22 research outputs found
The strong converse theorem for the product-state capacity of quantum channels with ergodic Markovian memory
Establishing the strong converse theorem for a communication channel confirms
that the capacity of that channel, that is, the maximum achievable rate of
reliable information communication, is the ultimate limit of communication over
that channel. Indeed, the strong converse theorem for a channel states that
coding at a rate above the capacity of the channel results in the convergence
of the error to its maximum value 1 and that there is no trade-off between
communication rate and decoding error. Here we prove that the strong converse
theorem holds for the product-state capacity of quantum channels with ergodic
Markovian correlated memory.Comment: 11 pages, single colum
Asymptotic Feynman-Kac formulae for large symmetrised systems of random walks
We study large deviations principles for random processes on the
lattice with finite time horizon under a symmetrised
measure where all initial and terminal points are uniformly given by a random
permutation. That is, given a permutation of elements and a
vector of initial points we let the random processes
terminate in the points and then sum over
all possible permutations and initial points, weighted with an initial
distribution. There is a two-level random mechanism and we prove two-level
large deviations principles for the mean of empirical path measures, for the
mean of paths and for the mean of occupation local times under this symmetrised
measure. The symmetrised measure cannot be written as any product of single
random process distributions. We show a couple of important applications of
these results in quantum statistical mechanics using the Feynman-Kac formulae
representing traces of certain trace class operators. In particular we prove a
non-commutative Varadhan Lemma for quantum spin systems with Bose-Einstein
statistics and mean field interactions.
A special case of our large deviations principle for the mean of occupation
local times of simple random walks has the Donsker-Varadhan rate function
as the rate function for the limit but for finite time . We give an interpretation in quantum statistical mechanics for this
surprising result
Correlation of clusters: Partially truncated correlation functions and their decay
In this article, we investigate partially truncated correlation functions
(PTCF) of infinite continuous systems of classical point particles with pair
interaction. We derive Kirkwood-Salsburg-type equations for the PTCF and write
the solutions of these equations as a sum of contributions labelled by certain
forests graphs, the connected components of which are tree graphs. We
generalize the method introduced by R.A. Minlos and S.K. Poghosyan (1977) in
the case of truncated correlations. These solutions make it possible to derive
strong cluster properties for PTCF which were obtained earlier for lattice spin
systems.Comment: 31 pages, 2 figures. 2nd revision. Misprints corrected and 1 figure
adde
The classical capacity of quantum channels with memory
We investigate the classical capacity of two quantum channels with memory: a
periodic channel with depolarizing channel branches, and a convex combination
of depolarizing channels. We prove that the capacity is additive in both cases.
As a result, the channel capacity is achieved without the use of entangled
input states. In the case of a convex combination of depolarizing channels the
proof provided can be extended to other quantum channels whose classical
capacity has been proved to be additive in the memoryless case.Comment: 6 double-column pages. Short note added on quantum memory channel
The Efficiency of Feynman's Quantum Computer
Feynman's circuit-to-Hamiltonian construction enables the mapping of a
quantum circuit to a time-independent Hamiltonian. Here we investigate the
efficiency of Feynman's quantum computer by analysing the time evolution
operator for Feynman's clock Hamiltonian . A general
formula is established for the probability, , that the desired
computation is complete at time for a quantum computer which executes an
arbitrary number of operations. The optimal stopping time, denoted by
, is defined as the time of the first local maximum of this probability.
We find numerically that there is a linear relationship between this optimal
stopping time and the number of operations, .
Theoretically, we corroborate this linear behaviour by showing that at , is approximately maximal. We also establish a
relationship between and in the limit of a large number
of operations. We show analytically that at the maximum, behaves
like . This is further proven numerically where we find the inverse
cubic root relationship . This is significantly
more efficient than paradigmatic models of quantum computation.Comment: 6 pages, 5 figure
The invalidity of a strong capacity for a quantum channel with memory
The strong capacity of a particular channel can be interpreted as a sharp
limit on the amount of information which can be transmitted reliably over that
channel. To evaluate the strong capacity of a particular channel one must prove
both the direct part of the channel coding theorem and the strong converse for
the channel. Here we consider the strong converse theorem for the periodic
quantum channel and show some rather surprising results. We first show that the
strong converse does not hold in general for this channel and therefore the
channel does not have a strong capacity. Instead, we find that there is a scale
of capacities corresponding to error probabilities between integer multiples of
the inverse of the periodicity of the channel. A similar scale also exists for
the random channel.Comment: 7 pages, double column. Comments welcome. Repeated equation removed
and one reference adde