67 research outputs found
Decomposability of Linear Maps under Tensor Products
Both completely positive and completely copositive maps stay decomposable
under tensor powers, i.e under tensoring the linear map with itself. But are
there other examples of maps with this property? We show that this is not the
case: Any decomposable map, that is neither completely positive nor completely
copositive, will lose decomposability eventually after taking enough tensor
powers. Moreover, we establish explicit bounds to quantify when this happens.
To prove these results we use a symmetrization technique from the theory of
entanglement distillation, and analyze when certain symmetric maps become
non-decomposable after taking tensor powers. Finally, we apply our results to
construct new examples of non-decomposable positive maps, and establish a
connection to the PPT squared conjecture.Comment: 26 pages, 3 figure
All unital qubit channels are -noisy operations
We show that any unital qubit channel can be implemented by letting the input
system interact unitarily with a -dimensional environment in the maximally
mixed state and then tracing out the environment. We also provide an example
where the dimension of such an environment has to be at least .Comment: 8 pages, no picture
On the monotonicity of a quantum optimal transport cost
We show that the quantum generalization of the -Wasserstein distance
proposed by Chakrabarti et al. is not monotone under partial traces. This
disproves a recent conjecture by Friedland et al. Finally, we propose a
stabilized version of the original definition, which we show to be monotone
under the application of general quantum channels.Comment: 9 pages. Comments are welcom
Cutting cakes and kissing circles
To divide a cake into equal sized pieces most people use a knife and a
mixture of luck and dexterity. These attempts are often met with varying
success. Through precise geometric constructions performed with the knife
replacing Euclid's straightedge and without using a compass we find methods for
solving certain cake-cutting problems exactly. Since it is impossible to
exactly bisect a circular cake when its center is not known, our constructions
need to use multiple cakes. Using three circular cakes we present a simple
method for bisecting each of them or to find their centers. Moreover, given a
cake with marked center we present methods to cut it into n pieces of equal
size for n=3,4 and 6. Our methods are based upon constructions by Steiner and
Cauer from the 19th and early 20th century.Comment: 9 pages, 11 figures. Simplified proof of main resul
Entropy Production of Doubly Stochastic Quantum Channels
We study the entropy increase of quantum systems evolving under primitive,
doubly stochastic Markovian noise and thus converging to the maximally mixed
state. This entropy increase can be quantified by a logarithmic-Sobolev
constant of the Liouvillian generating the noise. We prove a universal lower
bound on this constant that stays invariant under taking tensor-powers. Our
methods involve a new comparison method to relate logarithmic-Sobolev constants
of different Liouvillians and a technique to compute logarithmic-Sobolev
inequalities of Liouvillians with eigenvectors forming a projective
representation of a finite abelian group. Our bounds improve upon similar
results established before and as an application we prove an upper bound on
continuous-time quantum capacities. In the last part of this work we study
entropy production estimates of discrete-time doubly-stochastic quantum
channels by extending the framework of discrete-time logarithmic-Sobolev
inequalities to the quantum case.Comment: 24 page
Relative Entropy Convergence for Depolarizing Channels
We study the convergence of states under continuous-time depolarizing
channels with full rank fixed points in terms of the relative entropy. The
optimal exponent of an upper bound on the relative entropy in this case is
given by the log-Sobolev-1 constant. Our main result is the computation of this
constant. As an application we use the log-Sobolev-1 constant of the
depolarizing channels to improve the concavity inequality of the von-Neumann
entropy. This result is compared to similar bounds obtained recently by Kim et
al. and we show a version of Pinsker's inequality, which is optimal and tight
if we fix the second argument of the relative entropy. Finally, we consider the
log-Sobolev-1 constant of tensor-powers of the completely depolarizing channel
and use a quantum version of Shearer's inequality to prove a uniform lower
bound.Comment: 21 pages, 3 figure
Bi-PPT channels are entanglement breaking
In a recent paper, Hirche and Leditzky introduced the notion of bi-PPT
channels which are quantum channels that stay completely positive under
composition with a transposition and such that the same property holds for one
of their complementary channels. They asked whether there are examples of such
channels that are not antidegradable. We show that this is not the case, since
bi-PPT channels are always entanglement breaking. We also show that degradable
quantum channels staying completely positive under composition with a
transposition are entanglement breaking
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