Quantum Foundations

I will present recent work [1] on preparation by measurement of Greenberger–Horne–Zeilinger (GHZ) states in circuit quantum electrodynamics. In particular, for the 3-qubit case, when employing a nonlinear filter on the recorded homodyne signal the selected states are found to exhibit values of the Bell–Mermin operator exceeding 2 under realistic conditions. I will discuss the potential of the dispersive readout to demonstrate a violation of the Mermin bound, and present a measurement scheme avoiding the necessity for full detector tomography. [1] Lev S Bishop et al 2009 New J. Phys. 11 073040

We present a first-principles implementation of {\em spatial} scale invariance as a local gauge symmetry in geometry dynamics using the method of best matching. In addition to the 3-metric, the proposed scale invariant theory also contains a 3-vector potential A_k as a dynamical variable. Although some of the mathematics is similar to Weyl's ingenious, but physically questionable, theory, the equations of motion of this new theory are second order in time-derivatives. It is tempting to try to interpret the vector potential A_k as the electromagnetic field. We exhibit four independent reasons for not giving into this temptation. A more likely possibility is that it can play the role of ``dark matter''. Indeed, as noted in scale invariance seems to play a role in the MOND phenomenology. Spatial boundary conditions are derived from the free-endpoint variation method and a preliminary analysis of the constraints and their propagation in the Hamiltonian formulation is presented.

Betting (or gambling) is a useful tool for studying decision-making in the face of [classical] uncertainty. We would like to understand how a quantum "agent" would act when faced with uncertainty about its [quantum] environment. I will present a preliminary construction of a theory of quantum gambling, motivated by roulette and quantum optics. I'll begin by reviewing classical gambling and the Kelly Criterion for optimal betting. Then I'll demonstrate a quantum optical version of roulette, and discuss some of the challenges and pitfalls in designing such analogues. Quantum agents have access to many more strategies than classical agents. Quantum strategies provide no advantage in classical roulette, but I'll show that a quantum agent can outperform a classical agent in quantum roulette.

Violation of local realism can be probed by theory–independent tests, such as Bell’s inequality experiments. There, a common assumption is the existence of perfect, classical, reference frames, which allow for the specification of measurement settings with arbitrary precision. However, if the reference frames are ``bounded'', only limited precision can be attained. We expect then that the finiteness of the reference frames limits the observability of genuine quantum features. Using spin coherent states as reference frames, we determined their minimal size necessary to violate Bell’s inequalities in entangled systems ranging from qubits to macroscopic dimensions. In the latter, the reference frame’s size must be quadratically larger than that of the system. Lacking such large reference frames, precludes quantum phenomena from appearing in everyday experience.

Although most realistic approaches to quantum theory are based on classical particles, QFT reveals that classical fields are a much closer analog. And unlike quantum fields, classical fields can be extrapolated to curved spacetime without conceptual difficulty. These facts make it tempting to reconsider whether quantum theory might be reformulated on an underlying classical field structure. This seminar aims to demonstrate that by changing only how boundary conditions (BCs) are imposed on ordinary classical field equations, a psi-epistemic quantum theory naturally emerges. Uncertainty and basic quantization naturally result from imposing BCs on closed hypersurfaces (as in Lagrangian QFT); further quantization results from extending Hamilton's principle to restrict the BCs as well as the field equations. The partial dependence of field parameters on future BCs implies an effective contextuality, naturally avoiding the usual arguments against realistic quantum models. Successful applications to the relativistic scalar field will be presented, further motivating an ambitious research program of reformulating quantum theory in terms of ontic classical fields.

Researchers in quantum foundations claim (D'Ariano, Fuchs, ...): Quantum = probability theory + x and hence: x = Quantum - probability theory Guided by the metaphorical analogy: probability theory / x = flesh / bones we introduce a notion of quantum measurement within x, which, when flesing it with Hilbert spaces, provides orthodox quantum mechanical probability calculus.

We know the mathematical laws of quantum mechanics, but as yet we are not so sure why those laws should be inevitable. In the simpler but related environment of classical inference, we also know the laws (of probability). With better understanding of quantum mechanics as the eventual goal, Kevin Knuth and I have been probing the foundations of inference. The world we wish to infer is a partially-ordered set ('poset') of states, which may as often supposed be exclusive, but need not be (e.g. A might be a requirement for B). In inference, a state of mind about the world degrades from perfect knowledge through logical OR, which allows for uncertain alternatives. We don't need AND, and we don't need NOT; we just need OR. This display of acceptable states of mind is [close to] a mathematical 'lattice'. We find that the OR structure by itself (!) forces the ordinary rules of probability calculus. No other rules are compatible with the structure of a lattice, so the ordinary rules are inevitable. The standard Shannon information/entropy is likewise inevitable. Taking this idea further, the OR of states of mind gives a lattice of 'Questions' that might be useful for automated learning. Disconcertingly, this lattice is very much larger (in class aleph-2), and the natural valuations on it exhibit large range. I will present this extension, and ask whether we can rationally foresee its use in practical application.

