Quantum Foundations

The purpose of this talk is to describe bosonic fields and their Lagrangians in the causal set context. Spin-0 fields are defined to be real-valued functions on a causal set. Gauge fields are viewed as SU(n)-valued functions on the set of pairs of elements of a causal set, and gravity is viewed as the causal relation itself. The purpose of this talk is to come up with expressions for the Lagrangian densities of these fields in such a way that they approximate the Lagrangian densities expected from regular Quantum Field Theory on a differentiable manifold in the special case where the causal set is a random sprinkling of points in the manifold. I will then conjecture that that same expression is appropriate for an arbitrary causal set.

Mutually unbiased bases (MUBs) have attracted a lot of attention the last years. These bases are interesting for their potential use within quantum information processing and when trying to understand quantum state space. A central question is if there exists complete sets of N+1 MUBs in N-dimensional Hilbert space, as these are desired for quantum state tomography. Despite a lot of effort they are only known in prime power dimensions. I will describe in geometrical terms how a complete set of MUBs would sit in the set of density matrices and present a distance between bases–a measure of unbiasedness. Then I will explain the relation between MUBs and Hadamard matrices, and report on a search for MUB-sets in dimension N=6. In this case no sets of more than three MUBs are found, but there are several inequivalent triplets.

As physicists, we have become accustomed to the idea that a theory\\\'s content is always most transparent when written in coordinate-free language. But sometimes the choice of a good coordinate system is very useful for settling deep conceptual issues. Think of how Eddington-Finkelstein coordinates settled the longstanding question of whether the event horizon of a Schwarzschild black hole corresponds to a real spacetime singularity or not. Similarly we believe for an information-oriented or Bayesian approach to quantum foundations: That one good coordinate system may (eventually!) be worth more than a hundred blue-in-the-face arguments. This talk will motivate and chronicle the search for one such candidate coordinate system---the so-called Symmetric Informationally Complete Measurement---which has caught the attention of a handful of us here at PI and a handful of our visitors.

Bell\\\'s theorem is commonly understood to show that EPR correlations are not explainable via a local hidden variable theory. But Bell\\\'s theorem assumes that the initial state of the particles is independent of the final detector settings. It has been proposed that this independence assumption might be undermined by a relativistically-allowed form of \\\"backward causation\\\", thereby allowing construction of a local hidden-variable model after all. In this talk, I will show that there is no backward causation model which yields the desired correlations. However, there are other physical scenarios yielding nontrivial nonlocal correlations which violated Bell\\\'s independence assumption. I will present two.

In any attempt to construct a Quantum Theory of Gravity, one has to deal with the fact that Time in Quantum Mechanics appears to be very different from Time in General Relativity. This is the famous (or actually notorious!) \"Problem of Time\", and gives rise to both conceptual and technical problems. The decoherent histories approach to quantum theory, is an alternative formulation of quantum theory specially designed to deal with closed (no-external observer or environment) systems. This approach has been considered particularly promising, in dealing with the problem of time, since it puts space and time in equal footing (unlike standard QM) . This talk develops a particular implementation of the above expectations, i.e. we construct a general set of \"Class Operators\" corresponding to questions that appear to be \"Timeless\" (independent of the parameter time), but correspond to physically interesting questions. This is similar to finding a general enough set of timeless observables, in the evolving constants approach to the problem of time.

The descriptions of the quantum realm and the macroscopic classical world differ significantly not only in their mathematical formulations but also in their foundational concepts and philosophical consequences. When and how physical systems stop to behave quantumly and begin to behave classically is still heavily debated in the physics community and subject to theoretical and experimental research. Conceptually different from already existing models, we have developed a novel theoretical approach to understand this transition from the quantum to a macrorealistic world. It neither needs to refer to the environment of a system (decoherence) nor to change the quantum laws itself (collapse models) but puts the stress on the limits of observability of quantum phenomena due to our measurement apparatuses. First, we demonstrated that for unrestricted measurement accuracy a system’s time evolution cannot be described classically, not even if it is arbitrarily large and macroscopic. Under realistic conditions in every-day life, however, we are only able to perform coarse-grained measurements and do not resolve individual quantum levels of the macroscopic system. As we could show, it is this mere restriction to fuzzy measurements which is sufficient to see the natural emergence of macroscopic realism and even the classical Newtonian laws out of the full quantum laws: the system’s time evolution governed by the Schrödinger equation and the state projection induced by measurements. This resolves the apparent impossibility of how classical realism and deterministic laws can emerge out of fundamentally random quantum events. We find the sufficient condition for these classical evolutions for isolated systems under coarse-grained measurements. Then we demonstrate that nevertheless there exist ”non-classical” Hamiltonians which are in conflict with macroscopic realism. Thus, though at every instant of time the quantum state appears as a classical mixture, its time evolution cannot be understood classically. We argue why such Hamiltonians are unlikely to be realized in nature.

A conceptual framework is proposed for understanding the relationship between observables and operators in mechanics. We claim that the transformations generated by the objective properties of a physical system must be strictly interpreted as gauge transformations. It will be shown that this postulate cannot be consistently implemented in the framework of classical mechanics. We argue that the uncertainty principle is a consequence of the mutual intertwining between objective properties and gauge-dependant properties. Hence, in classical mechanics gauge-dependant properties are wrongly considered objective. It follows that the quantum description of objective physical states is not incomplete, but rather that the classical notion is overdetermined.

The spacetime or histories approach is a whole attitude to quantum mechanics in which histories are fundamental rather than states. In this talk we will review a suggested dynamics and a suggested interpretation in this framework, phrasing the dynamics of stochastic collapse models in the histories language then proceeding to explore a new realist interpretation suggested by Rafael Sorkin and examining its perspective on the Kochen-Specker result.

We describe the measurement statistics of the balanced homodyne detection scheme in terms of the moment operators of the associated positive operator measures. In particular, we give a mathematically rigorous proof for the fact that the high amplitude limit in the local oscillator leads to a measurement of a rotated quadrature operator of the signal _eld. Using these results, we also show that each covariant phase space observable can be measured with the eight-port homodyne detector.

First, a brief description of Bell\'s theorem will be given. It states states that there is no classical-like (local realistic) description of all quantum predictions. Next, a plausible class of non-local realistic models will be presented which is incompatible with quantum mechanics, as first shown by Leggett. Experiments confirming the incompatibility will be described. Finally, it will be argued that quantum mechanics can be seen as a theory of systems with limited information resources.

We derive a set of Bell inequalities using correlated random variables. Our inequalities are necessary conditions for the existence of a local realistic description of projective measurements on qubits. We analyze our inequalities for the case of two qubits and find that they are equivalent to the well known CHSH inequalities. We also discuss the sufficiency of our inequalities as well as their applicability to more than two qubits.