6,171 research outputs found
A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks
We consider two fundamental tasks in quantum information theory, data
compression with quantum side information as well as randomness extraction
against quantum side information. We characterize these tasks for general
sources using so-called one-shot entropies. We show that these
characterizations - in contrast to earlier results - enable us to derive tight
second order asymptotics for these tasks in the i.i.d. limit. More generally,
our derivation establishes a hierarchy of information quantities that can be
used to investigate information theoretic tasks in the quantum domain: The
one-shot entropies most accurately describe an operational quantity, yet they
tend to be difficult to calculate for large systems. We show that they
asymptotically agree up to logarithmic terms with entropies related to the
quantum and classical information spectrum, which are easier to calculate in
the i.i.d. limit. Our techniques also naturally yields bounds on operational
quantities for finite block lengths.Comment: See also arXiv:1208.1400, which independently derives part of our
result: the second order asymptotics for binary hypothesis testin
Non-Asymptotic Analysis of Privacy Amplification via Renyi Entropy and Inf-Spectral Entropy
This paper investigates the privacy amplification problem, and compares the
existing two bounds: the exponential bound derived by one of the authors and
the min-entropy bound derived by Renner. It turns out that the exponential
bound is better than the min-entropy bound when a security parameter is rather
small for a block length, and that the min-entropy bound is better than the
exponential bound when a security parameter is rather large for a block length.
Furthermore, we present another bound that interpolates the exponential bound
and the min-entropy bound by a hybrid use of the Renyi entropy and the
inf-spectral entropy.Comment: 6 pages, 4 figure
Macroscopic thermodynamic reversibility in quantum many-body systems
The resource theory of thermal operations, an established model for small-scale thermodynamics, provides an extension of equilibrium thermodynamics to nonequilibrium situations. On a lattice of any dimension with any translation-invariant local Hamiltonian, we identify a large set of translation-invariant states that can be reversibly converted to and from the thermal state with thermal operations and a small amount of coherence. These are the spatially ergodic states, i.e., states that have sharp statistics for any translation-invariant observable, and mixtures of such states with the same thermodynamic potential. As an intermediate result, we show for a general state that if the gap between the min- and the max-relative entropies to the thermal state is small, then the state can be approximately reversibly converted to and from the thermal state with thermal operations and a small source of coherence. Our proof provides a quantum version of the Shannon-McMillan-Breiman theorem for the relative entropy and a quantum Stein’s lemma for ergodic states and local Gibbs states. Our results provide a strong link between the abstract resource theory of thermodynamics and more realistic physical systems as we achieve a robust and operational characterization of the emergence of a thermodynamic potential in translation-invariant lattice systems
Fundamental Finite Key Limits for One-Way Information Reconciliation in Quantum Key Distribution
The security of quantum key distribution protocols is guaranteed by the laws
of quantum mechanics. However, a precise analysis of the security properties
requires tools from both classical cryptography and information theory. Here,
we employ recent results in non-asymptotic classical information theory to show
that one-way information reconciliation imposes fundamental limitations on the
amount of secret key that can be extracted in the finite key regime. In
particular, we find that an often used approximation for the information
leakage during information reconciliation is not generally valid. We propose an
improved approximation that takes into account finite key effects and
numerically test it against codes for two probability distributions, that we
call binary-binary and binary-Gaussian, that typically appear in quantum key
distribution protocols
Second-order coding rates for pure-loss bosonic channels
A pure-loss bosonic channel is a simple model for communication over
free-space or fiber-optic links. More generally, phase-insensitive bosonic
channels model other kinds of noise, such as thermalizing or amplifying
processes. Recent work has established the classical capacity of all of these
channels, and furthermore, it is now known that a strong converse theorem holds
for the classical capacity of these channels under a particular photon number
constraint. The goal of the present paper is to initiate the study of
second-order coding rates for these channels, by beginning with the simplest
one, the pure-loss bosonic channel. In a second-order analysis of
communication, one fixes the tolerable error probability and seeks to
understand the back-off from capacity for a sufficiently large yet finite
number of channel uses. We find a lower bound on the maximum achievable code
size for the pure-loss bosonic channel, in terms of the known expression for
its capacity and a quantity called channel dispersion. We accomplish this by
proving a general "one-shot" coding theorem for channels with classical inputs
and pure-state quantum outputs which reside in a separable Hilbert space. The
theorem leads to an optimal second-order characterization when the channel
output is finite-dimensional, and it remains an open question to determine
whether the characterization is optimal for the pure-loss bosonic channel.Comment: 18 pages, 3 figures; v3: final version accepted for publication in
Quantum Information Processin
Finite-key security analysis for multilevel quantum key distribution
We present a detailed security analysis of a d-dimensional quantum key
distribution protocol based on two and three mutually unbiased bases (MUBs)
both in an asymptotic and finite key length scenario. The finite secret key
rates are calculated as a function of the length of the sifted key by (i)
generalizing the uncertainly relation-based insight from BB84 to any d-level
2-MUB QKD protocol and (ii) by adopting recent advances in the second-order
asymptotics for finite block length quantum coding (for both d-level 2- and
3-MUB QKD protocols). Since the finite and asymptotic secret key rates increase
with d and the number of MUBs (together with the tolerable threshold) such QKD
schemes could in principle offer an important advantage over BB84. We discuss
the possibility of an experimental realization of the 3-MUB QKD protocol with
the orbital angular momentum degrees of freedom of photons.Comment: v4: close to the published versio
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