24 research outputs found
The C-Numerical Range in Infinite Dimensions
In infinite dimensions and on the level of trace-class operators rather
than matrices, we show that the closure of the -numerical range is
always star-shaped with respect to the set , where
denotes the essential numerical range of the bounded operator .
Moreover, the closure of is convex if either is normal with
collinear eigenvalues or if is essentially self-adjoint. In the case of
compact normal operators, the -spectrum of is a subset of the
-numerical range, which itself is a subset of the convex hull of the closure
of the -spectrum. This convex hull coincides with the closure of the
-numerical range if, in addition, the eigenvalues of or are
collinear.Comment: 31 pages, no figures; to appear in Linear and Multilinear Algebr
Quantum-Dynamical Semigroups and the Church of the Larger Hilbert Space
In this work we investigate Stinespring dilations of quantum-dynamical
semigroups, which are known to exist by means of a constructive proof given by
Davies in the early 70s. We show that if the semigroup describes an open
system, that is, if it does not consist of only unitary channels, then the
evolution of the dilated closed system has to be generated by an unbounded
Hamiltonian; subsequently the environment has to correspond to an
infinite-dimensional Hilbert space, regardless of the original system.
Moreover, we prove that the second derivative of Stinespring dilations with a
bounded total Hamiltonian yields the dissipative part of some quantum-dynamical
semigroup -- and vice versa. In particular this characterizes the generators of
quantum-dynamical semigroups via Stinespring dilations.Comment: 9+9 page
Progress on the Kretschmann-Schlingemann-Werner Conjecture
Given any pair of quantum channels such that at least one of
them has Kraus rank one, as well as any respective Stinespring isometries
, we prove that there exists a unitary on the environment such
that . Moreover, we provide a
simple example which shows that the factor on the right-hand side is
optimal, and we conjecture that this inequality holds for every pair of
channels.Comment: 8+2 pages, submitted to Quantum Inf. Compu
Progress on the Kretschmann-Schlingemann-Werner Conjecture
Given any pair of quantum channels Φ1, Φ2 such that at least one of them has Kraus
rank one, as well as any respective Stinespring isometries V1, V2, we prove that there
exists a unitary U on the environment such that ∥V1 − (1 ⊗ U )V2∥∞ ≤ p2∥Φ1 − Φ2∥⋄.
Moreover, we provide a simple example which shows that the factor √2 on the right-hand
side is optimal, and we conjecture that this inequality holds for every pair of channels
From Kraus Operators to the Stinespring Form of Quantum Maps: An Alternative Construction for Infinite Dimensions
We present an alternative (constructive) proof of the statement that for
every completely positive, trace-preserving map there exists an
auxiliary Hilbert space in a pure state
as well as a unitary operator on system plus environment such that
equals . The main tool of our proof
is Sz.-Nagy's dilation theorem applied to isometries defined on a subspace. In
our construction, the environment consists of a system of dimension "Kraus rank
of " together with a qubit, the latter only acting as a catalyst. In
contrast, the original proof of Hellwig & Kraus given in the 70s yields an
auxiliary system of dimension "Kraus rank plus one". We conclude by providing
an example which illustrates how the constructions differ from each other.Comment: 9 pages mai
Finite-Dimensional Stinespring Curves Can Approximate Any Dynamics
We generalize the recent result that all analytic quantum dynamics can be
represented exactly as the reduction of unitary dynamics generated by a
time-dependent Hamiltonian. More precisely, we prove that the partial trace
over analytic paths of unitaries can approximate any Lipschitz-continuous
quantum dynamics arbitrarily well. We conclude by discussing potential
improvements and generalizations of these results, their limitations, and the
general challenges one has to overcome when trying to relate dynamics to
quantities on the system-environment level.Comment: 12+7 pages, comments welcom
Strict Positivity and -Majorization
Motivated by quantum thermodynamics we first investigate the notion of strict
positivity, that is, linear maps which map positive definite states to
something positive definite again. We show that strict positivity is decided by
the action on any full-rank state, and that the image of non-strictly positive
maps lives inside a lower-dimensional subalgebra. This implies that the
distance of such maps to the identity channel is lower bounded by one.
The notion of strict positivity comes in handy when generalizing the
majorization ordering on real vectors with respect to a positive vector to
majorization on square matrices with respect to a positive definite matrix .
For the two-dimensional case we give a characterization of this ordering via
finitely many trace norm inequalities and, moreover, investigate some of its
order properties. In particular it admits a unique minimal and a maximal
element. The latter is unique as well if and only if minimal eigenvalue of
has multiplicity one.Comment: Supersedes arXiv:2003.0416
Exploring the Limits of Open Quantum Dynamics II: Gibbs-Preserving Maps from the Perspective of Majorization
Motivated by reachability questions in coherently controlled open quantum
systems coupled to a thermal bath, as well as recent progress in the field of
thermo-/vector--majorization we generalize classical majorization from
unital quantum channels to channels with an arbitrary fixed point of full
rank. Such channels preserve some Gibbs-state and thus play an important role
in the resource theory of quantum thermodynamics, in particular in
thermo-majorization.
Based on this we investigate -majorization on matrices in terms of its
topological and order properties, such as existence of unique maximal and
minimal elements, etc. Moreover we characterize D-majorization in the qubit
case via the trace norm and elaborate on why this is a challenging task when
going beyond two dimensions.Comment: Extended abstract, accepted into MTNS 202