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Bas Janssens

Publications and source records attributed to Bas Janssens.

30 records · Page 2Linked to original sources
arXiv2015

Projective unitary representations of infinite dimensional Lie groups

For an infinite dimensional Lie group $G$ modelled on a locally convex Lie algebra $\mathfrak{g}$, we prove that every smooth projective unitary representation of $G$ corresponds to a smooth linear unitary representation of a Lie group extension $G^{\sharp}$ of $G$. (The main point is the smooth structure on $G^{\sharp}$.) For infinite dimensional Lie groups $G$ which are 1-connected, regular, and modelled on a barrelled Lie algebra $\mathfrak{g}$, we characterize the unitary $\mathfrak{g}$-representations which integrate to $G$. Combining these results, we give a precise formulation of the correspondence between smooth projective unitary representations of $G$, smooth linear unitary representations of $G^{\sharp}$, and the appropriate unitary representations of its Lie algebra $\mathfrak{g}^{\sharp}$.

math.RT↗
arXiv2013

Norm continuous unitary representations of Lie algebras of smooth sections

We give a complete description of the bounded (i.e. norm continuous) unitary representations of the Fréchet-Lie algebra of all smooth sections, as well as of the LF-Lie algebra of compactly supported smooth sections, of a smooth Lie algebra bundle whose typical fiber is a compact Lie algebra. For the Lie algebra of all sections, bounded unitary irreducible representations are finite tensor products of so-called evaluation representations, hence in particular finite-dimensional. For the Lie algebra of compactly supported sections, bounded unitary irreducible (factor) representations are possibly infinite tensor products of evaluation representations, which reduces the classification problem to results of Glimm and Powers on irreducible (factor) representations of UHF C*-algebras. The key part in our proof is the classification of irreducible bounded unitary representations of Lie algebras that are the tensor product of a compact Lie algebra and a unital real continuous inverse algebra: every such representation is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component of the group of smooth gauge transformations of a principal fiber bundle with compact base and structure group, and the connected component of the group of special unitary n times n matrices with values in an involutive commutative continuous inverse algebra.

math.RT↗
arXiv2012

Universal Central Extensions of Gauge Algebras and Groups

We show that the canonical central extension of the group of sections of a Lie group bundle over a compact manifold, constructed in [NW09], is universal. In doing so, we prove universality of the corresponding central extension of Lie algebras in a slightly more general setting.

math.DG↗
arXiv2010

Transformation & Uncertainty. Some Thoughts on Quantum Probability Theory, Quantum Statistics, and Natural Bundles

This PHD thesis is concerned with uncertainty relations in quantum probability theory, state estimation in quantum stochastics, and natural bundles in differential geometry. After some comments on the nature and necessity of decoherence in open systems and its absence in closed ones, we prove sharp, state-independent inequalities reflecting the Heisenberg principle, the necessity of decoherence and the impossibility of perfect joint measurement. These bounds are used to judge how far a particular measurement is removed from the optimal one. We do this for a qubit interacting with the quantized EM field, continually probed using homodyne detection. We calculate to which extent this joint measurement is optimal. We then propose a two-step strategy to determine the (possibly mixed) state of n identically prepared qubits, and prove that it is asymptotically optimal in a local minimax sense, using `Quantum Local Asymptotic Normality' for qubits. We propose a physical implementation of QLAN, based on interaction with the quantized EM field. In differential geometry, a bundle is called `natural' if diffeomorphisms of the base lift to automorphisms of the bundle. We slightly extend this notion to `infinitesimal naturality', requiring only vector fields (infinitesimal diffeomorphisms) to lift. We classify these bundles. Physical fields that transform under (infinitesimal) space-time transformations must be described in terms of (infinitesimally) natural bundles. All spin structures are infinitesimally natural. Our framework thus encompasses Fermionic fields, for which the notion of a natural bundle is too restrictive. Interestingly, generalized spin structures (e.g. spin-c structures) are not always infinitesimally natural. We classify the ones that are. Depending on the gauge group at hand, this can significantly reduce the number of allowed space-time topologies.

math.DG↗
arXiv2009

Bundles with a lift of infinitesimal diffeomorphisms

We slightly extend the notion of a natural fibre bundle by requiring diffeomorphisms of the base to lift to automorphisms of the bundle only infinitesimally, i.e. at the level of the Lie algebra of vector fields. Spin structures are natural only in this extended sense. We classify fibre bundles with this property, assuming a finite dimensional structure group. This includes all spin structures, but only some generalised spin structures. This classification links the gauge group G to the topology of space-time.

