Quantum Foundations

I provide a reformulation of finite dimensional quantum theory in the circuit framework in terms of mathematical axioms, and a reconstruction of quantum theory from operational postulates. The mathematical axioms for quantum theory are the following: [Axiom 1] Operations correspond to operators. [Axiom 2] Every complete set of positive operators corresponds to a complete set of operations. The following operational postulates are shown to be equivalent to these mathematical axioms: [P1] Definiteness. Associated with any given pure state is a unique maximal effect giving probability equal to one. This maximal effect does not give probability equal to one for any other pure state. [P2] Information locality. A maximal measurement on a composite system is effected if we perform maximal measurements on each of the components. [P3] Tomographic locality. The state of a composite system can be determined from the statistics collected by making measurements on the components. [P4] Compound permutatability. There exists a compound reversible transformation on any system effecting any given permutation of any given maximal set of distinguishable states for that system. [P5] Preparability. Filters are non-mixing and non-flattening. Hence, from these postulates we can reconstruct all the usual features of quantum theory: States are represented by positive operators, transformations by completely positive trace non-increasing maps, and effects by positive operators. The Born rule (i.e. the trace rule) for calculating probabilities also follows. See arXiv:1104.2066 for more details. These operational postulates are deeper than those I gave ten years ago in quant-ph/0101012.

Quantum theory can be thought of a noncommutative generalization of classical probability and, from this perspective, it is puzzling that no quantum generalization of conditional probability is in widespread use. In this talk, I discuss one such generalization and show how it can unify the description of ensemble preparations of quantum states, POVM measurements and the description of correlations between quantum systems. The conditional states formalism allows for a description of prepare-and-measure experiments that is neutral with respect to the direction of inference, such that both the retrodictive formalism and the more usual predictive formalism are consequences of a more fundamental description in terms of a conditionally independent tripartite state, and the two formalisms are related by a quantum generalization of Bayes' rule. As an application, I give a generalized argument for the pooling rule proposed by Spekkens and Wiseman that is a direct analog of a result in classical supra-Bayesian pooling.

We will analyze different aspects of locality in causal operational probabilistic theories. We will first discuss the notion of local state and local objective information in operational probabilistic theories, and define an operational notion of discord that coincides with quantum discord in the case of quantum theory. Using such notion, we will show that the only theory in which all separable states have null discord is the classical one. We will then analyze locality of transformations, reviewing some general properties of no-signaling channels in causal theories. We will show that it is natural to define transformations on no-signaling channels that cannot be extended to all bipartite channels, and discuss the consequences of this fact on information processing.

We address the problem of testing the dimensionality of classical and quantum systems in a ‘black-box’ scenario. Imagine two uncharacterized devices. The first one allows an experimentalist to prepare a physical system in various ways. The second one allows the experimentalist to perform some measurement on the system. After collecting enough statistics, the experimentalist obtains a ‘data table’, featuring the probability distribution of the measurement outcomes for each choice of preparation (of the system) and of measurement. Here, we develop a general formalism to assess the minimal dimensionality of classical and quantum systems necessary to reproduce a given data table. To illustrate these ideas, we provide simple examples of classical and quantum ‘dimension witnesses’. In general quantum systems are more economical than classical ones in terms of dimensionality, in the sense that there exist data tables obtainable from quantum systems of dimension d which can only be generated from classical systems of dimension strictly greater than d. By drawing connections to communication complexity one can find data tables for which this classical/quantum separation is dramatic. Finally, these ideas can also be used to demonstrate security of one-way QKD in a semi-device-independent scenario, in which devices are uncharacterized, but only assumed to produce quantum systems of a given dimension.

A seminal work by Cleve, Høyer, Toner and Watrous (quant-ph/0404076) proposed a close connection between quantum nonlocality and computational complexity theory by considering nonlocal games and multi-prover interactive proof systems with entangled provers. It opened up the whole area of study of the computational nature of nonlocality. Since then, understanding nonlocality has been one of the major goals in computational complexity theory in the quantum setting. This talk gives a survey of this exciting area.

In this talk, I'll survey various "foils" of BQP (Bounded-Error Quantum Polynomial-Time) that have been proposed: that is, changes to the quantum model of computation that make it either more or less powerful. Possible topics include: postselected quantum computing, quantum computing with nonlinear Schrodinger equation, quantum computing with non-unitary linear transformations, quantum computing with hidden variables, linear-optical quantum computing, quantum computing with restricted gate sets, quantum computing with separable mixed states, quantum computing over finite fields, and more depending on audience interest.

