This series consists of talks in the area of Foundations of Quantum Theory. Seminar and group meetings will alternate.
We describe a notion of state for a quantum system which is given in terms of a collection of empirically realizable probability distributions and is formally analogous to the familiar concept of state from classical statistical mechanics. We first demonstrate the mathematical equivalence of this new notion to the standard quantum notion of density matrix. We identify the simple logical consistency condition (a generalization of the familiar no-signalling condition) which a collection of distributions must obey in order to reconstruct the unique quantum state from which they arise.
Quantum observables
are commonly described by self-adjoint operators on a Hilbert space H. I will
show that one can equivalently describe observables by real-valued functions on
the set P(H) of projections, which we call q-observable functions. If one regards
a quantum observable as a random variable, the corresponding q-observable
function can be understood as a quantum quantile function, generalising the
classical notion. I will briefly sketch how q-observable functions relate to
A century after the advent of Quantum Mechanics and General Relativity, both theories enjoy incredible empirical success, constituting the cornerstones of modern physics. Yet, paradoxically, they suffer from deep-rooted, so-far intractable, conflicts. Motivations for violations of the notion of
Physical theories ought to be built up from colloquial notions such as ’long bodies’, ’energetic sources’ etc. in terms of which one can define pre-theoretic ordering relations such as ’longer than’, ’more energetic than’. One of the questions addressed in previous work is how to make the transition from these pre-theoretic notions to quantification, such as making the transition from the ordering relation of ’longer than’ (if one body covers the other) to the notion of how much longer.
Weak values were introduced by Aharonov, Albert, and Vaidman 25 years ago, but it is only in the last 10 years that they have begun to enter into mainstream physics. I will introduce weak values as done by AAV, but then give them a modern definition in terms of generalized measurements. I will discuss their properties and their uses in experiment. Finally I will talk about what they have to contribute to quantum foundations.
The process of canonical quantization:is reexamined with the goal of
ensuring there is only one reality, where $\hbar>0$, in which classical
and quantum theories coexist. Two results are a clarification of the effect of
canonical coordinate transformations and the role of Cartesian coordinates.
Other results provide validation
The
"psi-epistemic" view is that the quantum state does not represent a
state of the world, but a state of knowledge about the world. It is
motivated, in part, by the observation of qualitative similarities between
characteristic properties of non-orthogonal quantum wavefunctions and between
overlapping classical probability distributions. It might be suggested
that this gives a natural explanation for these properties, which seem puzzling
for the alternative "psi-ontic" view. I will examine two such
The Wigner-Araki-Yanase (WAY) theorem delineates
circumstances under which a class of quantum measurements is ruled out.
Specifically, it states that any observable (given as a self adjoint operator)
not commuting with an additive conserved quantity of a quantum system and
measuring apparatus combined admits no repeatable measurements. I'll review the
content of this theorem and present some new work which generalises and
strengthens the existing results.
We
develop a theory for describing composite objects in physics. These can be
static objects, such as tables, or things that happen in spacetime (such as a
region of spacetime with fields on it regarded as being composed of smaller
such regions joined together). We propose certain fundamental axioms which, it
seems, should be satisfied in any theory of composition. A key axiom is the
order independence axiom which says we can describe the composition of a
composite object in any order. Then we provide a notation for describing
A quantum steering ellipsoid may be used to faithfully represent
a density matrix ? describing two qubits A and B. The ellipsoid is the
geometric set of states that Bob can steer Alice's qubit to when he implements
all possible measurements on his qubit. We show how the correlations between
qubits A and B manifest themselves in this paradigm, giving simple conditions
for when the state is entangled, or has discord. We will also present novel features
of the two qubit state that are revealed by the steering ellipsoid formalism,