math.DG↗
arXiv2007

Optimal estimation of qubit states with continuous time measurements

We propose an adaptive, two steps strategy, for the estimation of mixed qubit states. We show that the strategy is optimal in a local minimax sense for the trace norm distance as well as other locally quadratic figures of merit. Local minimax optimality means that given $n$ identical qubits, there exists no estimator which can perform better than the proposed estimator on a neighborhood of size $n^{-1/2}$ of an arbitrary state. In particular, it is asymptotically Bayesian optimal for a large class of prior distributions. We present a physical implementation of the optimal estimation strategy based on continuous time measurements in a field that couples with the qubits. The crucial ingredient of the result is the concept of local asymptotic normality (or LAN) for qubits. This means that, for large $n$, the statistical model described by $n$ identically prepared qubits is locally equivalent to a model with only a classical Gaussian distribution and a Gaussian state of a quantum harmonic oscillator. The term `local' refers to a shrinking neighborhood around a fixed state $ρ_{0}$. An essential result is that the neighborhood radius can be chosen arbitrarily close to $n^{-1/4}$. This allows us to use a two steps procedure by which we first localize the state within a smaller neighborhood of radius $n^{-1/2+ε}$, and then use LAN to perform optimal estimation.

quant-ph↗
arXiv2006

Classical Coding and the Cauchy-Schwarz Inequality

In classical coding, a single quantum state is encoded into classical information. Decoding this classical information in order to regain the original quantum state is known to be impossible. However, one can attempt to construct a state which comes as close as possible. We give bounds on the smallest possible trace distance between the original and the decoded state which can be reached. We give two approaches to the problem: one starting from Keyl and Werner's no-cloning theorem, and one starting from an operator-valued Cauchy-Schwarz inequality.

quant-ph↗
arXiv2006

Unifying Decoherence and the Heisenberg Principle

We exhibit three inequalities involving quantum measurement, all of which are sharp and state independent. The first inequality bounds the performance of joint measurement. The second quantifies the trade-off between the measurement quality and the disturbance caused on the measured system. Finally, the third inequality provides a sharp lower bound on the amount of decoherence in terms of the measurement quality. This gives a unified discription of both the Heisenberg principle and the collapse of the wave function.

quant-ph↗
arXiv2006

Information Transfer Implies State Collapse

We attempt to clarify certain puzzles concerning state collapse and decoherence. In open quantum systems decoherence is shown to be a necessary consequence of the transfer of information to the outside; we prove an upper bound for the amount of coherence which can survive such a transfer. We claim that in large closed systems decoherence has never been observed, but we will show that it is usually harmless to assume its occurrence. An independent postulate of state collapse over and above Schroedinger's equation and the probability interpretation of quantum states, is shown to be redundant.

quant-ph↗
arXiv2006

Optimal Pointers for Joint Measurement of sigma-x and sigma-z via Homodyne Detection

We study a model of a two-level system (i.e. a qubit) in interaction with the electromagnetic field. By means of homodyne detection, one field-quadrature is observed continuously in time. Due to the interaction, information about the initial state of the qubit is transferred into the field, thus influencing the homodyne measurement results. We construct random variables (pointers) on the probability space of homodyne measurement outcomes having distributions close to the initial distributions of sigma-x and sigma-z. Using variational calculus, we find the pointers that are optimal. These optimal pointers are very close to hitting the bound imposed by Heisenberg's uncertainty relation on joint measurement of two non-commuting observables. We close the paper by giving the probability densities of the pointers.

quant-ph↗
arXiv2005

Quantum Measurement, a coherent description

It is widely known that `collapse of the wave function' on a quantum system A may be brought about by an interaction with another quantum system B. We will prove that this is not just a possible, but a necessary consequence of information transfer from A to B. We generalize this in order to explain why coherences are normally not observed in macroscopic quantum systems. Finally, we provide a quantitative insight into the balance between information gain and state disturbance. We define the quality of an information transfer. For all information transfers of a certain quality, we find a minimum amount of state collapse. Along the way, we obtain generalizations of the Joint Measurement Theorem and of the Heisenberg Principle.

quant-ph↗