Quantum Physics

In this talk I would like to put forward Wasserstein-geometry as a natural language for Quantum hydrodynamics. Wasserstein-geometry is a formal, infinite dimensional, Riemannian manifold structure on the space of probability measures on a given Riemannian manifold. The basic equations of Quantum hydrodynamics on the other hand are given by the Madelung equations. In terms of Wasserstein-geometry, Madelung equations appear in the shape of Newton's second law of motion, in which the geodesics are disturbed by the influence of a quantum potential. This was pointed out in 2008 by Max. K. von Renesse. Finally, based on the notion of Wasserstein-distance, I will will briefly introduce a natural notion of Shape Space and some of its properties.

In quantum theory every state can be diagonalised, i.e. decomposed as a convex combination of perfectly distinguishable pure states. This fact is crucial in quantum statistical mechanics, as it provides the foundation for the notions of majorisation and entropy. A natural question then arises: can we give an operational characterisation of them? We address this question in the framework of general probabilistic theories, presenting a set of axioms that guarantee that every state can be diagonalised: Causality, Purity Preservation, Purification, and Pure Sharpness. If we add the Permutability and Strong Symmetry axioms, which are in fact completely equivalent in theories satisfying the other axioms, the diagonalisation result allows us to define a well-behaved majorisation preorder on states. Indeed this majorisation criterion fully captures the convertibility of states in the operational resource theory of purity where random reversible transformations are regarded as free operations. One can also put forward two alternative notions of purity as a resource: one where free operations are unital channels, and another where free operations are generated by reversible interactions with an environment in the invariant state. Under the validity of the above axioms, all these definitions are in fact equivalent, i.e. they all lead to the same preorder on states, which is given by majorisation, in the very same way as in quantum theory.

As discussed in last week’s colloquium, the use of the p-adic metric in state space provides a route to resolving the Bell Theorem in favour of realism and local causality, without fine tuning. Here the p-adic integers provide a natural way to describe the fractal geometry of Invariant Set Theory’s state space. In this talk I first explore the role of complex numbers in Invariant Set Theory (arXiv:1605.01051), and describe a novel realistic perspective on quantum interferometry. Then I will describe a programme of work to synthesise quantum and gravitational physics realistically and causally within the framework of Invariant Set Theory. I will describe a p-adic generalisation of the field equations of General Relativity, and discuss the consequent novel perspectives for understanding the dark (energy and matter) universe. 

In this talk I will: 1) review the results of my work on a geometric approach to foundations for a postquantum information theory; 2) discuss how it is related to other foundational approaches, including some resource theories of knowledge and quantum histories; 3) present some of my research on a category theoretic framework for a multi-agent information relativity. More details on part 1: this approach does not rely on probability theory, spectral theory, or Hilbert spaces. Normalisation of states, convexity, and tensor products are allowed but not assumed foundationally. Nonlinear generalisation of quantum kinematics and dynamics is constructed using geometric structures (quantum relative entropies and Banach Lie--Poisson structure) over the sets of quantum states on W*-algebras. In particular, unitary evolution is generalised to nonlinear hamiltonian flows, while Lueders' rules are generalised to constrained relative entropy maximisations. Combined together, they provide a framework for causal inference that is a generalisation and replacement for completely positive maps, with information dynamics determined directly by epistemic constraints, and no requirement for lack of correlation. Orthodox probability theory and quantum mechanics are special cases of this framework. I will also give the progress report on the reconstruction conjecture: given the category of sets of abstract "states" equipped with the suitably defined entropic distances and BLP structure, how one reconstructs the W*-algebraic case? The discussion of the consistent operational semantics for this approach will lead us to the parts 2 and 3.

In the study of closed quantum system, the simple harmonic oscillator is ubiquitous because all smooth potentials look quadratic locally, and exhaustively understanding it is very valuable because it is exactly solvable.  Although not widely appreciated, Markovian quantum Brownian motion (QBM) plays almost exactly the same role in the study of open quantum systems.  QBM is ubiquitous because it arises from only the Markov assumption and linear Lindblad operators, and it likewise has an elegant and transparent exact solution. QBM is often introduced with specific non-Markovian models like Caldeira-Leggett, but this makes it very difficult to see which phenomena are universal and which are idiosyncratic to the model.  Like frictionless classical mechanics or nonrenormalizable field theories, the exact Markov property is aphysical, but handling this subtlety is a small price to pay for the extreme generality.   The widest class of QBM dynamics is symplectic invariant and includes Einstein-Smoluchowski diffusion, damped harmonic oscillations, and pure spatial decoherence as special cases, whose close relationship is often obscured. 

