arXiv subjects

Oleg Szehr

Publications and source records attributed to Oleg Szehr.

29 records · Page 2Linked to original sources

arXiv2015

Connected components of irreducible maps and 1D quantum phases

We investigate elementary topological properties of sets of completely positive (CP) maps that arise in quantum Perron-Frobenius theory. We prove that the set of primitive CP maps of fixed Kraus rank is path-connected and we provide a complete classification of the connected components of irreducible CP maps at given Kraus rank and fixed peripheral spectrum in terms of a multiplicity index. These findings are then applied to analyse 1D quantum phases by studying equivalence classes of translational invariant Matrix Product States that correspond to the connected components of the respective CP maps. Our results extend the previously obtained picture in that they do not require blocking of physical sites, they lead to analytic paths and they allow to decompose into ergodic components and to study the breaking of translational symmetry.

math-ph↗

arXiv2015

Perturbation Theory for Parent Hamiltonians of Matrix Product States

This article investigates the stability of the ground state subspace of a canonical parent Hamiltonian of a Matrix product state against local perturbations. We prove that the spectral gap of such a Hamiltonian remains stable under weak local perturbations even in the thermodynamic limit, where the entire perturbation might not be bounded. Our discussion is based on preceding work by D.A. Yarotsky that develops a perturbation theory for relatively bounded quantum perturbation of classical Hamiltonians. We exploit a renormalization procedure, which on large scale transforms the parent Hamiltonian of a Matrix product state into a classical Hamiltonian plus some perturbation. We can thus extend D.A. Yarotsky's results to provide a perturbation theory for parent Hamiltonians of Matrix product states and recover some of the findings of independent contributions by S. Michalakis et al. [arXiv:1109.1588] and J. I. Cirac et al. [arXiv:1306.4003].

math-ph↗

arXiv2015

Maximum of the resolvent over matrices with given spectrum

In numerical analysis it is often necessary to estimate the condition number $CN(T)=||T||_{} \cdot||T^{-1}||_{}$ and the norm of the resolvent $||(ζ-T)^{-1}||_{}$ of a given $n\times n$ matrix $T$. We derive new spectral estimates for these quantities and compute explicit matrices that achieve our bounds. We recover the well-known fact that the supremum of $CN(T)$ over all matrices with $||T||_{} \leq1$ and minimal absolute eigenvalue $r=\min_{i=1,...,n}|λ_{i}|>0$ is the Kronecker bound $\frac{1}{r^{n}}$. This result is subsequently generalized by computing the corresponding supremum of $||(ζ-T)^{-1}||_{}$ for any $|ζ| \leq1$. We find that the supremum is attained by a triangular Toeplitz matrix. This provides a simple class of structured matrices on which condition numbers and resolvent norm bounds can be studied numerically. The occuring Toeplitz matrices are so-called model matrices, i.e. matrix representations of the compressed backward shift operator on the Hardy space $H_2$ to a finite-dimensional invariant subspace.

math.NA↗

arXiv2015

Eigenvalue estimates for the resolvent of a non-normal matrix

We investigate the relation between the spectrum of a non-normal matrix and the norm of its resolvent. We provide spectral estimates for the resolvent of matrices whose largest singular value is bounded by $1$ (so-called Hilbert space contractions) and for power-bounded matrices. In the first case our estimate is optimal and we present explicit matrices that achieve equality in the bound. This result recovers and generalizes previous estimates obtained by E.B. Davies and B. Simon in the study of orthogonal polynomials on the unit circle. In case of power-bounded matrices we achieve the strongest estimate so far. Our result unifies previous approaches, where the resolvent was estimated in certain restricted regions of the complex plane. To achieve our estimates we relate the problem of bounding the norm of a function of a matrix to a Nevanlinna-Pick interpolation problem in a corresponding function space. In case of Hilbert space contractions this problem is connected to the theory of compressed shift operators to which we contribute by providing explicit matrix representations for such operators. Finally, we apply our results to study the sensitivity of the stationary states of a classical or quantum Markov chain with respect to perturbations of the transition matrix.

math.SP↗

arXiv2014

Spectral convergence bounds for classical and quantum Markov processes

We introduce a new framework that yields spectral bounds on norms of functions of transition maps for finite, homogeneous Markov chains. The techniques employed work for bounded semigroups, in particular for classical as well as for quantum Markov chains and they do not require additional assumptions like detailed balance, irreducibility or aperiodicity. We use the method in order to derive convergence bounds that improve significantly upon known spectral bounds. The core technical observation is that power-boundedness of transition maps of Markov chains enables a Wiener algebra functional calculus in order to upper bound any norm of any holomorphic function of the transition map. Finally, we discuss how general detailed balance conditions for quantum Markov processes lead to spectral convergence bounds.

