7,501 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
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
Minimum-energy pulses for quantum logic cannot be shared
We show that if an electromagnetic energy pulse with average photon number
is used to carry out the same quantum logical operation on a set of N
atoms, either simultaneously or sequentially, the overall error probability in
the worst case scenario (i.e., maximized over all the possible initial atomic
states) scales as N^2/. This means that in order to keep the error
probability bounded by N\epsilon, with \epsilon ~ 1/, one needs to use
N/\epsilon photons, or equivalently N separate "minimum-energy'' pulses: in
this sense the pulses cannot, in general, be shared. The origin for this
phenomenon is found in atom-field entanglement. These results may have
important consequences for quantum logic and, in particular, for large-scale
quantum computation.Comment: To appear in Phys. Rev. A, Rapid Communication
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
Gate fidelity of arbitrary single-qubit gates constrained by conservation laws
Recent investigations show that conservation laws limit the accuracy of gate
operations in quantum computing. The inevitable error under the angular
momentum conservation law has been evaluated so far for the CNOT, Hadamard, and
NOT gates for spin 1/2 qubits, while the SWAP gate has no constraint. Here, we
extend the above results to general single-qubit gates. We obtain an upper
bound of the gate fidelity of arbitrary single-qubit gates implemented under
arbitrary conservation laws, determined by the geometry of the conservation law
and the gate operation on the Bloch sphere as well as the size of the ancilla.Comment: Title changed; to appear in J. Phys. A: Math. Theor.; 19 pages, 2
figure
Instruments and channels in quantum information theory
While a positive operator valued measure gives the probabilities in a quantum
measurement, an instrument gives both the probabilities and the a posteriori
states. By interpreting the instrument as a quantum channel and by using the
typical inequalities for the quantum and classical relative entropies, many
bounds on the classical information extracted in a quantum measurement, of the
type of Holevo's bound, are obtained in a unified manner.Comment: 12 pages, revtex
Noise and disturbance in quantum measurements: an information-theoretic approach
We introduce information-theoretic definitions for noise and disturbance in
quantum measurements and prove a state-independent noise-disturbance tradeoff
relation that these quantities have to satisfy in any conceivable setup.
Contrary to previous approaches, the information-theoretic quantities we define
are invariant under relabelling of outcomes, and allow for the possibility of
using quantum or classical operations to `correct' for the disturbance. We also
show how our bound implies strong tradeoff relations for mean square
deviations.Comment: v3: to appear on PRL (some issues fixed, supplemental material
expanded). v2: replaced with submitted version; 5 two-column pages + 6
one-column pages + 3 figures; one issue corrected and few references added.
v1: 17 pages, 3 figure
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
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