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The Data Compression Theorem for Ergodic Quantum Information Sources [article]

Igor Bjelakovic, Arleta Szkola
2003 arXiv   pre-print
We extend the data compression theorem to the case of ergodic quantum information sources. Moreover, we provide an asymptotically optimal compression scheme which is based on the concept of high probability subspaces. The rate of this compression scheme is equal to the von Neumann entropy rate.
arXiv:quant-ph/0301043v1 fatcat:ideltecajfdwneqijwijat7vze

Asymptotically Optimal Discrimination between Pure Quantum States [chapter]

Michael Nussbaum, Arleta Szkoła
2011 Lecture Notes in Computer Science  
We consider the decision problem between a finite number of states of a finite quantum system, when an arbitrarily large number of copies of the system is available for measurements. We provide an upper bound on the exponential rate of decay of the averaged probability of rejecting the true state. It represents a generalized quantum Chernoff distance of a finite set of states. As our main result we prove that the bound is sharp in the case of pure states.
doi:10.1007/978-3-642-18073-6_1 fatcat:3ikyzvcfhrfa5i3dsh43lah654

The Data Compression Theorem for Ergodic Quantum Information Sources

Igor Bjelaković, Arleta Szkola
2005 Quantum Information Processing  
We extend the data compression theorem to the case of ergodic quantum information sources. Moreover, we provide an asymptotically optimal compression scheme which is based on the concept of high probability subspaces. The rate of this compression scheme is equal to the von Neumann entropy rate. *
doi:10.1007/s11128-003-3195-1 fatcat:bj4uxclth5ewrguv7h5t4wcloy

Chained Typical Subspaces - a Quantum Version of Breiman's Theorem [article]

Igor Bjelakovic, Tyll Krueger, Rainer Siegmund-Schultze, Arleta Szkola
2003 arXiv   pre-print
We give an equivalent finitary reformulation of the classical Shannon-McMillan-Breiman theorem which has an immediate translation to the case of ergodic quantum lattice systems. This version of a quantum Breiman theorem can be derived from the proof of the quantum Shannon-McMillan theorem presented in our previous work (math.DS/0207121).
arXiv:quant-ph/0301177v2 fatcat:a2qwvhf34vfjhgdibeqynznmpm

Process Dimension of Classical and Non-Commutative Processes

Wolfgang Löhr, Arleta Szkoła, Nihat Ay
2012 Open systems & information dynamics  
We treat observable operator models (OOM) and their non-commutative generalisation, which we call NC-OOMs. A natural characteristic of a stochastic process in the context of classical OOM theory is the process dimension. We investigate its properties within the more general formulation, which allows to consider process dimension as a measure of complexity of non-commutative processes: We prove lower semi-continuity, and derive an ergodic decomposition formula. Further, we obtain results on the
more » ... lose relationship between the canonical OOM and the concept of causal states which underlies the definition of statistical complexity. In particular, the topological statistical complexity, i.e. the logarithm of the number of causal states, turns out to be an upper bound to the logarithm of process dimension.
doi:10.1142/s1230161212500072 fatcat:dnjsqg5bxbftjpsiwavgn3cr5i

Effective Complexity of Stationary Process Realizations

Nihat Ay, Markus Müller, Arleta Szkoła
2011 Entropy  
Effective complexity of stationary process realizations Nihat Ay, Markus Müller, Arleta Szkoła Abstract-The concept of effective complexity of an object as the minimal description length of its regularities  ... 
doi:10.3390/e13061200 fatcat:vkbizb3vrvdsreiuzarxp3mpru

A Quantum Version of Sanov's Theorem

Igor Bjelaković, Jean-Dominique Deuschel, Tyll Krüger, Ruedi Seiler, Rainer Siegmund-Schultze, Arleta Szkoła
2005 Communications in Mathematical Physics  
We present a quantum extension of a version of Sanov's theorem focussing on a hypothesis testing aspect of the theorem: There exists a sequence of typical subspaces for a given set Ψ of stationary quantum product states asymptotically separating them from another fixed stationary product state. Analogously to the classical case, the exponential separating rate is equal to the infimum of the quantum relative entropy with respect to the quantum reference state over the set Ψ. However, while in
more » ... classical case the separating subsets can be chosen universal, in the sense that they depend only on the chosen set of i.i.d. processes, in the quantum case the choice of the separating subspaces depends additionally on the reference state.
doi:10.1007/s00220-005-1426-2 fatcat:nzd6hhrvebdvbbv4ywde3cyyeq

Effective Complexity and Its Relation to Logical Depth

Nihat Ay, Markus Muller, Arleta Szkola
2010 IEEE Transactions on Information Theory  
Szkoła are with the Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany. e-mail: {nay,szkola}@mis.mpg.de, mueller@math.tu-berlin.de. M.  ... 
doi:10.1109/tit.2010.2053892 fatcat:ztklkea63nb2bfg7pirdybppea

The Shannon-McMillan Theorem for Ergodic Quantum Lattice Systems [article]

Igor Bjelakovic, Tyll Krueger, Rainer Siegmund-Schultze, Arleta Szkola
2002 arXiv   pre-print
We formulate and prove a quantum Shannon-McMillan theorem. The theorem demonstrates the significance of the von Neumann entropy for translation invariant ergodic quantum spin systems on n-dimensional lattices: the entropy gives the logarithm of the essential number of eigenvectors of the system on large boxes. The one-dimensional case covers quantum information sources and is basic for coding theorems.
arXiv:math/0207121v3 fatcat:d5cty6yu4vdzjibwifg6eu2ncq