In this talk I will introduce recent research into quantum clocks of finite dimension, with the focus on their accuracy, as determined by their dimension, coherence, and power consumption.

I will present arguments to bound the synchronization time of any quantum clock as a function of its dimension. In addition, quantum coherence appears to be necessary to saturate these bounds, as the synchronization time of incoherent clocks is seen to have a worse bound. In addition, I will review simple proposals for autonomous clocks built out of thermal machines, and demonstrate that the power consumption of thermal clocks determines the limit of their accuracy. Finally, I will introduce an example of a finite quantum clock that is able to control any quantum operation up to a calculable accuracy, and discuss whether it represents a best case scenario for quantum clocks.

In the de Broglie-Bohm pilot-wave formulation of quantum theory, standard quantum probabilities arise spontaneously through a process of dynamical relaxation that is broadly similar to thermal relaxation in classical physics. If we are to regard this process as the cause of the quantum probabilities we observe today, then we must infer a primordial ‘quantum nonequilibrium’ in the remote past. Such quantum nonequilibrium may have left observable traces today, perhaps in the cosmic microwave background, or in relic particle species that decoupled in the very early universe and that have been sufficiently minimally interacting ever since. The search for the dark matter–that we observe today only through its gravitational interactions–has provided a compendium of particle species that could at least in
principle carry quantum nonequilibrium. If they did indeed exist, nonequilibrium distributions would not only demonstrate the need to reevaluate the canonical quantum formalism, but also generate new phenomena that lie outside the domain of conventional quantum theory, potentially opening up a large domain for investigation. We will develop a simple, parameter free, quantum field theoretical model of spectral measurement, and use it to demonstrate some of the novel effects that could occur to the profiles of the line spectra of such relic particles. We find for instance, line broadening effects that scale with the resolution of the telescope, the possibility of line narrowing and other effects that could cause multiple bumps to form. We use this discussion to comment on possible implications on the indirect search for dark matter.

We give a new theoretical solution to a leading-edge experimental challenge, namely to the verification of quantum computations in the regime of high computational complexity. Our results are given in the language of quantum interactive proof systems. Specifically, we show that any language in BQP has a quantum interactive proof system with a polynomial-time classical verifier (who can also prepare random single-qubit pure states), and a quantum polynomial-time prover. Here, soundness is unconditional---i.e it holds even for computationally unbounded provers. Compared to prior work achieving similar results, our technique does not require the encoding of the input or of the computation; instead, we rely on encryption of the input (together with a method to perform computations on encrypted inputs), and show that the random choice between three types of input (defining a "computational run", versus two types of "test runs") suffice. As a proof technique, we use a reduction to an entanglement-based protocol; this enables a relatively simple analysis for a situation that has previously remained ambiguous in the literature.

Our physical theories often admit multiple formulations or variants.  Although these variants are generally empirically indistinguishable, they nonetheless appear to represent the world as having different structures.   In this talk, I will discuss several criteria for comparing empirically equivalent theories that may be used to identify (1) when one variant has more structure than another (i.e., when a formulation of a theory has “excess structure”) and (2) when two variants are theoretically equivalent, even though they appear to represent the world differently.  I will then discuss where this leaves the philosopher trying to use our empirically successful theories as a guide to the structure of the world.

Bell inequalities bound the strength of classical correlations arising between outcomes of measurements performed on subsystems of a shared physical system. The ability of quantum theory to violate Bell inequalities has been intensively studied for several decades. Recently, there has been an increased interest in studying physical correlations beyond the scenario of Bell inequalities, to more general network structures involving many sources of physical states and observers that may be measuring on subsystems of independent states. Much less is known about the nature of physical correlations in networks as compared to standard Bell inequalities. In this talk we will discuss the motivation and interest for studying such network correlations, review the recent progress in understanding such networks, and discuss the many open questions and new possible directions for research on this topic.