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My Publications
10 records found
Title
Author(s)
Year
A min-entropy uncertainty relation for finite size cryptography
H. Y. N. Ng; M. Berta; S. Wehner
2012
A min-entropy uncertainty relation for finite size cryptography
Type
Online Database
Author
H. Y. N. Ng; M. Berta; S. Wehner
Year
2012
Abstract
Apart from their foundational significance, entropic uncertainty relations play a central role in proving the security of quantum cryptographic protocols. Of particular interest are thereby relations in terms of the smooth min-entropy for BB84 and six-state encodings. Previously, strong uncertainty relations were obtained which are valid in the limit of large block lengths. Here, we prove a new uncertainty relation in terms of the smooth min-entropy that is only marginally less strong, but has the crucial property that it can be applied to rather small block lengths. This paves the way for a practical implementation of many cryptographic protocols. As part of our proof we show tight uncertainty relations for a family of Renyi entropies that may be of independent interest.
The Smooth Entropy Formalism on von Neumann Algebras
M. Berta; F. Furrer; V. B. Scholz
2011
The Smooth Entropy Formalism on von Neumann Algebras
Type
Online Database
Author
M. Berta; F. Furrer; V. B. Scholz
Year
2011
Abstract
We discuss quantum information theoretical concepts on von Neumann algebras and lift the smooth entropy formalism to the most general quantum setting. For the smooth min- and max-entropies we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-dimensional case. We generalize the entropic uncertainty relation with quantum side information of Tomamichel and Renner and sketch possible applications to continuous variable quantum cryptography. In particular, we prove the possibility to perform privacy amplification and classical data compression with quantum side information modeled by a von Neumann algebra. From this we generalize the formula of Renes and Renner characterizing the optimal length of a distillable secure finite-key. We also elaborate on the question when the formalism of von Neumann algebras is of advantage in the description of quantum systems with an infinite number of degrees of freedom.
Even though randomness is an essential resource for many information processing tasks, it is not easily found in nature. The goal of randomness extraction is to distill (almost) perfect randomness from a weak source of randomness. When the source yields a classical string X, many extractor constructions are known. Yet, when considering a physical randomness source, X is itself ultimately the result of a measurement on an underlying quantum system. When characterizing the power of a source to supply randomness it is hence a natural question to ask, how much classical randomness we can extract from a quantum state. To tackle this question we here take on the study of quantum-to-classical randomness extractors (QC-extractors). We provide constructions of QC-extractors based on measurements in a full set of mutually unbiased bases (MUBs), and certain single qubit measurements. As the first application, we show that any QC-extractor gives rise to entropic uncertainty relations with respect to quantum side information. Such relations were previously only known for two measurements. As the second application, we resolve the central open question in the noisy-storage model [Wehner et al., PRL 100, 220502 (2008)] by linking security to the quantum capacity of the adversary's storage device.
A natural question in characterizing the information theoretic power of quantum channels is to ask at what rate entanglement is needed in order to asymptotically simulate a quantum channel in the presence of free classical communication. We call this the entanglement cost of a channel, and prove a formula describing it for all channels. We discuss two applications. Firstly, our result has consequences for the study of the strong converse property of the quantum capacity. More precisely, we show that any coding scheme sending quantum information through a quantum channel at a rate larger than the entanglement cost of the channel is exponentially'bad'in the number of channel uses. Secondly, and independently of the first application, we are able to link the security in the noisy-storage model to a problem of sending quantum rather than classical information through the adversary's storage device. This not only greatly improves the range of parameters where security could be shown previously, but allows us to prove security for storage devices for which no non-trivial statements were known before.
Continuous Variable Quantum Key Distribution: Finite-Key Analysis of Composable Security against Coherent Attacks
F. Furrer; T. Franz; M. Berta; V. B. Scholz; M. Tomamichel; R. F. Werner
2011
Continuous Variable Quantum Key Distribution: Finite-Key Analysis of Composable Security against Coherent Attacks
Type
Online Database
Author
F. Furrer; T. Franz; M. Berta; V. B. Scholz; M. Tomamichel; R. F. Werner
Year
2011
Abstract
We provide a security analysis for continuous variable quantum key distribution protocols based on the transmission of squeezed vacuum states measured via homodyne detection. We employ a version of the entropic uncertainty relation for smooth entropies to give a lower bound on the number of secret bits which can be extracted from a finite number of runs of the protocol. This bound is valid under general coherent attacks, and gives rise to keys which are composably secure. For comparison, we also give a lower bound valid under the assumption of collective attacks. For both scenarios, we find positive key rates using experimental parameters reachable today.