math-ph↗

arXiv2014

Variations on Classical and Quantum Extractors

Many constructions of randomness extractors are known to work in the presence of quantum side information, but there also exist extractors which do not [Gavinsky {\it et al.}, STOC'07]. Here we find that spectral extractors $ψ$ with a bound on the second largest eigenvalue $λ_{2}(ψ^{\dagger}\circψ)$ are quantum-proof. We then discuss fully quantum extractors and call constructions that also work in the presence of quantum correlations decoupling. As in the classical case we show that spectral extractors are decoupling. The drawback of classical and quantum spectral extractors is that they always have a long seed, whereas there exist classical extractors with exponentially smaller seed size. For the quantum case, we show that there exists an extractor with extremely short seed size $d=O(\log(1/ε))$, where $ε>0$ denotes the quality of the randomness. In contrast to the classical case this is independent of the input size and min-entropy and matches the simple lower bound $d\geq\log(1/ε)$.

quant-ph↗

arXiv2014

On quantum Renyi entropies: a new generalization and some properties

The Renyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of Renyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Renyi entropies that contains the von Neumann entropy, min-entropy, collision entropy and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.

quant-ph↗

arXiv2013

Decoupling with unitary approximate two-designs

Consider a bipartite system, of which one subsystem, A, undergoes a physical evolution separated from the other subsystem, R. One may ask under which conditions this evolution destroys all initial correlations between the subsystems A and R, i.e. decouples the subsystems. A quantitative answer to this question is provided by decoupling theorems, which have been developed recently in the area of quantum information theory. This paper builds on preceding work, which shows that decoupling is achieved if the evolution on A consists of a typical unitary, chosen with respect to the Haar measure, followed by a process that adds sufficient decoherence. Here, we prove a generalized decoupling theorem for the case where the unitary is chosen from an approximate two-design. A main implication of this result is that decoupling is physical, in the sense that it occurs already for short sequences of random two-body interactions, which can be modeled as efficient circuits. Our decoupling result is independent of the dimension of the R system, which shows that approximate 2-designs are appropriate for decoupling even if the dimension of this system is large.

quant-ph↗

arXiv2013

Perturbation Bounds for Quantum Markov Processes and their Fixed Points

We investigate the stability of quantum Markov processes with respect to perturbations of their transition maps. In the first part, we introduce a condition number that measures the sensitivity of fixed points of a quantum channel to perturbations. We establish upper and lower bounds on this condition number in terms of subdominant eigenvalues of the transition map. In the second part, we consider quantum Markov processes that converge to a unique stationary state and we analyze the stability of the evolution at finite times. In this way we obtain a linear relation between the mixing time of a quantum Markov process and the sensitivity of its fixed point with respect to perturbations of the transition map.

math-ph↗

arXiv2012

Decoupling Theorems

Decoupling theorems have proven useful in various applications in the area of quantum information theory. This thesis builds upon preceding work by Frédéric Dupuis [arXiv:1012.6044v1], where a general decoupling theorem is obtained and its implications for quantum coding theory are studied. At first we generalize this theorem to the case where the average is taken over an approximate unitary 2-design. The second part of this project tackles the question whether or not it is possible to decorrelate CQ-states with classical operations. We obtain results similar to the pivotal Leftover Hash Lemma. Finally we analyze the decoupling power of permutation operators in a fully quantum context and show a general procedure that yields decoupling theorems with permutations.

quant-ph↗

arXiv2012

A decoupling approach to classical data transmission over quantum channels

Most coding theorems in quantum Shannon theory can be proven using the decoupling technique: to send data through a channel, one guarantees that the environment gets no information about it; Uhlmann's theorem then ensures that the receiver must be able to decode. While a wide range of problems can be solved this way, one of the most basic coding problems remains impervious to a direct application of this method: sending classical information through a quantum channel. We will show that this problem can, in fact, be solved using decoupling ideas, specifically by proving a "dequantizing" theorem, which ensures that the environment is only classically correlated with the sent data. Our techniques naturally yield a generalization of the Holevo-Schumacher-Westmoreland Theorem to the one-shot scenario, where a quantum channel can be applied only once.

quant-ph↗