59,662 research outputs found
The structure of degradable quantum channels
Degradable quantum channels are among the only channels whose quantum and
private classical capacities are known. As such, determining the structure of
these channels is a pressing open question in quantum information theory. We
give a comprehensive review of what is currently known about the structure of
degradable quantum channels, including a number of new results as well as
alternate proofs of some known results. In the case of qubits, we provide a
complete characterization of all degradable channels with two dimensional
output, give a new proof that a qubit channel with two Kraus operators is
either degradable or anti-degradable and present a complete description of
anti-degradable unital qubit channels with a new proof.
For higher output dimensions we explore the relationship between the output
and environment dimensions ( and respectively) of degradable
channels. For several broad classes of channels we show that they can be
modeled with a environment that is "small" in the sense . Perhaps
surprisingly, we also present examples of degradable channels with ``large''
environments, in the sense that the minimal dimension . Indeed, one
can have .
In the case of channels with diagonal Kraus operators, we describe the
subclass which are complements of entanglement breaking channels. We also
obtain a number of results for channels in the convex hull of conjugations with
generalized Pauli matrices. However, a number of open questions remain about
these channels and the more general case of random unitary channels.Comment: 42 pages, 3 figures, Web and paper abstract differ; (v2 contains only
minor typo corrections
An application of decomposable maps in proving multiplicativity of low dimensional maps
In this paper we present a class of maps for which the multiplicativity of
the maximal output p-norm holds when p is 2 and p is larger than or equal to 4.
The class includes all positive trace-preserving maps from the matrix algebra
on the three-dimensional space to that on the two-dimensional.Comment: 9 page
Maximization of capacity and p-norms for some product channels
It is conjectured that the Holevo capacity of a product channel \Omega
\otimes \Phi is achieved when product states are used as input. Amosov, Holevo
and Werner have also conjectured that the maximal p-norm of a product channel
is achieved with product input states. In this paper we establish both of these
conjectures in the case that \Omega is arbitrary and \Phi is a CQ or QC channel
(as defined by Holevo). We also establish the Amosov, Holevo and Werner
conjecture when \Omega is arbitrary and either \Phi is a qubit channel and p=2,
or \Phi is a unital qubit channel and p is integer. Our proofs involve a new
conjecture for the norm of an output state of the half-noisy channel I \otimes
\Phi, when \Phi is a qubit channel. We show that this conjecture in some cases
also implies additivity of the Holevo capacity
Qubit Channels Can Require More Than Two Inputs to Achieve Capacity
We give examples of qubit channels that require three input states in order
to achieve the Holevo capacity.Comment: RevTex, 5 page, 4 figures
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