9,194 research outputs found
Coexistence of qubit effects
Two quantum events, represented by positive operators (effects), are coexistent if they can occur as possible outcomes in a single measurement scheme. Equivalently, the corresponding effects are coexistent if and only if they are contained in the ranges of a single (joint) observable. Here we give several equivalent characterizations of coexistent pairs of qubit effects. We also establish the equivalence between our results and those obtained independently by other authors. Our approach makes explicit use of the Minkowski space geometry inherent in the four-dimensional real vector space of selfadjoint operators in a two-dimensional complex Hilbert space
Quantum Mechanics as a Framework for Dealing with Uncertainty
Quantum uncertainty is described here in two guises: indeterminacy with its
concomitant indeterminism of measurement outcomes, and fuzziness, or
unsharpness. Both features were long seen as obstructions of experimental
possibilities that were available in the realm of classical physics. The birth
of quantum information science was due to the realization that such
obstructions can be turned into powerful resources. Here we review how the
utilization of quantum fuzziness makes room for a notion of approximate joint
measurement of noncommuting observables. We also show how from a classical
perspective quantum uncertainty is due to a limitation of measurability
reflected in a fuzzy event structure -- all quantum events are fundamentally
unsharp.Comment: Plenary Lecture, Central European Workshop on Quantum Optics, Turku
2009
Approximating incompatible von Neumann measurements simultaneously
We study the problem of performing orthogonal qubit measurements
simultaneously. Since these measurements are incompatible, one has to accept
additional imprecision. An optimal joint measurement is the one with the least
possible imprecision. All earlier considerations of this problem have concerned
only joint measurability of observables, while in this work we also take into
account conditional state transformations (i.e., instruments). We characterize
the optimal joint instrument for two orthogonal von Neumann instruments as
being the Luders instrument of the optimal joint observable.Comment: 9 pages, 4 figures; v2 has a more extensive introduction + other
minor correction
Uncertainty reconciles complementarity with joint measurability
The fundamental principles of complementarity and uncertainty are shown to be
related to the possibility of joint unsharp measurements of pairs of
noncommuting quantum observables. A new joint measurement scheme for
complementary observables is proposed. The measured observables are represented
as positive operator valued measures (POVMs), whose intrinsic fuzziness
parameters are found to satisfy an intriguing pay-off relation reflecting the
complementarity. At the same time, this relation represents an instance of a
Heisenberg uncertainty relation for measurement imprecisions. A
model-independent consideration show that this uncertainty relation is
logically connected with the joint measurability of the POVMs in question.Comment: 4 pages, RevTeX. Title of previous version: "Complementarity and
uncertainty - entangled in joint path-interference measurements". This new
version focuses on the "measurement uncertainty relation" and its role,
disentangling this issue from the special context of path interference
duality. See also http://www.vjquantuminfo.org (October 2003
Unsharp Quantum Reality
The positive operator (valued) measures (POMs) allow one to generalize the notion of observable beyond the traditional one based on projection valued measures (PVMs). Here, we argue that this generalized conception of observable enables a consistent notion of unsharp reality and with it an adequate concept of joint properties. A sharp or unsharp property manifests itself as an element of sharp or unsharp reality by its tendency to become actual or to actualize a specific measurement outcome. This actualization tendency-or potentiality-of a property is quantified by the associated quantum probability. The resulting single-case interpretation of probability as a degree of reality will be explained in detail and its role in addressing the tensions between quantum and classical accounts of the physical world will be elucidated. It will be shown that potentiality can be viewed as a causal agency that evolves in a well-defined way
Maintaining Quantum Coherence in the Presence of Noise through State Monitoring
Unsharp POVM measurements allow the estimation and tracking of quantum
wavefunctions in real-time with minimal disruption of the dynamics. Here we
demonstrate that high fidelity state monitoring, and hence quantum control, is
possible even in the presence of classical dephasing and amplitude noise, by
simulating such measurements on a two-level system undergoing Rabi
oscillations. Finite estimation fidelity is found to persist indefinitely long
after the decoherence times set by the noise fields in the absence of
measurement.Comment: 5 pages, 4 figure
A dilemma in representing observables in quantum mechanics
There are self-adjoint operators which determine both spectral and
semispectral measures. These measures have very different commutativity and
covariance properties. This fact poses a serious question on the physical
meaning of such a self-adjoint operator and its associated operator measures.Comment: 10 page
The structure of classical extensions of quantum probability theory
On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called MisraâBugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hidden-variable model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed
Universal joint-measurement uncertainty relation for error bars
We formulate and prove a new, universally valid uncertainty relation for the necessary error bar widths in any approximate joint measurement of position and momentum
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