arXiv subjects

Karol Zyczkowski

Publications and source records attributed to Karol Zyczkowski.

At least 37 records · Page 2Linked to original sources

arXiv2013

Unitary quantum gates, perfect entanglers and unistochastic maps

Non-local properties of ensembles of quantum gates induced by the Haar measure on the unitary group are investigated. We analyze the entropy of entanglement of a unitary matrix U equal to the Shannon entropy of the vector of singular values of the reshuffled matrix. Averaging the entropy over the Haar measure on U(N^2) we find its asymptotic behaviour. For two--qubit quantum gates we derive the induced probability distribution of the interaction content and show that the relative volume of the set of perfect entanglers reads 8/3 π\approx 0.85. We establish explicit conditions under which a given one-qubit bistochastic map is unistochastic, so it can be obtained by partial trace over a one--qubit environment initially prepared in the maximally mixed state.

quant-ph↗

arXiv2012

Densities of the Raney distributions

We prove that if $p\ge 1$ and $0< r\le p$ then the sequence $\binom{mp+r}{m}\frac{r}{mp+r}$, $m=0,1,2,...$, is positive definite, more precisely, is the moment sequence of a probability measure $μ(p,r)$ with compact support contained in $[0,+\infty)$. This family of measures encompasses the multiplicative free powers of the Marchenko-Pastur distribution as well as the Wigner's semicircle distribution centered at $x=2$. We show that if $p>1$ is a rational number, $0<r\le p$, then $μ(p,r)$ is absolutely continuous and its density $W_{p,r}(x)$ can be expressed in terms of the Meijer and the generalized hypergeometric functions. In some cases, including the multiplicative free square and the multiplicative free square root of the Marchenko-Pastur measure, $W_{p,r}(x)$ turns out to be an elementary function.

math.PR↗

arXiv2012

Joint numerical ranges, quantum maps, and joint numerical shadows

We associate with k hermitian N\times N matrices a probability measure on R^k. It is supported on the joint numerical range of the k-tuple of matrices. We call this measure the joint numerical shadow of these matrices. Let k=2. A pair of hermitian N\times N matrices defines a complex N\times N matrix. The joint numerical range and the joint numerical shadow of the pair of hermitian matrices coincide with the numerical range and the numerical shadow, respectively, of this complex matrix. We study relationships between the dynamics of quantum maps on the set of quantum states, on one hand, and the numerical ranges, on the other hand. In particular, we show that under the identity resolution assumption on Kraus operators defining the quantum map, the dynamics shrinks numerical ranges.

quant-ph↗

arXiv2012

Almost Hadamard matrices: general theory and examples

We develop a general theory of "almost Hadamard matrices". These are by definition the matrices $H\in M_N(\mathbb R)$ having the property that $U=H/\sqrt{N}$ is orthogonal, and is a local maximum of the 1-norm on O(N). Our study includes a detailed discussion of the circulant case ($H_{ij}=γ_{j-i}$) and of the two-entry case ($H_{ij}\in{x,y}$), with the construction of several families of examples, and some 1-norm computations.

math.CO↗

arXiv2012

Square root voting system, optimal threshold and π

The problem of designing an optimal weighted voting system for the two-tier voting, applicable in the case of the Council of Ministers of the European Union (EU), is investigated. Various arguments in favour of the square root voting system, where the voting weights of member states are proportional to the square root of their population are discussed and a link between this solution and the random walk in the one-dimensional lattice is established. It is known that the voting power of every member state is approximately equal to its voting weight, if the threshold q for the qualified majority in the voting body is optimally chosen. We analyze the square root voting system for a generic 'union' of M states and derive in this case an explicit approximate formula for the level of the optimal threshold: q \simeq 1/2+1/\sqrt{π M}. The prefactor 1/\sqrt{π} appears here as a result of averaging over the ensemble of unions with random populations.

physics.soc-ph↗

arXiv2012

Tensor Products of Random Unitary Matrices

Tensor products of M random unitary matrices of size N from the circular unitary ensemble are investigated. We show that the spectral statistics of the tensor product of random matrices becomes Poissonian if M=2, N become large or M become large and N=2.

math.PR↗

arXiv2012

Mathematical aspects of degressive proportionality

We analyze properties of apportionment functions in context of the problem of allocating seats in the European Parliament. Necessary and sufficient conditions for apportionment functions are investigated. Some exemplary families of apportionment functions are specified and the corresponding partitions of the seats in the European Parliament among the Member States of the European Union are presented. Although the choice of the allocation functions is theoretically unlimited, we show that the constraints are so strong that the acceptable functions lead to rather similar solutions.

