321 research outputs found
Comment on "Optimum Quantum Error Recovery using Semidefinite Programming"
In a recent paper ([1]=quant-ph/0606035) it is shown how the optimal recovery
operation in an error correction scheme can be considered as a semidefinite
program. As a possible future improvement it is noted that still better error
correction might be obtained by optimizing the encoding as well. In this note
we present the result of such an improvement, specifically for the four-bit
correction of an amplitude damping channel considered in [1]. We get a strict
improvement for almost all values of the damping parameter. The method (and the
computer code) is taken from our earlier study of such correction schemes
(quant-ph/0307138).Comment: 2 pages, 1 figur
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
Effect of nonnegativity on estimation errors in one-qubit state tomography with finite data
We analyze the behavior of estimation errors evaluated by two loss functions,
the Hilbert-Schmidt distance and infidelity, in one-qubit state tomography with
finite data. We show numerically that there can be a large gap between the
estimation errors and those predicted by an asymptotic analysis. The origin of
this discrepancy is the existence of the boundary in the state space imposed by
the requirement that density matrices be nonnegative (positive semidefinite).
We derive an explicit form of a function reproducing the behavior of the
estimation errors with high accuracy by introducing two approximations: a
Gaussian approximation of the multinomial distributions of outcomes, and
linearizing the boundary. This function gives us an intuition for the behavior
of the expected losses for finite data sets. We show that this function can be
used to determine the amount of data necessary for the estimation to be treated
reliably with the asymptotic theory. We give an explicit expression for this
amount, which exhibits strong sensitivity to the true quantum state as well as
the choice of measurement.Comment: 9 pages, 4 figures, One figure (FIG. 1) is added to the previous
version, and some typos are correcte
Binegativity and geometry of entangled states in two qubits
We prove that the binegativity is always positive for any two-qubit state. As
a result, as suggested by the previous works, the asymptotic relative entropy
of entanglement in two qubits does not exceed the Rains bound, and the
PPT-entanglement cost for any two-qubit state is determined to be the
logarithmic negativity of the state. Further, the proof reveals some
geometrical characteristics of the entangled states, and shows that the partial
transposition can give another separable approximation of the entangled state
in two qubits.Comment: 5 pages, 3 figures. I made the proof more transparen
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