4,597 research outputs found
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
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
Solution to the Mean King's problem with mutually unbiased bases for arbitrary levels
The Mean King's problem with mutually unbiased bases is reconsidered for
arbitrary d-level systems. Hayashi, Horibe and Hashimoto [Phys. Rev. A 71,
052331 (2005)] related the problem to the existence of a maximal set of d-1
mutually orthogonal Latin squares, in their restricted setting that allows only
measurements of projection-valued measures. However, we then cannot find a
solution to the problem when e.g., d=6 or d=10. In contrast to their result, we
show that the King's problem always has a solution for arbitrary levels if we
also allow positive operator-valued measures. In constructing the solution, we
use orthogonal arrays in combinatorial design theory.Comment: REVTeX4, 4 page
Instabilities in Zakharov Equations for Laser Propagation in a Plasma
F.Linares, G.Ponce, J-C.Saut have proved that a non-fully dispersive Zakharov
system arising in the study of Laser-plasma interaction, is locally well posed
in the whole space, for fields vanishing at infinity. Here we show that in the
periodic case, seen as a model for fields non-vanishing at infinity, the system
develops strong instabilities of Hadamard's type, implying that the Cauchy
problem is strongly ill-posed
Perforated exit regions for the reduction of micro-pressure waves from tunnels
The authors are grateful to the following bodies that provided financial support for the project: (i) China Scholarship Council (20117 00029), (ii) National Natural Science Foundation of China (Grant no. U1334201) and (iii) UK Engineering and Physical Sciences Research Council (Grant no. EP/G069441/1).Peer reviewedPostprin
Conservative Quantum Computing
Conservation laws limit the accuracy of physical implementations of
elementary quantum logic gates. If the computational basis is represented by a
component of spin and physical implementations obey the angular momentum
conservation law, any physically realizable unitary operators with size less
than n qubits cannot implement the controlled-NOT gate within the error
probability 1/(4n^2), where the size is defined as the total number of the
computational qubits and the ancilla qubits. An analogous limit for bosonic
ancillae is also obtained to show that the lower bound of the error probability
is inversely proportional to the average number of photons. Any set of
universal gates inevitably obeys a related limitation with error probability
O(1/n^2)$. To circumvent the above or related limitations yielded by
conservation laws, it is recommended that the computational basis should be
chosen as the one commuting with the additively conserved quantities.Comment: 5 pages, RevTex. Corrected to include a new statement that for
bosonic ancillae the lower bound of the error probability is inversely
proportional to the average number of photons, kindly suggested by Julio
Gea-Banacloch
- …