119 research outputs found
The Convex Closure of the Output Entropy of Infinite Dimensional Channels and the Additivity Problem
The continuity properties of the convex closure of the output entropy of
infinite dimensional channels and their applications to the additivity problem
are considered.
The main result of this paper is the statement that the superadditivity of
the convex closure of the output entropy for all finite dimensional channels
implies the superadditivity of the convex closure of the output entropy for all
infinite dimensional channels, which provides the analogous statements for the
strong superadditivity of the EoF and for the additivity of the minimal output
entropy.
The above result also provides infinite dimensional generalization of Shor's
theorem stated equivalence of different additivity properties.
The superadditivity of the convex closure of the output entropy (and hence
the additivity of the minimal output entropy) for two infinite dimensional
channels with one of them a direct sum of noiseless and entanglement-breaking
channels are derived from the corresponding finite dimensional results.
In the context of the additivity problem some observations concerning
complementary infinite dimensional channels are considered.Comment: 24 page
A generalization of the Entropy Power Inequality to Bosonic Quantum Systems
In most communication schemes information is transmitted via travelling modes
of electromagnetic radiation. These modes are unavoidably subject to
environmental noise along any physical transmission medium and the quality of
the communication channel strongly depends on the minimum noise achievable at
the output. For classical signals such noise can be rigorously quantified in
terms of the associated Shannon entropy and it is subject to a fundamental
lower bound called entropy power inequality. Electromagnetic fields are however
quantum mechanical systems and then, especially in low intensity signals, the
quantum nature of the information carrier cannot be neglected and many
important results derived within classical information theory require
non-trivial extensions to the quantum regime. Here we prove one possible
generalization of the Entropy Power Inequality to quantum bosonic systems. The
impact of this inequality in quantum information theory is potentially large
and some relevant implications are considered in this work
Quantum concentration inequalities
We establish transportation cost inequalities (TCI) with respect to the
quantum Wasserstein distance by introducing quantum extensions of well-known
classical methods: first, using a non-commutative version of Ollivier's coarse
Ricci curvature, we prove that high temperature Gibbs states of commuting
Hamiltonians on arbitrary hypergraphs satisfy a TCI with constant
scaling as . Second, we argue that the temperature range for which the
TCI holds can be enlarged by relating it to recently established modified
logarithmic Sobolev inequalities. Third, we prove that the inequality still
holds for fixed points of arbitrary reversible local quantum Markov semigroups
on regular lattices, albeit with slightly worsened constants, under a seemingly
weaker condition of local indistinguishability of the fixed points. Finally, we
use our framework to prove Gaussian concentration bounds for the distribution
of eigenvalues of quasi-local observables and argue the usefulness of the TCI
in proving the equivalence of the canonical and microcanonical ensembles and an
exponential improvement over the weak Eigenstate Thermalization Hypothesis.Comment: 31 pages, one figur
Quantum Concentration Inequalities
We establish Transportation Cost Inequalities (TCIs) with respect to the quantum Wasserstein distance by introducing quantum extensions of well-known classical methods: First, we generalize the Dobrushin uniqueness condition to prove that Gibbs states of 1D commuting Hamiltonians satisfy a TCI at any positive temperature and provide conditions under which this first result can be extended to non-commuting Hamiltonians. Next, using a non-commutative version of Ollivier’s coarse Ricci curvature, we prove that high temperature Gibbs states of commuting Hamiltonians on arbitrary hypergraphs H= (V, E) satisfy a TCI with constant scaling as O(|V|). Third, we argue that the temperature range for which the TCI holds can be enlarged by relating it to recently established modified logarithmic Sobolev inequalities. Fourth, we prove that the inequality still holds for fixed points of arbitrary reversible local quantum Markov semigroups on regular lattices, albeit with slightly worsened constants, under a seemingly weaker condition of local indistinguishability of the fixed points. Finally, we use our framework to prove Gaussian concentration bounds for the distribution of eigenvalues of quasi-local observables and argue the usefulness of the TCI in proving the equivalence of the canonical and microcanonical ensembles and an exponential improvement over the weak Eigenstate Thermalization Hypothesis
Classical communication over a quantum interference channel
Calculating the capacity of interference channels is a notorious open problem in classical information theory. Such channels have two senders and two receivers, and each sender would like to communicate with a partner receiver. The capacity of such channels is known exactly in the settings of very strong and strong interference, while the Han-Kobayashi coding strategy gives the best known achievable rate region in the general case. Here, we introduce and study the quantum interference channel, a natural generalization of the interference channel to the setting of quantum information theory. We restrict ourselves for the most part to channels with two classical inputs and two quantum outputs in order to simplify the presentation of our results (though generalizations of our results to channels with quantum inputs are straightforward). We are able to determine the exact classical capacity of this channel in the settings of very strong and strong interference, by exploiting Winter\u27s successive decoding strategy and a novel two-sender quantum simultaneous decoder, respectively. We provide a proof that a Han-Kobayashi strategy is achievable with Holevo information rates, up to a conjecture regarding the existence of a three-sender quantum simultaneous decoder. This conjecture holds for a special class of quantum multiple-access channels with average output states that commute, and we discuss some other variations of the conjecture that hold. Finally, we detail a connection between the quantum interference channel and prior work on the capacity of bipartite unitary gates. © 2012 IEEE
Classical codes for quantum broadcast channels
We present two approaches for transmitting classical information over quantum
broadcast channels. The first technique is a quantum generalization of the
superposition coding scheme for the classical broadcast channel. We use a
quantum simultaneous nonunique decoder and obtain a proof of the rate region
stated in [Yard et al., IEEE Trans. Inf. Theory 57 (10), 2011]. Our second
result is a quantum generalization of the Marton coding scheme. The error
analysis for the quantum Marton region makes use of ideas in our earlier work
and an idea recently presented by Radhakrishnan et al. in arXiv:1410.3248. Both
results exploit recent advances in quantum simultaneous decoding developed in
the context of quantum interference channels.Comment: v4: 20 pages, final version to appear in IEEE Transactions on
Information Theor
Classical codes for quantum broadcast channels
We discuss two techniques for transmitting classical information over quantum broadcast channels. The first technique is a quantum generalization of the superposition coding scheme for the classical broadcast channel. We use a quantum simultaneous nonunique decoder and obtain a simpler proof of the rate region recently published by Yard et al. in independent work. Our second result is a quantum Marton coding scheme, which gives the best known achievable rate region for quantum broadcast channels. Both results exploit recent advances in quantum simultaneous decoding developed in the context of quantum interference channels. © 2012 IEEE
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