2,574 research outputs found
Quantum Measurement, Information, and Completely Positive Maps
Axiomatic approach to measurement theory is developed. All the possible
statistical properties of apparatuses measuring an observable with
nondegenerate spectrum allowed in standard quantum mechanics are characterized.Comment: 9 pages, LaTeX, invited talk at 5th Int'l Conf. on Quantum
Communication, Measurement and Computing (Capri, Iatly, July 3--8, 2000
Realization of Measurement and the Standard Quantum Limit
This paper, following [M. Ozawa, Phys. Rev. Lett. 60, 385 (1988)], reports a
refutation of the claim that for monitoring the position of a free mass such as
gravitational-wave interferometers the sensitivity is limited by the so called
standard quantum limit (SQL) due to the uncertainty principle. The latest proof
of the SQL is analyzed to revleal an unsupported assumption on quantum
measurements. Quantum measurement theory is introduced to give a general
criterion for physically realizable measurements in quantum mechanics. A theory
of approximate position measurements is developed to obtain a rigorous
condition for the SQL and also to show that a precise position measurement can
leave the object in an arbitrary family of states independent of the input
state. This concludes that Yuen's proposal of breaking the SQL by a contractive
state measurement, a measurement of the position leaving the free mass in a
state with the position uncertainty decreasing in time, is physically
realizable in principle. To enforce this conclusion, a model for error-free
position measurement that leaves the object in a contractive state is
constructed with a solvable Hamiltonian for measuring interaction. Finally,
this model is shown to break the SQL with arbitrary accuracy.Comment: 24 pages, based on the author's invited address at a NATO Advanced
Research Workshop on Squeezed and Non-Classical Light, held January 25-29,
1988, in Cortina d' Ampezzo, Italy. in Squeezed and Nonclassical Light
(edited by P. Tombesi and R. Pike), NATO ASI Series Volume 190 (Plenum, New
York, 1989), pp. 263-28
Measurability and Computability
The conceptual relation between the measurability of quantum mechanical
observables and the computability of numerical functions is re-examined. A new
formulation is given for the notion of measurability with finite precision in
order to reconcile the conflict alleged by M. A. Nielsen [Phys. Rev. Lett. 79,
2915 (1997)] that the measurability of a certain observable contradicts the
Church-Turing thesis. It is argued that any function computable by a quantum
algorithm is a recursive function obeying the Church-Turing thesis, whereas any
observable can be measured in principle.Comment: 8 pages, RevTe
Soundness and completeness of quantum root-mean-square errors
Defining and measuring the error of a measurement is one of the most
fundamental activities in experimental science. However, quantum theory shows a
peculiar difficulty in extending the classical notion of root-mean-square (rms)
error to quantum measurements. A straightforward generalization based on the
noise-operator was used to reformulate Heisenberg's uncertainty relation for
the accuracy of simultaneous measurements to be universally valid and made the
conventional formulation testable to observe its violation. Recently, its
reliability was examined based on an anomaly that the error vanishes for some
imprecise measurements, in which the meter does not commute with the measured
observable. Here, we propose an improved definition for a quantum
generalization of the classical rms error, which is state-dependent,
operationally definable, and perfectly characterizes precise measurements.
