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DOI: A direct computation of these exponents is however often infeasible. By establishing a link between Lyapunov exponents and an information theoretic tool called entropy accumulation theorem we derive an upper and a lower bound for the maximal and minimal Lyapunov exponent, respectively.

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The bounds assume independence of the random matrices, are analytical, and are tight in the commutative case as well as in other scenarios. They can be expressed in terms of an optimization problem that only involves single matrices rather than large products. The upper bound for the maximal Lyapunov exponent can be evaluated efficiently via the theory of convex optimization.

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Lim arXiv, pp. Here, we provide a general finite-size security proof that can simultaneously encompass these approaches, using tighter finite-size bounds than previous analyses.

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In doing so, we develop a method to compute tight lower bounds on the asymptotic keyrate for any such DIQKD protocol with binary inputs and outputs. With this, we show that positive asymptotic keyrates are achievable up to depolarizing noise values of 9. We also develop a modification to random-key-measurement protocols, using a pre-shared seed followed by a "seed recovery" step, which yields peev bitcoin higher net key generation rates by essentially removing the sifting factor.

Some of our results may also improve the keyrates of device-independent randomness expansion.

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While no perfect universal quantum processor can exist, approximate processors have been proposed in the past two decades.

A fundamental open question is how the size of the smallest quantum program scales with the approximation error. Here we answer the question, by proving a bound on the size of the program and designing a concrete protocol that attains the bound in the asymptotic limit.

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Our result is based on a connection between optimal programming and the Heisenberg limit of quantum metrology, and establishes an asymptotic equivalence between the tasks of programming, learning, and estimating unitary gates. The Asymptotic Equipartition Property implies that this is indeed the case to first order in n—under the assumption that the parts Ai are identical and independent of each other.


Here we show that entropy accumulation occurs more generally, i. The analysis of a large system can hence be reduced to the study of its parts.

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This is relevant for applications. In device-independent cryptography, for instance, the approach yields essentially optimal security bounds valid for general attacks, as shown by Arnon-Friedman et al.

Here we study the relation between the quality of a time signal and the physics of the system that generates it. According to quantum theory, any time signal can be decomposed into individual quanta that lead to single detection events. Our main result is a bound on peev bitcoin sharply peaked in time these events can be, which depends on the dimension of the signal pionex robot. This result promises applications in various directions, including information peev bitcoin, quantum clocks, and process simulation.

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We have also found minor non-random contributions from imperfections in detector electronics and an internal processing algorithm. Our work shows that the design quality of a commercial quantum-optical randomness source can be verified without cooperation of the manufacturer and without access to the engineering documentation. Tan, Charles C. Trenkwalder, Henrik Wilming, Hans J. Briegel and Renato Renner arXiv, pp. The representations learnt by most current machine learning techniques reflect statistical structure present in the training data; however, these methods do not allow us to specify explicit and operationally meaningful requirements on peev bitcoin representation.

Here, we present a neural peev bitcoin architecture based on the notion that agents dealing with different aspects of a physical system should be able to communicate relevant information as efficiently as possible to one another. This produces representations that separate different parameters which are useful for making statements about the physical system in different experimental settings. We present examples involving both classical and quantum physics.

For instance, our architecture finds a compact peev bitcoin of an arbitrary two-qubit system that separates local parameters from parameters describing quantum correlations.

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We further show that this method can be combined with reinforcement peev bitcoin to enable representation learning within interactive scenarios where agents need to explore experimental settings to identify relevant variables. Are quantum cryptographic security claims vacuous?

Joseph M. Renes and Renato Renner arXiv, pp.

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This claim has recently been challenged by Bernstein arXiv He argues that the laws of physics may also entail an unavoidable leakage of any classical information encoded in physical carriers. The security claim of quantum key distribution would then be vacuous, as the computation of the final secret key would leak its value.

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However, as we explain in this short note, Bernstein's reasoning is based on a too "classical" understanding of physics. It follows from known theorems about fault-tolerant quantum computation that quantum physics avoids his conclusion.

In the simplest scenario, these guarantees are derived from the CHSH score, which is a simple linear combination of four correlation functions. We here derive a security proof from a generalisation of the CHSH score, which effectively takes into account the individual values of two correlation functions.

We show that this additional information, which is anyway available in practice, allows one to get higher key rates than with the CHSH score. We discuss the potential advantage of this technique for realistic photonic implementations of device-independent quantum key distribution.

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