1,106 research outputs found
Fisher information in quantum statistics
Braunstein and Caves (1994) proposed to use Helstrom's {\em quantum
information} number to define, meaningfully, a metric on the set of all
possible states of a given quantum system. They showed that the quantum
information is nothing else than the maximal Fisher information in a
measurement of the quantum system, maximized over all possible measurements.
Combining this fact with classical statistical results, they argued that the
quantum information determines the asymptotically optimal rate at which
neighbouring states on some smooth curve can be distinguished, based on
arbitrary measurements on identical copies of the given quantum system.
We show that the measurement which maximizes the Fisher information typically
depends on the true, unknown, state of the quantum system. We close the
resulting loophole in the argument by showing that one can still achieve the
same, optimal, rate of distinguishability, by a two stage adaptive measurement
procedure.
When we consider states lying not on a smooth curve, but on a manifold of
higher dimension, the situation becomes much more complex. We show that the
notion of ``distinguishability of close-by states'' depends strongly on the
measurement resources one allows oneself, and on a further specification of the
task at hand. The quantum information matrix no longer seems to play a central
role.Comment: This version replaces the previous versions of February 1999 (titled
'An Example of Non-Attainability of Expected Quantum Information') and that
of November 1999. Proofs and results are much improved. To appear in J. Phys.
Derivative pricing under the possibility of long memory in the supOU stochastic volatility model
We consider the supOU stochastic volatility model which is able to exhibit
long-range dependence. For this model we give conditions for the discounted
stock price to be a martingale, calculate the characteristic function, give a
strip where it is analytic and discuss the use of Fourier pricing techniques.
Finally, we present a concrete specification with polynomially decaying
autocorrelations and calibrate it to observed market prices of plain vanilla
options
Option Pricing in Multivariate Stochastic Volatility Models of OU Type
We present a multivariate stochastic volatility model with leverage, which is
flexible enough to recapture the individual dynamics as well as the
interdependencies between several assets while still being highly analytically
tractable.
First we derive the characteristic function and give conditions that ensure
its analyticity and absolute integrability in some open complex strip around
zero. Therefore we can use Fourier methods to compute the prices of multi-asset
options efficiently. To show the applicability of our results, we propose a
concrete specification, the OU-Wishart model, where the dynamics of each
individual asset coincide with the popular Gamma-OU BNS model. This model can
be well calibrated to market prices, which we illustrate with an example using
options on the exchange rates of some major currencies. Finally, we show that
covariance swaps can also be priced in closed form.Comment: 28 pages, 5 figures, to appear in SIAM Journal on Financial
Mathematic
The Wishart short rate model
We consider a short rate model, driven by a stochastic process on the cone of
positive semidefinite matrices. We derive sufficient conditions ensuring that
the model replicates normal, inverse or humped yield curves
Analysis of a convenient information bound for general quantum channels
Open questions from Sarovar and Milburn (2006 J.Phys. A: Math. Gen. 39 8487)
are answered. Sarovar and Milburn derived a convenient upper bound for the
Fisher information of a one-parameter quantum channel. They showed that for
quasi-classical models their bound is achievable and they gave a necessary and
sufficient condition for positive operator-valued measures (POVMs) attaining
this bound. They asked (i) whether their bound is attainable more generally,
(ii) whether explicit expressions for optimal POVMs can be derived from the
attainability condition. We show that the symmetric logarithmic derivative
(SLD) quantum information is less than or equal to the SM bound, i.e.\
and we find conditions for equality. As
the Fisher information is less than or equal to the SLD quantum information,
i.e. , we can deduce when equality holds in
. Equality does not hold for all
channels. As a consequence, the attainability condition cannot be used to test
for optimal POVMs for all channels. These results are extended to
multi-parameter channels.Comment: 16 pages. Published version. Some of the lemmas have been corrected.
New resuts have been added. Proofs are more rigorou
Measuring Polynomial Invariants of Multi-Party Quantum States
We present networks for directly estimating the polynomial invariants of
multi-party quantum states under local transformations. The structure of these
networks is closely related to the structure of the invariants themselves and
this lends a physical interpretation to these otherwise abstract mathematical
quantities. Specifically, our networks estimate the invariants under local
unitary (LU) transformations and under stochastic local operations and
classical communication (SLOCC). Our networks can estimate the LU invariants
for multi-party states, where each party can have a Hilbert space of arbitrary
dimension and the SLOCC invariants for multi-qubit states. We analyze the
statistical efficiency of our networks compared to methods based on estimating
the state coefficients and calculating the invariants.Comment: 8 pages, 4 figures, RevTex4, v2 references update
Preferred Measurements: Optimality and Stability in Quantum Parameter Estimation
We explore precision in a measurement process incorporating pure probe
states, unitary dynamics and complete measurements via a simple formalism. The
concept of `information complement' is introduced. It undermines measurement
precision and its minimization reveals the system properties at an optimal
point. Maximally precise measurements can exhibit independence from the true
value of the estimated parameter, but demanding this severely restricts the
type of viable probe and dynamics, including the requirement that the
Hamiltonian be block-diagonal in a basis of preferred measurements. The
curvature of the information complement near a globally optimal point provides
a new quantification of measurement stability.Comment: 4 pages, 2 figures, in submission. Substantial Extension and
replacement of arXiv:0902.3260v1 in response to Referees' remark
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