456 research outputs found
Uncertainty relations for general phase spaces
We describe a setup for obtaining uncertainty relations for arbitrary pairs
of observables related by Fourier transform. The physical examples discussed
here are standard position and momentum, number and angle, finite qudit
systems, and strings of qubits for quantum information applications. The
uncertainty relations allow an arbitrary choice of metric for the distance of
outcomes, and the choice of an exponent distinguishing e.g., absolute or root
mean square deviations. The emphasis of the article is on developing a unified
treatment, in which one observable takes values in an arbitrary locally compact
abelian group and the other in the dual group. In all cases the phase space
symmetry implies the equality of measurement uncertainty bounds and preparation
uncertainty bounds, and there is a straightforward method for determining the
optimal bounds.Comment: For the proceedings of QCMC 201
Uncertainty from Heisenberg to Today
We explore the different meanings of "quantum uncertainty" contained in
Heisenberg's seminal paper from 1927, and also some of the precise definitions
that were explored later. We recount the controversy about "Anschaulichkeit",
visualizability of the theory, which Heisenberg claims to resolve. Moreover, we
consider Heisenberg's programme of operational analysis of concepts, in which
he sees himself as following Einstein. Heisenberg's work is marked by the
tensions between semiclassical arguments and the emerging modern quantum
theory, between intuition and rigour, and between shaky arguments and
overarching claims. Nevertheless, the main message can be taken into the new
quantum theory, and can be brought into the form of general theorems. They come
in two kinds, not distinguished by Heisenberg. These are, on one hand,
constraints on preparations, like the usual textbook uncertainty relation, and,
on the other, constraints on joint measurability, including trade-offs between
accuracy and disturbance.Comment: 36 pages, 1 figur
Heisenberg uncertainty for qubit measurements
Reports on experiments recently performed in Vienna [Erhard et al, Nature
Phys. 8, 185 (2012)] and Toronto [Rozema et al, Phys. Rev. Lett. 109, 100404
(2012)] include claims of a violation of Heisenberg's error-disturbance
relation. In contrast, we have presented and proven a Heisenberg-type relation
for joint measurements of position and momentum [Phys. Rev. Lett. 111, 160405
(2013)]. To resolve the apparent conflict, we formulate here a new general
trade-off relation for errors in qubit measurements, using the same concepts as
we did in the position-momentum case. We show that the combined errors in an
approximate joint measurement of a pair of +/-1 valued observables A,B are
tightly bounded from below by a quantity that measures the degree of
incompatibility of A and B. The claim of a violation of Heisenberg is shown to
fail as it is based on unsuitable measures of error and disturbance. Finally we
show how the experiments mentioned may directly be used to test our error
inequality.Comment: Version 3 contains further clarifications in our argument refuting
the alleged violation of Heisenberg's error-disturbance relation. Some new
material added on the connection between preparation uncertainty and
approximation error relation
A Continuity Theorem for Stinespring's Dilation
We show a continuity theorem for Stinespring's dilation: two completely
positive maps between arbitrary C*-algebras are close in cb-norm iff we can
find corresponding dilations that are close in operator norm. The proof
establishes the equivalence of the cb-norm distance and the Bures distance for
completely positive maps. We briefly discuss applications to quantum
information theory.Comment: 18 pages, no figure
Uncertainty Relations for Angular Momentum
In this work we study various notions of uncertainty for angular momentum in
the spin-s representation of SU(2). We characterize the "uncertainty regions''
given by all vectors, whose components are specified by the variances of the
three angular momentum components. A basic feature of this set is a lower bound
for the sum of the three variances. We give a method for obtaining optimal
lower bounds for uncertainty regions for general operator triples, and evaluate
these for small s. Further lower bounds are derived by generalizing the
technique by which Robertson obtained his state-dependent lower bound. These
are optimal for large s, since they are saturated by states taken from the
Holstein-Primakoff approximation. We show that, for all s, all variances are
consistent with the so-called vector model, i.e., they can also be realized by
a classical probability measure on a sphere of radius sqrt(s(s+1)). Entropic
uncertainty relations can be discussed similarly, but are minimized by
different states than those minimizing the variances for small s. For large s
the Maassen-Uffink bound becomes sharp and we explicitly describe the
extremalizing states. Measurement uncertainty, as recently discussed by Busch,
Lahti and Werner for position and momentum, is introduced and a generalized
observable (POVM) which minimizes the worst case measurement uncertainty of all
angular momentum components is explicitly determined, along with the minimal
uncertainty. The output vectors for the optimal measurement all have the same
length r(s), where r(s)/s goes to 1 as s tends to infinity.Comment: 30 pages, 22 figures, 1 cut-out paper model, video abstract available
on https://youtu.be/h01pHekcwF
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