric concepts and measures. Thus the geometry of space, which is assumed to be fundamental, emerges as a result of quantifying a partially ordered set. More importantly, this proposed foundation of geometry is entirely observer-based, which may provide a natural way toward integration with quantum mechanics.   

Quantum state space as a maximal consistent set

Measurement statistics in quantum theory are obtained from the Born rule and the uniqueness of the probability measure it assigns through quantum states is guaranteed by Gleason's theorem. Thus, a possible systematic way of exploring the geometry of quantum state space expresses quantum states in terms of outcome probabilities of a symmetric informationally complete measurement. This specific choice for representing quantum states is motivated by how the associated probability space provides a natural venue for characterizing the set of quantum states as a geometric construct called a maximal consistent set. We define the conditions for consistency and maximality of a set, provide some examples of maximal consistent sets and attempt to deduce the steps for building up a maximal consistent set of probability distributions equivalent to Hilbert space. In particular, we demonstrate how the reconstruction procedure works for qutrits and observe how it reveals an elegant underlying symmetry among five SIC-POVMs and a complete set of mutually unbiased bases, known in finite affine geometry as the Hesse configuration.   

SIC-POVMs and Lie Algebras

A symmetric informationally complete positive operator valued measure (SIC-POVM) is usually thought of as a highly symmetric structure in quantum state space. However Appleby, Flammia and Fuchs (J. Math. Phys. \textbf{52}, 022202, 2011) have shown that the existence of a SIC-POVM in dimension $d$ is equivalent to a proposition concerning the Lie Algebra $\mathrm{gl}(d,C)$. Related to this they show that there is, associated to each SIC-POVM, a rich and intricate geometric structure in the adjoint representation space of $\mathrm{gl}(d,C)$. In this talk we present a deeper exploration of this structure.   

The Galois Group of Symmetric Measurements

The problem of proving (or disproving) the existence of symmetric informationally complete positive operator valued measures (SICs) has been the focus of much effort in the quantum information community during the last 12 years. In this talk we describe the Galois invariances of Weyl-Heisenberg covariant SICs (the class which has been most intensively studied). It is a striking fact that the published exact solutions (in dimensions 2--16, 19, 24, 35 and 48) are all expressible in terms of radicals, implying that the associated Galois groups must be solvable. Building on the work of Scott and Grassl (\emph{J. Math. Phys.}\ \textbf{51}, 042203 (2010)) we investigate the Galois group in more detail. We show that there is an intriguing interplay between the Galois and Clifford group symmetries. We also show that there are a number of interesting regularities in the Galois group structure for the cases we have examined. We conclude with some speculations about the bearing this may have on the SIC existence problem.   

Quantum Nonlocal Boxes Exhibit Stronger Distillability

Given the apparent limited distillability of nonlocal boxes (NLBs), we initiate a study of the distillation of correlations for NLBs that output quantum states rather than classical bits. We propose a new non-adaptive protocol for nonlocality distillation which asymptotically distills correlated quantum nonlocal boxes to the value 3.098, whereas in contrast, the optimal non-adaptive parity protocol for classical NLBs asymptotically distills to the value 3.0. The protocol is also proven to be an optimal non-adaptive protocol for 1, 2 and 3 copies by formulating nonlocality distillation as a semi-definite programming optimization problem. Even if we restrict out attention to non-adaptive protocols, qNLBs offer improved distillation over NLBs. A generalization of our SDP approach that allows for adaptive protocols may reveal a similar improvement in general. This may imply distillability for nonlocal correlations that are currently not known to be distillable. As a consequence of the work on nonlocality distillation we provide numerical evidence that correlations with non-trivial marginals which are not known to satisfy the macroscopic locality principle may be distillable even when corresponding correlations with trivial marginals are not.   

The necessity of entanglement and the equivalency of Bell's theorem with the second law of thermodynamics

We demonstrate that both Wigner's form of Bell's inequalities as well as a form of the second law of thermodynamics, as manifest in Carath\'{e}odory's principle, can be derived from the same assumptions. The results suggest that Bell's theorem is merely a well-disguised statement of the second law. It also suggests that entanglement is necessary for quantum theory to be in full accord with the second law.