1,076 research outputs found
Vanishing quantum discord is necessary and sufficient for completely positive maps
Two long standing open problems in quantum theory are to characterize the
class of initial system-bath states for which quantum dynamics is equivalent to
(1) a map between the initial and final system states, and (2) a completely
positive (CP) map. The CP map problem is especially important, due to the
widespread use of such maps in quantum information processing and open quantum
systems theory. Here we settle both these questions by showing that the answer
to the first is "all", with the resulting map being Hermitian, and that the
answer to the second is that CP maps arise exclusively from the class of
separable states with vanishing quantum discord.Comment: 4 pages, no figures. v2: Accepted for publication in Phys. Rev. Let
Retrodiction of Generalised Measurement Outcomes
If a generalised measurement is performed on a quantum system and we do not
know the outcome, are we able to retrodict it with a second measurement? We
obtain a necessary and sufficient condition for perfect retrodiction of the
outcome of a known generalised measurement, given the final state, for an
arbitrary initial state. From this, we deduce that, when the input and output
Hilbert spaces have equal (finite) dimension, it is impossible to perfectly
retrodict the outcome of any fine-grained measurement (where each POVM element
corresponds to a single Kraus operator) for all initial states unless the
measurement is unitarily equivalent to a projective measurement. It also
enables us to show that every POVM can be realised in such a way that perfect
outcome retrodiction is possible for an arbitrary initial state when the number
of outcomes does not exceed the output Hilbert space dimension. We then
consider the situation where the initial state is not arbitrary, though it may
be entangled, and describe the conditions under which unambiguous outcome
retrodiction is possible for a fine-grained generalised measurement. We find
that this is possible for some state if the Kraus operators are linearly
independent. This condition is also necessary when the Kraus operators are
non-singular. From this, we deduce that every trace-preserving quantum
operation is associated with a generalised measurement whose outcome is
unambiguously retrodictable for some initial state, and also that a set of
unitary operators can be unambiguously discriminated iff they are linearly
independent. We then examine the issue of unambiguous outcome retrodiction
without entanglement. This has important connections with the theory of locally
linearly dependent and locally linearly independent operators.Comment: To appear in Physical Review
Characterizations of linear sufficient statistics
A surjective bounded linear operator T from a Banach space X to a Banach space Y must be a sufficient statistic for a dominated family of probability measures defined on the Borel sets of X. These results were applied, so that they characterize linear sufficient statistics for families of the exponential type, including as special cases the Wishart and multivariate normal distributions. The latter result was used to establish precisely which procedures for sampling from a normal population had the property that the sample mean was a sufficient statistic
Ampleness of Automorphic Line Bundles on Shimura Varieties
Let be a totally real field in which is unramfied and let denote
the integral model of the Hilbert modular variety with good reduction at .
Consider the usual automorphic line bundle over . On the
generic fiber, it is well known that is ample if and only if all
the coefficients are positive. On the special fiber, it is conjectured in
\citep{Tian-Xiao} that is ample if and only if the coefficients
satisfy certain inequalities. We prove this conjecture for Shimura
varieties in this paper and deduce a similar statement for Hilbert modular
varieties from this.Comment: 29 pages, 0 figure
On Quantum Statistical Inference, I
Recent developments in the mathematical foundations of quantum mechanics have
brought the theory closer to that of classical probability and statistics. On
the other hand, the unique character of quantum physics sets many of the
questions addressed apart from those met classically in stochastics.
Furthermore, concurrent advances in experimental techniques and in the theory
of quantum computation have led to a strong interest in questions of quantum
information, in particular in the sense of the amount of information about
unknown parameters in given observational data or accessible through various
possible types of measurements. This scenery is outlined (with an audience of
statisticians and probabilists in mind).Comment: A shorter version containing some different material will appear
(2003), with discussion, in J. Roy. Statist. Soc. B, and is archived as
quant-ph/030719
Equilibrium stresses and rigidity for infinite tensegrities and frameworks
Asymptotic equilibrium stresses are defined for countably infinite
tensegrities and generalisations of the Roth-Whiteley characterisation of
first-order rigidity are obtained. Generalisations of prestress stability and
second order rigidity are given for countably infinite bar-joint frameworks and
are shown to give sufficient conditions for continuous rigidity relative to
certain prescribed motions. The proofs are based on a new short proof for
finite frameworks that prestress stability ensures continuous rigidity.Comment: Journal accepted version, to appear in the Journal of Mathematical
Analysis and Applications. 19 pages, 8 figures. More detail and examples
given. The implication "BPS implies BSR (boundedly smoothly rigid)" has been
strengthened to "BPS implies DCR (directedly continuously rigid)
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