20 research outputs found
Exact treatment of operator difference equations with nonconstant and noncommutative coefficients
We study a homogeneous linear second-order difference equation with nonconstant and noncommuting
operator coefficients in a vector space. We build its exact resolutive formula consisting of the explicit noniterative
expression of a generic term of the unknown sequence of vectors. Some nontrivial applications are reported in order
to show the usefulness and the broad applicability of the resul
Positive reduction from spectra
We study the problem of whether all bipartite quantum states having a
prescribed spectrum remain positive under the reduction map applied to one
subsystem. We provide necessary and sufficient conditions, in the form of a
family of linear inequalities, which the spectrum has to verify. Our conditions
become explicit when one of the two subsystems is a qubit, as well as for
further sets of states. Finally, we introduce a family of simple entanglement
criteria for spectra, closely related to the reduction and positive partial
transpose criteria, which also provide new insight into the set of spectra that
guarantee separability or positivity of the partial transpose.Comment: Linear Algebra and its Applications (2015
Order preserving maps on quantum measurements
We study the partially ordered set of equivalence classes of quantum mea-surements endowed with the post-processing partial order. The post-processing order is fundamental as it enables to compare measurements by their intrinsic noise and it gives grounds to define the important concept of quantum incompatibility. Our approach is based on mapping this set into a simpler partially ordered set using an order preserving map and investigating the resulting image. The aim is to ignore unnecessary details while keeping the essential structure, thereby simplifying e.g. detection of incompatibility. One possible choice is the map based on Fisher information introduced by Huangjun Zhu, known to be an order morphism taking values in the cone of positive semidefinite matrices. We explore the properties of that construction and improve Zhu's incompatibility criterion by adding a constraint depending on the number of measurement out-comes. We generalize this type of construction to other ordered vector spaces and we show that this map is optimal among all quadratic maps
Approximating incompatible von Neumann measurements simultaneously
We study the problem of performing orthogonal qubit measurements
simultaneously. Since these measurements are incompatible, one has to accept
additional imprecision. An optimal joint measurement is the one with the least
possible imprecision. All earlier considerations of this problem have concerned
only joint measurability of observables, while in this work we also take into
account conditional state transformations (i.e., instruments). We characterize
the optimal joint instrument for two orthogonal von Neumann instruments as
being the Luders instrument of the optimal joint observable.Comment: 9 pages, 4 figures; v2 has a more extensive introduction + other
minor correction