Moreover, it is shown that the new notion maintains the previously obtained
universally valid uncertainty relations and their experimental confirmations
without changing their forms and interpretations, in contrast to a prevailing
view that a state-dependent formulation for measurement uncertainty relation is
not tenable.Comment: 11 pages, late
Quantum Limits of Measurement and Computing Induced by Conservation Laws and Uncertainty Relations
A quantitative extension of the Wigner-Araki-Yanase theorem is obtained on
the limitation on precise, non-disturbing measurements of observables which do
not commute with additive conserved quantities, and applied to obtaining a
limitation on the accuracy of quantum computing with computational bases which
do not commute with angular momentum.Comment: 6 pages, LaTe
Quantum Limits of Measurements and Uncertainty Principle
In this paper, we show how the Robertson uncertainty relation gives certain
intrinsic quantum limits of measurements in the most general and rigorous
mathematical treatment. A general lower bound for the product of the
root-mean-square measurement errors arising in joint measurements of
noncommuting observables is established. We give a rigorous condition for
holding of the standard quantum limit (SQL) for repeated measurements, and
prove that if a measuring instrument has no larger root-mean-square
preparational error than the root-mean-square measurement errors then it obeys
the SQL. As shown previously, we can even construct many linear models of
position measurement which circumvent this condition for the SQL.Comment: 15 pages, presented at the International Workshop on Quantum Aspects
of Optical Communications, C.N.R.S. Paris, France, November 26-28, 199
Transfer principle in quantum set theory
In 1981, Takeuti introduced quantum set theory as the quantum counterpart of
Boolean valued models of set theory by constructing a model of set theory based
on quantum logic represented by the lattice of closed subspaces in a Hilbert
space and showed that appropriate quantum counterparts of ZFC axioms hold in
the model. Here, Takeuti's formulation is extended to construct a model of set
theory based on the logic represented by the lattice of projections in an
arbitrary von Neumann algebra. A transfer principle is established that enables
us to transfer theorems of ZFC to their quantum counterparts holding in the
model. The set of real numbers in the model is shown to be in one-to-one
correspondence with the set of self-adjoint operators affiliated with the von
Neumann algebra generated by the logic. Despite the difficulty pointed out by
Takeuti that equality axioms do not generally hold in quantum set theory, it is
shown that equality axioms hold for any real numbers in the model. It is also
shown that any observational proposition in quantum mechanics can be
represented by a corresponding statement for real numbers in the model with the
truth value consistent with the standard formulation of quantum mechanics, and
that the equality relation between two real numbers in the model is equivalent
with the notion of perfect correlation between corresponding observables
(self-adjoint operators) in quantum mechanics. The paper is concluded with some
remarks on the relevance to quantum set theory of the choice of the implication
connective in quantum logic.Comment: 25 pages, to appear in JS
Uncertainty principle for quantum instruments and computing
Recently, universally valid uncertainty relations have been established to
set a precision limit for any instruments given a disturbance constraint in a
form more general than the one originally proposed by Heisenberg. One of them
leads to a quantitative generalization of the Wigner-Araki-Yanase theorem on
the precision limit of measurements under conservation laws. Applying this, a
rigorous lower bound is obtained for the gate error probability of physical
implementations of Hadamard gates on a standard qubit of a spin 1/2 system by
interactions with control fields or ancilla systems obeying the angular
momentum conservation law.Comment: 13 pages, Revtex
Comment on "Proof of Heisenberg's error-disturbance principle"
Recently, Kosugi [arXiv:1504.03779v2 [quant-ph]] argued that Heisenberg's
error-disturbance relation (EDR) must be interpreted as being between the
resolution, the preparational error for the post-measurement observable, and
the disturbance. He further claimed that Heisenberg's EDR can be proven to hold
true in general, when the meter observable is modified as one of its functions.
Here, some comments are given to suggest that the above claims are not
supported.Comment: 1 pag
Error-disturbance relations in mixed states
Heisenberg's uncertainty principle was originally formulated in 1927 as a
quantitative relation between the "mean error" of a measurement of one
observable and the disturbance thereby caused on another observable. Heisenberg
derived this famous relation under an additional assumption on quantum
measurements that has been abandoned in the modern theory, and its universal
validity was questioned in a debate on the sensitivity limit to
gravitational-wave detectors in 1980s. A universally valid form of the
error-disturbance relation was shown to be derived in the modern framework of
general quantum measurements in 2003. We have experienced a considerable
progress in theoretical and experimental study of error-disturbance relations
in the last decade. In 2013 Branciard showed a new stronger form of universally
valid error-disturbance relations, one of which is proved tight for spin
measurements carried out in "pure" states. Nevertheless, a recent
information-theoretical study of error-disturbance relations has suggested that
Branciard relations can be considerably strengthened for measurements in mixed
states. Here, we show a method for strengthening Branciard relations in mixed
states and derive several new universally valid and stronger error-disturbance
relations in mixed states. In particular, it is proved that one of them gives
an ultimate error-disturbance relation for spin measurements, which is tight in
any state. The new relations will play an important role in applications to
state estimation problems including quantum cryptographic scenarios.Comment: 11 pages, late
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