Eugene Wigner and Hermann Weyl led the way in applying the theory of group representations to the newly formulated theory of quantum mechanics starting in 1927. My talk will focus, first, on two aspects of this early work. Physicists had long exploited symmetries as a way of simplifying problems within classical physics. Wigner recognized that the theory of group representations would similarly have enormous payoff in quantum mechanics, allowing him to solve problems in atomic spectroscopy ``almost without calculation.'' Here I will describe the novel aspects of symmetry in QM that Wigner clarified in the series of papers leading up to his 1931 textbook (Wigner's theorem, projective representations, etc.). The second aspect is less well-known: Weyl (1927) argued that group theory could also be used to address foundational questions in quantum mechanics, leading to a reformulation of the classical commutation relations and a proposal for quantization. Weyl's program had much less immediate impact, although it led to the Stone-von Neumann theorem and to Mackey's imprimitivity theorem. As a final historical point, I argue that in this early work the theory of group representations was optional (as emphasized by Slater and others) in a sense that it was not in particle physics in the 60s. The closing section of the talk turns to philosophical morals that have been drawn from this historical episode, in particular claims regarding ontic structural realism (French, Ladyman) and the group-theoretic constitution of objects (Castellani).

Quantum foundations in the light of gauge theories We will present the conjecture according to which the fact that q and p cannot be both ``observables'' of the same quantum system indicates that there is a remnant universal symmetry acting on classical states. In order to unpack this claim we will generalize to unconstrained systems the gauge correspondence between properties defined by first-class constraints and gauge symmetries generated by these constraints. As we shall see, this means that the uncertainty principle might be encoded in the very definition of the canonical variables q and p. According to the ontology of quantum objects that stems from this analysis, the quantum-mechanical description of physical objects is complete.

Quantum mechanics is a non-classical probability theory, but hardly the most general one imaginable: any compact convex set can serve as the state space for an abstract probabilistic model (classical models corresponding to simplices). From this altitude, one sees that many phenomena commonly regarded as ``characteristically quantum' are in fact generically ``non-classical'. In this talk, I'll show that almost any non-classical probabilistic theory shares with quantum mechanics a notion of entanglement and, with this, a version of the so-called measurement problem. I'll then discuss what's required for an abstract probabilistic theory to admit a somewhat simplified version of Everett's response to this problem -- an exercise that turns out to be instructive in several ways.

A standard canonical quantization of general relativity yields a time-independent Schroedinger equation whose solutions are static wavefunctions on configuration space. Naively this is in contradiction with the real world where things do change. Broadly speaking, the problem how to reconcile a theory which contains no concept of time with a changing world is called 'the problem of time'. In this seminar we shall study this problem using a reformulation of Newtonian mechanics due to Jacobi (Jacobi's timeless mechanics) which allows one to study the problem of time without all the technical difficulties present in quantized general relativity. We show explicitly that Jacobi's timeless mechanics is a straightforward counterexample to the claim that all first class constraints generate gauge transformations, i.e. physically indistinguishable states. The implications of this is unclear. By making use of deBroglie-Bohm trajectories we derive a necessary and sufficient condition for a time-dependent Schroedinger equation to emerge for subsystems. The importance of 'strong' entanglement between subsystem and environment for the emergence of time is stressed.

The essential ingredients of a quantum theory are usually a Hilbert space of states and an algebra of operators encoding observables. The mathematical operations available with these structures translate fairly well into physical operations (preparation, measurement etc.) in a non-relativistic world. This correspondence weakens in quantum field theory, where the direct operational meaning of the observable algebra structure (encoded usually through commutators) is lost. The situation becomes even worse when we want to give a more dynamical role to spacetime as for example in attempts to formulate a quantum theory of gravity. I argue that a revision of the structures that we think of as fundamental in a quantum theory is in order. I go on to outline a proposal in this direction, based on the so called 'general boundary formulation', emphasizing the operational meaning of the ingredients. If time permits I will also comment on the relation to the framework of algebraic quantum field theory.