Quantum Physics

The amount of nonlocality in the GHZ state can be quantified by determining how much classical communication is required to bring a local-hidden-variable model into agreement with the predictions of quantum mechanics. It turns out that one bit suffices, and, of course, nothing less will do. I will discuss generalizations of this result to graph states and its relation to the stabilizer formalism.

Quantum information methods have been recently used for studying the properties of ground state entanglement in several many body and field theory systems. We will discuss a thought experiment wherein entanglement can be extracted from the vacuum of a relativistic field theory into a pair of arbitrarily spatially separated atoms. In order to simulate the detection process, we will consider the ground state of a linear chain of cooled trapped ions, and discuss a scheme for detecting the entanglement between the ion's motional degrees of freedom.

Collapse models are one of the most promising attempts to overcome the measurement problem of quanum mechanics: they descibe, within one single framework, both the quantum properties of microscopic systems and the classical properties of macroscopic objects, and in particular they explain why measurements always have definite outcomes, distributed according to the Born probability rule. We will discuss some recent developments in this field: i) we will show how it is possible to formulate collapse models in such a way that the mean energy of physical system does non increse indefinitely, a typical feature of the models first proposed in the literature; ii) we will discuss recent experiments aiming at testing the validity of the superposition principle, thus of collapse models, at the mesoscopic level.

We will postulate a novel notion of probability; this will involve introducing an extra axiom of probability that seems natural from a Bayesian perspective. We will then provide an analogue of Gleason's theorem for these probabilities. We will also discuss why this approach may be useful for generalizations of quantum theory such as quantum gravity theories; this will involve discussing an analogy between Bayesian approaches and relational approaches.

Asymptotic statements like the almost-equi-partition law, the theorm of Shannon Mc -Millan-Breiman, the theorem of Sanov have all natural quantum analogs. They all talk about the thermodynamik limit of quantum spin systems. I will try to summarize these results and sketch the main ideas of proof.

I look at the information-processing involved in a quantum computation, in terms of the difference between the Boolean logic underlying a classical computation and the non-Boolean logic represented by the projective geometry of Hilbert space, in which the subspace structure of Hilbert space replaces the set-theoretic structure of classical logic. I show that the original Deutsch XOR algorithm, Simon's algorithm, and Shor's algorithm all involve a similar geometric formulation. In terms of this picture, I consider the question of where the speedup relative to classical algorithms comes from.

It is shown that inflationary cosmology may be used to test the statistical predictions of quantum theory at very short distances. Hidden-variables theories, such as the pilot-wave theory of de Broglie and Bohm, allow the existence of vacuum states with non-standard field fluctuations (“quantum non-equilibrium”). It is shown that such non-equilibrium vacua lead to statistical anomalies, such as a breaking of scale invariance for the primordial power spectrum. The results depend only weakly on the details of the de Broglie-Bohm dynamics. Recent observations of the cosmic microwave background are used to set limits on violations of quantum theory in the early universe.

The problem of associating beables (hidden variables) to QFT, in the spirit of what Bohm did for nonrelativistic QM, is not trivial. In 1984, John Bell suggested a way of solving the problem, according to which the beables are the positions of fermions, in a discretized version of QFT, and obey a stochastic evolution that simulates all predictions of QFT. In the continuum limit, it will be shown that the Bell model becomes deterministic and that it is related to the choice of the charge density as a beable. Moreover, the charge superselection rule is a consequence of the Bell model. The non-relativistic limit and the derivation of Bohm's first quantized interpretation in this limit are also studied. I will also consider whether the Bell model can be applied to bosons.

The information spectrum approach gives general formulae for optimal rates of codes in many areas of information theory. In this talk I shall relate the information spectrum approach to Shannon information theory and explore its relationship to ``entropic'' properties including subadditivity, chain rules, Araki-Lieb inequlities, and monotonicity.

Space-time measurements and gravitational experiments are made by the mutual relations between objects, fields, particles etc... Any operationally meaningful assertion about spacetime is therefore intrinsic to the degrees of freedom of the matter (i.e. non-gravitational) fields and concepts such as ``locality'' and ``proximity'' should, at least in principle, be definible entirely within the dynamics of the matter fields. We propose to consider the regions of space just as general ``subsystems''. By writing the Hilbert space of the matter fields as a generic tensor product of subsystems we analyse the evolution of a state vector on an information theoretical basis and discuss general principles to recover a posteriori the usual space-time relations. We apply such principles to generic interacting second quantized models with a finite number of fermionic degrees of freedom. Finally, we discuss the possible role of gravity in this framework.

We consider the problem of bounded-error quantum state identification: given one of two known states, what is the optimal probability with which we can identify the given state, subject to our guess being correct with high probability (but we are permitted to output "don't know" instead of a guess). We prove a direct product theorem for this problem. Our proof is based on semidefinite programming duality and the technique may be of wider interest. Using this result, we present two new exponential separations in the simultaneous message passing model of communication complexity. Both are shown in the strongest possible sense: -- we describe a relation that can be computed with O(log n) classical bits of communication in the presence of shared randomness, but needs n^(1/3) communication if the parties don't share randomness, even if communication is quantum; -- we describe a relation that can be computed with O(log n) classical bits of communication in the presence of shared entanglement, but needs (almost) n^(1/3) communication if the parties share randomness but no entanglement, even if communication is quantum.