18 research outputs found
L\"uders' and quantum Jeffrey's rules as entropic projections
We prove that the standard quantum mechanical description of a quantum state
change due to measurement, given by Lueders' rules, is a special case of the
constrained maximisation of a quantum relative entropy functional. This result
is a quantum analogue of the derivation of the Bayes--Laplace rule as a special
case of the constrained maximisation of relative entropy. The proof is provided
for the Umegaki relative entropy of density operators over a Hilbert space as
well as for the Araki relative entropy of normal states over a W*-algebra. We
also introduce a quantum analogue of Jeffrey's rule, derive it in the same way
as above, and discuss the meaning of these results for quantum bayesianism
Postquantum Br\`{e}gman relative entropies and nonlinear resource theories
We introduce the family of postquantum Br\`{e}gman relative entropies, based
on nonlinear embeddings into reflexive Banach spaces (with examples given by
reflexive noncommutative Orlicz spaces over semi-finite W*-algebras,
nonassociative L spaces over semi-finite JBW-algebras, and noncommutative
L spaces over arbitrary W*-algebras). This allows us to define a class of
geometric categories for nonlinear postquantum inference theory (providing an
extension of Chencov's approach to foundations of statistical inference), with
constrained maximisations of Br\`{e}gman relative entropies as morphisms and
nonlinear images of closed convex sets as objects. Further generalisation to a
framework for nonlinear convex operational theories is developed using a larger
class of morphisms, determined by Br\`{e}gman nonexpansive operations (which
provide a well-behaved family of Mielnik's nonlinear transmitters). As an
application, we derive a range of nonlinear postquantum resource theories
determined in terms of this class of operations.Comment: v2: several corrections and improvements, including an extension to
the postquantum (generally) and JBW-algebraic (specifically) cases, a section
on nonlinear resource theories, and more informative paper's titl
On principles of inductive inference
We propose an intersubjective epistemic approach to foundations of
probability theory and statistical inference, based on relative entropy and
category theory, and aimed to bypass the mathematical and conceptual problems
of existing foundational approaches.Comment: To appear in: Goyal P. (ed.), Proceedings of the 31th International
Workshop on Bayesian Inference and Maximum Entropy Methods in Science and
Engineering, 10-15 July 2011, Waterloo, AIP Conf. Proc., Springer, Berli
Information dynamics and new geometric foundations of quantum theory
We discuss new approach to mathematical foundations of quantum theory, which
is completely independent of Hilbert spaces and measure spaces. New kinematics
is defined by non-linear geometry of spaces of integrals on abstract
non-commutative algebras. New dynamics is defined by constrained maximisation
of quantum relative entropy. We recover Hilbert space based approach (including
unitary evolution and the von Neumann--L\"{u}ders rule) and measure theoretic
approach to probability theory (including Bayes' rule) as special cases of our
approach.Comment: To appear in: Khrennikov A. (ed.), Proceedings of the Foundations of
Probability and Physics 6 conference, Linneuniversitetet, Vaxjo, June 13-16,
2011, AIP Conf. Proc., Springer, Berli
Quantum collapse rules from the maximum relative entropy principle
We show that the von Neumann--Lueders collapse rules in quantum mechanics
always select the unique state that maximises the quantum relative entropy with
respect to the premeasurement state, subject to the constraint that the
postmeasurement state has to be compatible with the knowledge gained in the
measurement. This way we provide an information theoretic characterisation of
quantum collapse rules by means of the maximum relative entropy principle.Comment: v2: some references added, improved presentation, result generalised
to cover nonfaithful states; v3: cross-ref to arXiv:1408.3502 added, v4: some
small corrections plus reference to a published version adde