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Stefan Weigert

Publications and source records attributed to Stefan Weigert.

At least 19 recordsLinked to original sources
arXiv2025

Gleason's Theorem for a Qubit as Part of a Composite System

We extend Gleason's theorem to the two-dimensional Hilbert space of a qubit by invoking the standard axiom that describes composite quantum systems. The tensor-product structure allows us to derive density matrices and Born's rule for $d=2$ from a simple requirement: the probabilities assigned to measurement outcomes must not depend on whether a system is considered on its own or as a subsystem of a larger one. In line with Gleason's original theorem, our approach assigns probabilities only to projection-valued measures, while other known extensions rely on considering more general classes of measurements. This extension of Gleason's theorem to two-dimensional systems is shown to remain valid for some foil theories of quantum theory.

quant-ph↗
arXiv2025

Beyond the Projection Postulate and Back: Quantum Theories with Generalised State-Update Rules

Are there consistent and physically reasonable alternatives to the projection postulate? Does it have unique properties compared with acceptable alternatives? We answer these questions by systematically investigating hypothetical state-update rules for quantum systems that nature could have chosen over the L\"uders rule. Among other basic properties, any prospective rule must define unique post-measurement states and not allow for superluminal signalling. Particular attention will be paid to consistently defining post-measurement states when performing local measurements in composite systems. Explicit examples of valid unconventional update rules are presented, each resulting in a distinct, well-defined foil of quantum theory. This framework of state-update rules allows us to identify operational properties that distinguish the projective update rule from all others and to put earlier derivations of the projection postulate into perspective.

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arXiv2024

Mutually Unbiased Bases in Composite Dimensions -- A Review

Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.

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arXiv2024

A refined Frauchiger--Renner paradox based on strong contextuality

The Frauchiger--Renner paradox derives an inconsistency when quantum theory is used to describe the use of itself, by means of a scenario where agents model other agents quantumly and reason about each other's knowledge. We observe that logical contextuality (\`a la Hardy) is the key ingredient of the FR paradox, and we provide a stronger paradox based on the strongly contextual GHZ--Mermin scenario. In contrast to the FR paradox, this GHZ--FR paradox neither requires post-selection nor any reasoning by observers who are modelled quantumly. If one accepts the universality of quantum theory including superobservers, we propose a natural extension of Peres's dictum to resolve these extended Wigner's friend paradoxes.

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arXiv2023

Relativity and Quantum Theory: Under the Spell of Today's Paradigms

Thomas S. Kuhn interprets the development of the (natural) sciences as a specific dynamical process. Periods of piecemeal growth of knowledge based on widely accepted paradigms are interrupted by bursts of revolutionary changes which lead to new paradigms incommensurate with the earlier ones. This process is briefly illustrated by recalling the changes to classical physics brought about by Einstein's theory of relativity on the one hand, and by quantum theory on the other. Both theories represent fundamental paradigms of contemporary physics. They appear unshakable to the working physicist but according to Kuhn their paradigmatic status is of a temporary nature only. Does Kuhn's framework help us to identify potential future revolutions?

physics.hist-ph↗
arXiv2023

A Quantum Theory with Non-Collapsing Measurements

A collapse-free version of quantum theory is examined to systematically study the role of the projection postulate. This foil theory assumes "passive" measurements that do not update quantum states although measurement outcomes still occur probabilistically, and in accordance with Born's rule. The Hilbert space setting of quantum theory is retained. "Passive quantum theory" is shown to reproduce preparational uncertainty relations, the no-cloning theorem, and no-signalling, among other properties. Striking differences occur, however, if protocols involve post-measurement states. For example, a single system, rather than an ensemble, is sufficient to reconstruct the state of the system. The possibility to "observe" a state increases the computational power of some quantum algorithms. Passive quantum theory is not locally tomographic but capable of "simulating" quantum measurements modulo a finite delay. Outcome probabilities for composite systems may violate Bell inequalities, without however entailing an argument against local hidden variables.

quant-ph↗
arXiv2023

Uncertainty relations for the support of quantum states

Given a narrow signal over the real line, there is a limit to the localisation of its Fourier transform. In spaces of prime dimensions, Tao derived a sharp state-independent uncertainty relation which holds for the support sizes of a pure qudit state in two bases related by a discrete Fourier transform. We generalise Tao's uncertainty relation to complete sets of mutually unbiased bases in spaces of prime dimensions. The bound we obtain appears to be sharp for dimension three only. Analytic and numerical results for prime dimensions up to nineteen suggest that the bound cannot be saturated in general. For prime dimensions two to seven we construct sharp bounds on the support sizes in $(d+1)$ mutually unbiased bases and identify some of the states achieving them.

quant-ph↗
arXiv2021

General Probabilistic Theories with a Gleason-type Theorem

Gleason-type theorems for quantum theory allow one to recover the quantum state space by assuming that (i) states consistently assign probabilities to measurement outcomes and that (ii) there is a unique state for every such assignment. We identify the class of general probabilistic theories which also admit Gleason-type theorems. It contains theories satisfying the no-restriction hypothesis as well as others which can simulate such an unrestricted theory arbitrarily well when allowing for post-selection on measurement outcomes. Our result also implies that the standard no-restriction hypothesis applied to effects is not equivalent to the dual no-restriction hypothesis applied to states which is found to be less restrictive.