The Quantum Reverse Shannon Theorem based on One-Shot Information Theory
M. Berta; M. Christandl; R. Renner
2011
The Quantum Reverse Shannon Theorem based on One-Shot Information Theory
Type
Journal Article
Author
M. Berta; M. Christandl; R. Renner
Year
2011
Journal
Communications in Mathematical Physics
Abstract
The Quantum Reverse Shannon Theorem states that any quantum channel can be simulated by an unlimited amount of shared entanglement and an amount of classical communication equal to the channel’s entanglement assisted classical capacity. In this paper, we provide a new proof of this theorem, which has previously been proved by Bennett, Devetak, Harrow, Shor, and Winter. Our proof has a clear structure being based on two recent information-theoretic results: one-shot Quantum State Merging and the Post-Selection Technique for quantum channels.
If a quantum system A, which is initially correlated to another system, E, undergoes an evolution separated from E, then the correlation to E generally decreases. Here, we study the conditions under which the correlation disappears completely, resulting in a decoupling of A from E. We give a criterion for decoupling in terms of two smooth entropies, one quantifying the amount of initial correlation between A and E, and the other characterizing the mapping that describes the evolution of A. The criterion applies to arbitrary such mappings and is tight if the mapping satisfies certain natural conditions. Decoupling has a number of applications both in physics and information theory, e.g., as a building block for quantum information processing protocols. As an example, we give a one-shot state merging protocol and show that it is essentially optimal in terms of its entanglement consumption/production.
The uncertainty principle in the presence of quantum memory
M. Berta; M. Christandl; R. Colbeck; J. M. Renes; R. Renner
2010
The uncertainty principle in the presence of quantum memory
Type
Journal Article
Author
M. Berta; M. Christandl; R. Colbeck; J. M. Renes; R. Renner
Year
2010
Journal
Nature Physics
Abstract
The uncertainty principle, originally formulated by Heisenberg, clearly illustrates the difference between classical and quantum mechanics. The principle bounds the uncertainties about the outcomes of two incompatible measurements, such as position and momentum, on a particle. It implies that one cannot predict the outcomes for both possible choices of measurement to arbitrary precision, even if information about the preparation of the particle is available in a classical memory. However, if the particle is prepared entangled with a quantum memory, a device that might be available in the not-too-distant future, it is possible to predict the outcomes for both measurement choices precisely. Here, we extend the uncertainty principle to incorporate this case, providing a lower bound on the uncertainties, which depends on the amount of entanglement between the particle and the quantum memory. We detail the application of our result to witnessing entanglement and to quantum key distribution.
A Conceptually Simple Proof of the Quantum Reverse Shannon Theorem
M. Berta; M. Christandl; R. Renner
2010
A Conceptually Simple Proof of the Quantum Reverse Shannon Theorem
Type
Conference Proceedings
Author
M. Berta; M. Christandl; R. Renner
Year of Conference
2010
Publisher
Springer
Conference Name
Theory of Quantum Computation, Communication, and Cryptography - TQC 2010
Abstract
The Quantum Reverse Shannon Theorem states that any quantum channel can be simulated by an unlimited amount of shared entanglement and an amount of classical communication equal to the channel’s entanglement assisted classical capacity. In this extended ab- stract, we summarize a new and conceptually simple proof of this theorem [journal reference: arXiv.org:quant-ph/0912.3805], which has previously been proved in [Bennett et al., arXiv.org:quant-ph/0912.5537]. Our proof is based on optimal one-shot Quantum State Merging and the Post-Selection Technique for quantum channels.
We consider an unknown quantum state shared between two parties, Alice and Bob, and ask how much quantum communication is needed to transfer the full state to Bob. This problem is known as state merging and was introduced in [Horodecki et al., Nature, 436, 673 (2005)]. It has been shown that for free classical communication the minimal number of quantum bits that need to be sent from Alice to Bob is given by the conditional von Neumann entropy. However this result only holds asymptotically (in the sense that Alice and Bob share initially many identical copies of the state) and it was unclear how much quantum communication is necessary to merge a single copy. We show that the minimal amount of quantum communication needed to achieve this single-shot state merging is given by minus the smooth conditional min-entropy of Alice conditioned on the environment. This gives an operational meaning to the smooth conditional min-entropy.
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