Hanna Wojewódka

Realistic noise-tolerant randomness amplification using finite number of devices.

Randomness is a fundamental concept, with implications from security of modern data systems, to fundamental laws of nature and even the philosophy of science. Randomness is called certified if it describes events that cannot be pre-determined by an external adversary. It is known that weak certified randomness can be amplified to nearly ideal randomness using quantum-mechanical systems. However, so far, it was unclear whether randomness amplification is a realistic task, as the existing proposals either do not tolerate noise or require an unbounded number of different devices. Here we provide an error-tolerant protocol using a finite number of devices for amplifying arbitrary weak randomness into nearly perfect random bits, which are secure against a no-signalling adversary. The correctness of the protocol is assessed by violating a Bell inequality, with the degree of violation determining the noise tolerance threshold. An experimental realization of the protocol is within reach of current technology.Fernando G.S.L. Brandão,1 Ravishankar Ramanathan,2, 3 Andrzej Grudka,4 Karol Horodecki,2, 5 Michał Horodecki,2, 3 and Paweł Horodecki2, 6 1Department of Computer Science, University College London 2National Quantum Information Center of Gdańsk, 81-824 Sopot, Poland 3Institute of Theoretical Physics and Astrophysics, University of Gdańsk, 80-952 Gdańsk, Poland 4Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland 5Institute of Informatics, University of Gdańsk, 80-952 Gdańsk, Poland 6Faculty of Applied Physics and Mathematics, Technical University of Gdańsk, 80-233 Gdańsk, Poland (Dated: October 18, 2013)

Randomness amplification against no-signaling adversaries using two devices

Ravishankar Ramanathan,1, 2 Fernando G.S.L. Brandão,3, 4 Karol Horodecki,1, 5 Michał Horodecki,1, 2 Paweł Horodecki,1, 6 and Hanna Wojewódka1, 2, 7 National Quantum Information Center of Gdańsk, 81-824 Sopot, Poland Institute of Theoretical Physics and Astrophysics, University of Gdańsk, 80-952 Gdańsk, Poland Quantum Architectures and Computations Group, Microsoft Research, Redmond, WA (USA) Department of Computer Science, University College London Institute of Informatics, University of Gdańsk, 80-952 Gdańsk, Poland Faculty of Applied Physics and Mathematics, Technical University of Gdańsk, 80-233 Gdańsk, Poland Institute of Mathematics, University of Gdańsk, 80-952 Gdańsk, Poland (Dated: April 24, 2015)

Exponential rate of convergence for some Markov operators

Abstract The exponential rate of convergence for Markov operators is established. The operators correspond to continuous iterated function systems which are a very useful tool in some cell cycle models.

Limit theorems for some Markov chains

Abstract The exponential rate of convergence and the Central Limit Theorem for some Markov operators are established. These operators were efficiently used in some biological models (see Hille, Horbacz & Szarek (2015) [8] ), which generalize the cell cycle model given by Lasota & Mackey (1999) [12] .

Amplifying the Randomness of Weak Sources Correlated With Devices

The problem of device-independent randomness amplification against no-signaling adversaries has so far been studied under the assumption that the weak source of randomness is uncorrelated with the (quantum) devices used in the amplification procedure. In this paper, we relax this assumption, and reconsider the original protocol of Colbeck and Renner using a Santha-Vazirani (SV) source. To do so, we introduce an SV-like condition for devices, namely that any string of SV source bits remains weakly random conditioned upon any other bit string from the same SV source and the outputs obtained when this further string is input into the devices. Assuming this condition, we show that a quantum device using a singlet state to violate the chained Bell inequalities leads to full randomness in the asymptotic scenario of a large number of settings, for a restricted set of SV sources (with

Law of the iterated logarithm for some Markov operators

0 \leq \varepsilon < (2^{(1/12)} - 1)/(2(2^{(1/12)} + 1)) \approx 0.0144

Limit theorems for certain stochastic models for gene regulatory networks

). We also study a device-independent protocol that allows for correlations between the sequence of boxes used in the protocol and the SV source bits used to choose the particular box from whose output the randomness is obtained. Assuming the SV-like condition for devices, we show that the honest parties can achieve amplification of the weak source, for the parameter range

The strong law of large numbers for certain piecewise-deterministic Markov processes with application to a gene expression model

0 \leq \varepsilon <0.0132

A Useful Version of the Central Limit Theorem for a General Class of Markov Chains.

, against a class of attacks given as a mixture of product box sequences, made of extremal no-signaling boxes, with additional symmetry conditions. Composable security proof against this class of attacks is provided.

The Strassen Invariance Principle for Certain Non-stationary Markov-Feller Chains.

The Law of the Iterated Logarithm for some Markov operators, which converge exponentially to the invariant measure, is established. The operators correspond to iterated function systems which, for example, may be used to generalize the cell cycle model examined by A. Lasota and M.C. Mackey, J. Math. Biol. (1999).