quant-ph↗
arXiv2020

Optimal Detection of Rotations about Unknown Axes by Coherent and Anticoherent States

Coherent and anticoherent states of spin systems up to spin j=2 are known to be optimal in order to detect rotations by a known angle but unknown rotation axis. These optimal quantum rotosensors are characterized by minimal fidelity, given by the overlap of a state before and after a rotation, averaged over all directions in space. We calculate a closed-form expression for the average fidelity in terms of anticoherent measures, valid for arbitrary values of the quantum number j. We identify optimal rotosensors (i) for arbitrary rotation angles in the case of spin quantum numbers up to j=7/2 and (ii) for small rotation angles in the case of spin quantum numbers up to j=5. The closed-form expression we derive allows us to explain the central role of anticoherence measures in the problem of optimal detection of rotation angles for arbitrary values of j.

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arXiv2019

Gleason-type Theorems from Cauchy's Functional Equation

Gleason-type theorems derive the density operator and the Born rule formalism of quantum theory from the measurement postulate, by considering additive functions which assign probabilities to measurement outcomes. Additivity is also the defining property of solutions to Cauchy's functional equation. This observation suggests an alternative proof of the strongest known Gleason-type theorem, based on techniques used to solve functional equations.

quant-ph↗
arXiv2018

A Gleason-type theorem for qubits based on mixtures of projective measurements

We derive Born's rule and the density-operator formalism for quantum systems with Hilbert spaces of dimension two or larger. 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.

quant-ph↗
arXiv2017

Geometry of Uncertainty Relations for Linear Combinations of Position and Momentum

For a quantum particle with a single degree of freedom, we derive preparational sum and product uncertainty relations satisfied by $N$ linear combinations of position and momentum observables. The state-independent bounds depend on their degree of incompatibility defined by the area of a parallelogram in an $N$-dimensional coefficient space. Maximal incompatibility occurs if the observables give rise to regular polygons in phase space. We also conjecture a Hirschman-type uncertainty relation for N observables linear in position and momentum, generalizing the original relation which lower-bounds the sum of the position and momentum Shannon entropies of the particle.

quant-ph↗
arXiv2016

Universality in Uncertainty Relations for a Quantum Particle

A general theory of preparational uncertainty relations for a quantum particle in one spatial dimension is developed. We derive conditions which determine whether a given smooth function of the particle's variances and its covariance is bounded from below. Whenever a global minimum exists, an uncertainty relation has been obtained. The squeezed number states of a harmonic oscillator are found to be universal: no other pure or mixed states will saturate any such relation. Geometrically, we identify a convex uncertainty region in the space of second moments which is bounded by the inequality derived by Robertson and Schrödinger. Our approach not only unifies existing uncertainty relations but also leads to new inequalities for second moments.

quant-ph↗
arXiv2016

Preparational Uncertainty Relations for $N$ Continuous Variables

A smooth function of the second moments of $N$ continuous variables gives rise to an uncertainty relation if it is bounded from below. We present a method to systematically derive such bounds by generalizing an approach applied previously to a single continuous variable. New uncertainty relations are obtained for multi-partite systems which allow one to distinguish entangled from separable states. We also investigate the geometry of the "uncertainty region" in the $N(2N+1)$-dimensional space of moments. It is shown to be a convex set for any number continuous variables, and the points on its boundary found to be in one-to-one correspondence with pure Gaussian states of minimal uncertainty. For a single degree of freedom, the boundary can be visualized as one sheet of a "Lorentz-invariant" hyperboloid in the three-dimensional pace of second moments.

quant-ph↗
arXiv2016

Mutually Unbiased Product Bases for Multiple Qudits

We investigate the interplay between mutual unbiasedness and product bases for multiple qudits of possibly different dimensions. A product state of such a system is shown to be mutually unbiased to a product basis only if each of its factors is mutually unbiased to all the states which occur in the corresponding factors of the product basis. This result implies both a tight limit on the number of mutually unbiased product bases which the system can support and a complete classification of mutually unbiased product bases for multiple qubits or qutrits. In addition, only maximally entangled states can be mutually unbiased to a maximal set of mutually unbiased product bases.

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arXiv2015

Friction Causing Unpredictability

The periodic motion of a classical point particle in a one-dimensional double-well potential acquires a surprising degree of complexity if friction is added. Finite uncertainty in the initial state can make it impossible to predict in which of the two wells the particle will finally settle. For two models of friction, we exhibit the structure of the basins of attraction in phase space which causes the final-state sensitivity. Adding friction to an integrable system with more than one stable equilibrium emerges as a possible "route to chaos" whenever initial conditions can be specified with finite accuracy only.

nlin.CD↗
arXiv2014

Heisenberg Uncertainty Relation for Three Canonical Observables

Uncertainty relations provide fundamental limits on what can be said about the properties of quantum systems. For a quantum particle, the commutation relation of position and momentum observables entails Heisenberg's uncertainty relation. A third observable is presented which satisfies canonical commutation relations with both position and momentum. The resulting triple of pairwise canonical observables gives rise to a Heisenberg-type uncertainty relation for the product of three standard deviations. We derive the smallest possible value of this bound and determine the specific squeezed state which saturates the triple uncertainty relation. Quantum optical experiments are proposed to verify our findings.

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