40 research outputs found
On Weyl channels being covariant with respect to the maximum commutative group of unitaries
We investigate the Weyl channels being covariant with respect to the maximum
commutative group of unitary operators. This class includes the quantum
depolarizing channel and the "two-Pauli" channel as well. Then, we show that
our estimation of the output entropy for a tensor product of the phase damping
channel and the identity channel based upon the decreasing property of the
relative entropy allows to prove the additivity conjecture for the minimal
output entropy for the quantum depolarizing channel in any prime dimesnsion and
for the "two Pauli" channel in the qubit case.Comment: A completely revised version, 20 page
Counterexample to an additivity conjecture for output purity of quantum channels
A conjecture arising naturally in the investigation of additivity of
classical information capacity of quantum channels states that the maximal
purity of outputs from a quantum channel, as measured by the p-norm, should be
multiplicative with respect to the tensor product of channels. We disprove this
conjecture for p>4.79. The same example (with p=infinity) also disproves a
conjecture for the multiplicativity of the injective norm of Hilbert space
tensor products.Comment: 3 pages, 3 figures, revte
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
Multiplicativity of maximal output purities of Gaussian channels under Gaussian inputs
We address the question of the multiplicativity of the maximal p-norm output
purities of bosonic Gaussian channels under Gaussian inputs. We focus on
general Gaussian channels resulting from the reduction of unitary dynamics in
larger Hilbert spaces. It is shown that the maximal output purity of tensor
products of single-mode channels under Gaussian inputs is multiplicative for
any p>1 for products of arbitrary identical channels as well as for a large
class of products of different channels. In the case of p=2 multiplicativity is
shown to be true for arbitrary products of generic channels acting on any
number of modes.Comment: 9 page
Transmitting qudits through larger quantum channels
We address the problem of transmitting states belonging to finite dimensional
Hilbert space through a quantum channel associated with a larger (even infinite
dimensional) Hilbert space.Comment: 5 pages, ReVTeX, minor changes, to appear in J. Phys.
Notes on multiplicativity of maximal output purity for completely positive qubit maps
A problem in quantum information theory that has received considerable
attention in recent years is the question of multiplicativity of the so-called
maximal output purity (MOP) of a quantum channel. This quantity is defined as
the maximum value of the purity one can get at the output of a channel by
varying over all physical input states, when purity is measured by the Schatten
-norm, and is denoted by . The multiplicativity problem is the
question whether two channels used in parallel have a combined that is
the product of the of the two channels. A positive answer would imply a
number of other additivity results in QIT.
Very recently, P. Hayden has found counterexamples for every value of .
Nevertheless, these counterexamples require that the dimension of these
channels increases with and therefore do not rule out multiplicativity
for in intervals with depending on the channel dimension. I
argue that this would be enough to prove additivity of entanglement of
formation and of the classical capacity of quantum channels.
More importantly, no counterexamples have as yet been found in the important
special case where one of the channels is a qubit-channel, i.e. its input
states are 2-dimensional. In this paper I focus attention to this qubit case
and I rephrase the multiplicativity conjecture in the language of block
matrices and prove the conjecture in a number of special cases.Comment: Manuscript for a talk presented at the SSPCM07 conference in
Myczkowce, Poland, 10/09/2007. 12 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
Conditions for the multiplicativity of maximal l_p-norms of channels for fixed integer p
We introduce a condition for memoryless quantum channels which, when
satisfied guarantees the multiplicativity of the maximal l_p-norm with p a
fixed integer. By applying the condition to qubit channels, it can be shown
that it is not a necessary condition, although some known results for qubits
can be recovered. When applied to the Werner-Holevo channel, which is known to
violate multiplicativity when p is large relative to the dimension d, the
condition suggests that multiplicativity holds when . This
conjecture is proved explicitly for p=2, 3, 4. Finally, a new class of channels
is considered which generalizes the depolarizing channel to maps which are
combinations of the identity channel and a noisy one whose image is an
arbitrary density matrix. It is shown that these channels are multiplicative
for p = 2.Comment: 20 pages: new material added including additional details supporting
the conjecture that the Werner-Holevo channel is multiplicative for all p
when d > 2^{p-1}; typos corrected. To appear on JM