Exponential error rates in multiple state discrimination on a quantum spin chain

Michael Nussbaum, Arleta Szkoła
2010 Journal of Mathematical Physics  
We consider decision problems on finite sets of hypotheses represented by pairwise different shift-invariant states on a quantum spin chain. The decision in favor of one of the hypotheses is based on outputs of generalized measurements performed on local states on blocks of finite size. We assume existence of the mean quantum Chernoff distances of any pair of states from the given set and refer to the minimum of them as the mean generalized quantum Chernoff distance. We establish that this
more » ... um specifies an asymptotic bound on the exponential decay of the averaged probability of rejecting the true state in increasing block size, if the mean quantum Chernoff distance of any pair of the hypothetic states is achievable as an asymptotic error exponent in the corresponding binary problem. This assumption is in particular fulfiled by shift-invariant product states (i.i.d. states). Further, we provide a constructive proof for the existence of a sequence of quantum tests in increasing block size, which achieves an asymptotic error exponent which is equal to the mean generalized quantum Chernoff distance of the given set of states up to a factor, which depends on the set itself. It can be arbitrary close to 1 and is not less than 1/m for m being the number of different pairs of states from the set considered.
doi:10.1063/1.3451110 fatcat:tx4oc72nwjag7n4xivx5vnyowe

The Chernoff lower bound for symmetric quantum hypothesis testing

Michael Nussbaum, Arleta Szkoła
2009 Annals of Statistics  
We consider symmetric hypothesis testing in quantum statistics, where the hypotheses are density operators on a finite-dimensional complex Hilbert space, representing states of a finite quantum system. We prove a lower bound on the asymptotic rate exponents of Bayesian error probabilities. The bound represents a quantum extension of the Chernoff bound, which gives the best asymptotically achievable error exponent in classical discrimination between two probability measures on a finite set. In
more » ... r framework, the classical result is reproduced if the two hypothetic density operators commute. Recently, it has been shown elsewhere [Phys. Rev. Lett. 98 (2007) 160504] that the lower bound is achievable also in the generic quantum (noncommutative) case. This implies that our result is one part of the definitive quantum Chernoff bound.
doi:10.1214/08-aos593 fatcat:jlxzvmbnlnbzhgtqfmbnhvayxy

An asymptotic error bound for testing multiple quantum hypotheses

Michael Nussbaum, Arleta Szkoła
2011 Annals of Statistics  
We consider the problem of detecting the true quantum state among r possible ones, based of measurements performed on n copies of a finite-dimensional quantum system. A special case is the problem of discriminating between r probability measures on a finite sample space, using n i.i.d. observations. In this classical setting, it is known that the averaged error probability decreases exponentially with exponent given by the worst case binary Chernoff bound between any possible pair of the r
more » ... bility measures. Define analogously the multiple quantum Chernoff bound, considering all possible pairs of states. Recently, it has been shown that this asymptotic error bound is attainable in the case of r pure states, and that it is unimprovable in general. Here we extend the attainability result to a larger class of r-tuples of states which are possibly mixed, but pairwise linearly independent. We also construct a quantum detector which universally attains the multiple quantum Chernoff bound up to a factor 1/3.
doi:10.1214/11-aos933 fatcat:whzxwvxhnfamlffgdtiqy7eksy

Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach

Leiba Rodman, Ilya M. Spitkovsky, Arleta Szkoła, Stephan Weis
2016 Journal of Mathematical Physics  
We study the continuity of an abstract generalization of the maximum-entropy inference - a maximizer. It is defined as a right-inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations. Further, we
more » ... use numerical range methods in the case of quantum inference which refers to two observables. One result is a complete characterization of points of discontinuity for 3× 3 matrices.
doi:10.1063/1.4926965 fatcat:kom77yd6hncofpfytjsg3uemta

Typical Support and Sanov Large Deviations of Correlated States

Igor Bjelaković, Jean-Dominique Deuschel, Tyll Krüger, Ruedi Seiler, Rainer Siegmund-Schultze, Arleta Szkoła
2008 Communications in Mathematical Physics  
Tyll Krüger, Rainer Siegmund-Schultze and Arleta Szkoła are particularly grateful to Nihat Ay for his constant encouragement during the preparation of the manuscript.  ... 
doi:10.1007/s00220-008-0440-6 fatcat:ch45z6ztwrh7nd3hlapgfilqxa

Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno's Theorem

Fabio Benatti, Tyll Krüger, Markus Müller, Rainer Siegmund-Schultze, Arleta Szkoła
2006 Communications in Mathematical Physics  
In classical information theory, entropy rate and Kolmogorov complexity per symbol are related by a theorem of Brudno. In this paper, we prove a quantum version of this theorem, connecting the von Neumann entropy rate and two notions of quantum Kolmogorov complexity, both based on the shortest qubit descriptions of qubit strings that, run by a universal quantum Turing machine, reproduce them as outputs.
doi:10.1007/s00220-006-0027-z fatcat:ef4pbhu5azbwzbuipjqcr3ahky
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