8,634 research outputs found
Quantum Reality and Measurement: A Quantum Logical Approach
The recently established universal uncertainty principle revealed that two
nowhere commuting observables can be measured simultaneously in some state,
whereas they have no joint probability distribution in any state. Thus, one
measuring apparatus can simultaneously measure two observables that have no
simultaneous reality. In order to reconcile this discrepancy, an approach based
on quantum logic is proposed to establish the relation between quantum reality
and measurement. We provide a language speaking of values of observables
independent of measurement based on quantum logic and we construct in this
language the state-dependent notions of joint determinateness, value identity,
and simultaneous measurability. This naturally provides a contextual
interpretation, in which we can safely claim such a statement that one
measuring apparatus measures one observable in one context and simultaneously
it measures another nowhere commuting observable in another incompatible
context.Comment: 16 pages, Latex. Presented at the Conference "Quantum Theory:
Reconsideration of Foundations, 5 (QTRF5)," Vaxjo, Sweden, 15 June 2009. To
appear in Foundations of Physics
Universal Uncertainty Principle in the Measurement Operator Formalism
Heisenberg's uncertainty principle has been understood to set a limitation on
measurements; however, the long-standing mathematical formulation established
by Heisenberg, Kennard, and Robertson does not allow such an interpretation.
Recently, a new relation was found to give a universally valid relation between
noise and disturbance in general quantum measurements, and it has become clear
that the new relation plays a role of the first principle to derive various
quantum limits on measurement and information processing in a unified
treatment. This paper examines the above development on the noise-disturbance
uncertainty principle in the model-independent approach based on the
measurement operator formalism, which is widely accepted to describe a class of
generalized measurements in the field of quantum information. We obtain
explicit formulas for the noise and disturbance of measurements given by the
measurement operators, and show that projective measurements do not satisfy the
Heisenberg-type noise-disturbance relation that is typical in the gamma-ray
microscope thought experiments. We also show that the disturbance on a Pauli
operator of a projective measurement of another Pauli operator constantly
equals the square root of 2, and examine how this measurement violates the
Heisenberg-type relation but satisfies the new noise-disturbance relation.Comment: 11 pages. Based on the author's invited talk at the 9th International
Conference on Squeezed States and Uncertainty Relations (ICSSUR'2005),
Besancon, France, May 2-6, 200
Digraph Complexity Measures and Applications in Formal Language Theory
We investigate structural complexity measures on digraphs, in particular the
cycle rank. This concept is intimately related to a classical topic in formal
language theory, namely the star height of regular languages. We explore this
connection, and obtain several new algorithmic insights regarding both cycle
rank and star height. Among other results, we show that computing the cycle
rank is NP-complete, even for sparse digraphs of maximum outdegree 2.
Notwithstanding, we provide both a polynomial-time approximation algorithm and
an exponential-time exact algorithm for this problem. The former algorithm
yields an O((log n)^(3/2))- approximation in polynomial time, whereas the
latter yields the optimum solution, and runs in time and space O*(1.9129^n) on
digraphs of maximum outdegree at most two. Regarding the star height problem,
we identify a subclass of the regular languages for which we can precisely
determine the computational complexity of the star height problem. Namely, the
star height problem for bideterministic languages is NP-complete, and this
holds already for binary alphabets. Then we translate the algorithmic results
concerning cycle rank to the bideterministic star height problem, thus giving a
polynomial-time approximation as well as a reasonably fast exact exponential
algorithm for bideterministic star height.Comment: 19 pages, 1 figur
Conservation laws, uncertainty relations, and quantum limits of measurements
The uncertainty relation between the noise operator and the conserved
quantity leads to a bound for the accuracy of general measurements. The bound
extends the assertion by Wigner, Araki, and Yanase that conservation laws limit
the accuracy of ``repeatable'', or ``nondisturbing'', measurements to general
measurements, and improves the one previously obtained by Yanase for spin
measurements. The bound also sets an obstacle to making a small quantum
computer.Comment: 4 pages, RevTex, to appear in PR
Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement
The Heisenberg uncertainty principle states that the product of the noise in
a position measurement and the momentum disturbance caused by that measurement
should be no less than the limit set by Planck's constant, hbar/2, as
demonstrated by Heisenberg's thought experiment using a gamma-ray microscope.