Operational theories [1], defined in terms of the actions and observations of an experimenter, have been extremely successful as foils to quantum mechanics, providing a generic framework in which families of theories may be compared and classified. One area of particular interest has been in the non-classical correlations (often referred to non-locality) which can arise in quantum (and generalised) theories, when measurements are space-like separated. In the context of non-locality, one usually considers the correlations in separated measurements on isolated systems. A similar setting arises in In quantum computation theory, in measurement-based quantum computation, a model of computation of equivalent power to standard circuit model quantum computation. Measurements are made on isolated non-interacting quantum systems, and the non-classical correlations which arise embody (in some loose sense) the mechanism via which the computation is executed. These measurements are adaptive, meaning that bases are chosen dependent upon the outcome of prior measurements, but apart from this, the setting is essentially identical to a multi-party Bell non-locality experiment (e.g. [2]). In this talk I will review some recent work [3] in which Bell-type correlations are studied from the perspective of computation - in particular drawing parallels with measurement-based quantum computation. In particular, I shall give examples of results [3] which appear naturally in this setting, while being not so self-evident in more conventional approaches. Finally, I shall discuss approaches to and challenges in developing non-trivial models of correlation-based quantum computation in general operational theories. [1] See e.g. H. Barnum, J. Barrett, M. Leifer and A. Wilce, Phys. Rev. Lett., 99, 240501 (2007). [2] See e.g. R. F. Werner and M. M. Wolf, Phys. Rev. A 64, 032112 (2001), M. Zukowski, C. Brukner, Phys. Rev. Lett. 88 210401 (2002). [3] M.J. Hoban and D.E. Browne, http://arxiv.org/abs/1102.1438, M.J. Hoban et al, http://arxiv.org/abs/1009.5213, J. Anders and D.E. Browne http://arxiv.org/abs/0805.1002 .

David Deutsch re-formulated the Church-Turing thesis as a physical principle, asserting that "every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means". Such principle can be regarded as a new theoretical paradigm, whereby the entire Physics is emerging from a quantum computation. But for a theory to be a good one, it must explain a large class of phenomena based on few general principles. Taking as a general principle the topological homogeneity of the computational network with graph-dimension equal to the space-time dimension corresponds to replacing quantum field theory (QFT) with a numerable set of quantum systems in local interaction. This means to consider QFT as a kind of Fermi-scale thermodynamic" limit of a deeper Planck-scale theory, with the quantum field replaced by a giant quantum computer. In the talk, I will illustrate mechanisms of emergence of physics from the quantum computation in 1+1 dimensions. We will see that Dirac's is just the equation describing the free flow of information, leading to an informational definition of inertial mass and Planck constant. I will then illustrate the emergence mechanism of Minkowsian space-time from the computation, how the field Hamiltonian comes out, and how quantum fields are actually eliminated in favor of qubits. We will see that the digital nature of the field leads to an in-principle observable consequence in terms of a mass-dependent refraction index of vacuum, with the information becoming stationary at the Planck mass. Such refraction index of vacuum is a general phenomenon due to unitariety in the discrete, and can also help in solving the speed-of-light isotropy conundrum posed by digitalization of the field in more than 1 space dimensions. We will also see how the quantum nature of the processed information plays a crucial role in other practical informational issues, e.g. the possibility of driving the information in different directions, without the need of increasing the complexity of the circuit. Finally I will briefly comment about gravity as emergent from the quantum computation, and the connection with Verlinde-Jacobson approach.

A central question in our understanding of the physical world is how our knowledge of the whole relates to our knowledge of the individual parts. One aspect of this question is the following: to what extent does ignorance about a whole preclude knowledge of at least one of its parts? Relying purely on classical intuition, one would certainly be inclined to conjecture that a strong ignorance of the whole cannot come without significant ignorance of at least one of its parts. Indeed, we show that this reasoning holds in any non-contextual hidden variable model (NC-HV). Curiously, however, such a conjecture is false in quantum theory: we provide an explicit example where a large ignorance about the whole can coexist with an almost perfect knowledge of each of its parts. More specifically, we provide a simple information-theoretic inequality satisfied in any NC-HV, but which can be arbitrarily violated by quantum mechanics. Our inequality has interesting implications for quantum cryptography.

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.

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 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.