Quantum Foundations

Bell inequalities are important tools in contrasting classical and quantum behaviors. To date, most Bell inequalities are linear combinations of statistical correlations between remote parties. Nevertheless, finding the classical and quantum mechanical (Tsirelson) bounds for a given Bell inequality in a general scenario is a difficult task which rarely leads to closed-form solutions. Here we introduce a new class of Bell inequalities based on products of correlators that alleviate these issues. Each such Bell inequality is associated with a non-cooperative coordination game. In the simplest case, Alice and Bob, each  having two random variables, attempt to maximize the area of a rectangle and the rectangle’s area is represented by a certain parameter. This parameter, which is a function of the correlations between their random variables, is shown to be a Bell parameter, i.e. the achievable bound using only classical correlations is strictly smaller than the achievable bound using non-local quantum correlations We continue by generalizing to the case in which Alice and Bob, each having now n random variables, wish to maximize a certain volume in n-dimensional space. We term this parameter a multiplicative Bell parameter and prove its Tsirelson bound. Finally, we investigate the case of local hidden variables and show that for any deterministic strategy of one of the players the Bell parameter is a harmonic function whose maximum approaches the Tsirelson bound as the number of measurement devices increases. Some implications of these results are discussed.

The no-signalling principle, preventing superluminal communication and the consequent logical paradoxes, is typically formulated within the information-theoretic framework in terms of admissible correlations in composite systems. In my talk, I will present its complementary incarnation associated with dynamics of single systems subject to invasive measurements. The `dynamical no-signalling principle' applies to any theory with well defined rules of calculating detection statistics in spacetime. It thus offers a new framework, based on measure theory, for studying ``post-quantum'' theories in spacetime. I will show that, strikingly, the `dynamical no-signalling' principle rules out some of the well know models of quantum wave dynamics.

Randomness is a valuable resource in both classical and quantum networks and we wish to generate desired probability distributions as cheaply as possible. If we are allowed to slightly change the distribution under some tolerance level, we can sometimes greatly reduce the cardinality of the randomness or the dimension of the entanglement. By studying statistical inequalities, we show how to upper bound of the amount of randomness required for any given classical network and tolerance level. We also present a problem we encounter when compressing the randomness in a quantum network.

Canonical quantization is not well suited to quantize gravity, while affine quantization is. For those unfamiliar with affine quantization the talk will include a primer. This procedure is then applied to deal with three nonrenormalizable, field theoretical, problems of increasing difficulty, the last one being general relativity itself.

The need for a time-shift invariant formulation of quantum theory arises from fundamental symmetry principles as well as heuristic cosmological considerations. Such a description then leaves open the question of how to reconcile global invariance with the perception of change, locally. By introducing relative time observables, we are able to make rigorous the Page-Wootters conditional probability formalism to show how local Heisenberg evolution is compatible with global invariance.

We derive Born’s rule and the density-operator formalism for quantum systems with Hilbert spaces of finite dimension. Our extension of Gleason’s theorem only relies upon the consistent assignment of probabilities to the outcomes of projective measurements and their classical mixtures. This assumption is significantly weaker than those required for existing Gleason-type theorems valid in dimension two.

The theory of relativity associates a proper time with each moving object via its spacetime trajectory. In quantum theory on the other hand, such trajectories are forbidden. I will discuss an operation approach to exploring this conflict, considering the average time measured by a quantum clock in the weak-field, low-velocity limit. Considering the role of the clock’s state of motion, one finds that all ``good'' quantum clocks experience the time dilation prescribed by general relativity for the most classical states of motion. For nonclassical states of motion, on the other hand, one finds that quantum interference effects give rise to a discrepancy between the proper time and the time measured by the clock. I will also describe how ignorance of the clock's state of motion leads to a larger uncertainty in the time as measured by the clock, a consequence of entanglement between the clock time and its center-of-mass degrees of freedom.