Here I show that this common assumption is false: a universally valid trade-off
relation between the noise and the disturbance has an additional correlation
term, which is redundant when the intervention brought by the measurement is
independent of the measured object, but which allows the noise-disturbance
product much below Planck's constant when the intervention is dependent. A
model of measuring interaction with dependent intervention shows that
Heisenberg's lower bound for the noise-disturbance product is violated even by
a nearly nondisturbing, precise position measuring instrument. An experimental
implementation is also proposed to realize the above model in the context of
optical quadrature measurement with currently available linear optical devices.Comment: Revtex, 6 page
Quantum Limits of Measurements Induced by Multiplicative Conservation Laws: Extension of the Wigner-Araki-Yanase Theorem
The Wigner-Araki-Yanase (WAY) theorem shows that additive conservation laws
limit the accuracy of measurements. Recently, various quantitative expressions
have been found for quantum limits on measurements induced by additive
conservation laws, and have been applied to the study of fundamental limits on
quantum information processing. Here, we investigate generalizations of the WAY
theorem to multiplicative conservation laws. The WAY theorem is extended to
show that an observable not commuting with the modulus of, or equivalently the
square of, a multiplicatively conserved quantity cannot be precisely measured.
We also obtain a lower bound for the mean-square noise of a measurement in the
presence of a multiplicatively conserved quantity. To overcome this noise it is
necessary to make large the coefficient of variation (the so-called relative
fluctuation), instead of the variance as is the case for additive conservation
laws, of the conserved quantity in the apparatus.Comment: 8 pages, REVTEX; typo added, to appear in PR
The modern tools of quantum mechanics (A tutorial on quantum states, measurements, and operations)
This tutorial is devoted to review the modern tools of quantum mechanics,
which are suitable to describe states, measurements, and operations of
realistic, not isolated, systems in interaction with their environment, and
with any kind of measuring and processing devices. We underline the central
role of the Born rule and and illustrate how the notion of density operator
naturally emerges, together the concept of purification of a mixed state. In
reexamining the postulates of standard quantum measurement theory, we
investigate how they may formally generalized, going beyond the description in
terms of selfadjoint operators and projective measurements, and how this leads
to the introduction of generalized measurements, probability operator-valued
measures (POVM) and detection operators. We then state and prove the Naimark
theorem, which elucidates the connections between generalized and standard
measurements and illustrates how a generalized measurement may be physically
implemented. The "impossibility" of a joint measurement of two non commuting
observables is revisited and its canonical implementations as a generalized
measurement is described in some details. Finally, we address the basic
properties, usually captured by the request of unitarity, that a map
transforming quantum states into quantum states should satisfy to be physically
admissible, and introduce the notion of complete positivity (CP). We then state
and prove the Stinespring/Kraus-Choi-Sudarshan dilation theorem and elucidate
the connections between the CP-maps description of quantum operations, together
with their operator-sum representation, and the customary unitary description
of quantum evolution. We also address transposition as an example of positive
map which is not completely positive, and provide some examples of generalized
measurements and quantum operations.Comment: Tutorial. 26 pages, 1 figure. Published in a special issue of EPJ -
ST devoted to the memory of Federico Casagrand
Quantum Mechanics with a Momentum-Space Artificial Magnetic Field
The Berry curvature is a geometrical property of an energy band which acts as
a momentum space magnetic field in the effective Hamiltonian describing
single-particle quantum dynamics. We show how this perspective may be exploited
to study systems directly relevant to ultracold gases and photonics. Given the
exchanged roles of momentum and position, we demonstrate that the global
topology of momentum space is crucially important. We propose an experiment to
study the Harper-Hofstadter Hamiltonian with a harmonic trap that will
illustrate the advantages of this approach and that will also constitute the
first realization of magnetism on a torus
Momentum-space Harper-Hofstadter model
We show how the weakly trapped Harper-Hofstadter model can be mapped onto a
Harper-Hofstadter model in momentum space. In this momentum-space model, the
band dispersion plays the role of the periodic potential, the Berry curvature
plays the role of an effective magnetic field, the real-space harmonic trap
provides the momentum-space kinetic energy responsible for the hopping, and the
trap position sets the boundary conditions around the magnetic Brillouin zone.
Spatially local interactions translate into nonlocal interactions in momentum
space: within a mean-field approximation, we show that increasing interparticle
interactions leads to a structural change of the ground state, from a single
rotationally symmetric ground state to degenerate ground states that
spontaneously break rotational symmetry.Comment: 10 pages, 7 figure
- …