physics.soc-ph↗

arXiv2011

Composition of quantum operations and products of random matrices

Spectral properties of evolution operators corresponding to random maps and quantized chaotic systems strongly interacting with an environment can be described by the ensemble of non-hermitian random matrices from the real Ginibre ensemble. We analyze evolution operators Psi=Psi_s...Psi_1 representing the composition of s random maps and demonstrate that their complex eigenvalues are asymptotically described by the law of Burda et al. obtained for a product of s independent random complex Ginibre matrices. Numerical data support the conjecture that the same results are applicable to characterize the distribution of eigenvalues of the s-th power of a random Ginibre matrix. Squared singular values of Psi are shown to be described by the Fuss-Catalan distribution of order s. Results obtained for products of random Ginibre matrices are also capable to describe the s-step evolution operator for a model deterministic dynamical system - a generalized quantum baker map subjected to strong interaction with an environment.

quant-ph↗

arXiv2011

Product of Ginibre matrices: Fuss-Catalan and Raney distributions

Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distribution P_s(x), such that their moments are equal to the Fuss-Catalan numbers or order s. We find a representation of the Fuss--Catalan distributions P_s(x) in terms of a combination of s hypergeometric functions of the type sF_{s-1}. The explicit formula derived here is exact for an arbitrary positive integer s and for s=1 it reduces to the Marchenko--Pastur distribution. Using similar techniques, involving Mellin transform and the Meijer G-function, we find exact expressions for the Raney probability distributions, the moments of which are given by a two parameter generalization of the Fuss-Catalan numbers. These distributions can also be considered as a two parameter generalization of the Wigner semicircle law.

math-ph↗

arXiv2011

Matrices of fidelities for ensembles of quantum states and the Holevo quantity

The entropy of the Gram matrix of a joint purification of an ensemble of K mixed states yields an upper bound for the Holevo information Chi of the ensemble. In this work we combine geometrical and probabilistic aspects of the ensemble in order to obtain useful bounds for Chi. This is done by constructing various correlation matrices involving fidelities between every pair of states from the ensemble. For K=3 quantum states we design a matrix of root fidelities that is positive and the entropy of which is conjectured to upper bound Chi. Slightly weaker bounds are established for arbitrary ensembles. Finally, we investigate correlation matrices involving multi-state fidelities in relation to the Holevo quantity.

quant-ph↗

arXiv2011

Generating random density matrices

We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi--partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N \to \infty, by the Marchenko-Pastur distribution.

quant-ph↗

arXiv2011

Entropic characterization of quantum operations

We investigate decoherence induced by a quantum channel in terms of minimal output entropy and of map entropy. The latter is the von Neumann entropy of the Jamiolkowski state of the channel. Both quantities admit q-Renyi versions. We prove additivity of the map entropy for all q. For the case q = 2, we show that the depolarizing channel has the smallest map entropy among all channels with a given minimal output Renyi entropy of order two. This allows us to characterize pairs of channels such that the output entropy of their tensor product acting on a maximally entangled input state is larger than the sum of the minimal output entropies of the individual channels. We conjecture that for any channel Φ1 acting on a finite dimensional system there exists a class of channels Φ2 sufficiently close to a unitary map such that additivity of minimal output entropy for Ψ1 x Ψ2 holds.

quant-ph↗

arXiv2010

Truncations of Random Orthogonal Matrices

Statistical properties of non--symmetric real random matrices of size $M$, obtained as truncations of random orthogonal $N\times N$ matrices are investigated. We derive an exact formula for the density of eigenvalues which consists of two components: finite fraction of eigenvalues are real, while the remaining part of the spectrum is located inside the unit disk symmetrically with respect to the real axis. In the case of strong non--orthogonality, $M/N=$const, the behavior typical to real Ginibre ensemble is found. In the case $M=N-L$ with fixed $L$, a universal distribution of resonance widths is recovered.

cond-mat.stat-mech↗

arXiv2010

How often is a random quantum state k-entangled?

The set of trace preserving, positive maps acting on density matrices of size d forms a convex body. We investigate its nested subsets consisting of k-positive maps, where k=2,...,d. Working with the measure induced by the Hilbert-Schmidt distance we derive asymptotically tight bounds for the volumes of these sets. Our results strongly suggest that the inner set of (k+1)-positive maps forms a small fraction of the outer set of k-positive maps. These results are related to analogous bounds for the relative volume of the sets of k-entangled states describing a bipartite d X d system.

quant-ph↗

arXiv2010

Universal bounds for the Holevo quantity, coherent information \\ and the Jensen-Shannon divergence

The Holevo quantity provides an upper bound for the mutual information between the sender of a classical message encoded in quantum carriers and the receiver. Applying the strong sub-additivity of entropy we prove that the Holevo quantity associated with an initial state and a given quantum operation represented in its Kraus form is not larger than the exchange entropy. This implies upper bounds for the coherent information and for the quantum Jensen--Shannon divergence. Restricting our attention to classical information we bound the transmission distance between any two probability distributions by the entropic distance, which is a concave function of the Hellinger distance.

quant-ph↗

arXiv2010

Random graph states, maximal flow and Fuss-Catalan distributions

For any graph consisting of $k$ vertices and $m$ edges we construct an ensemble of random pure quantum states which describe a system composed of $2m$ subsystems. Each edge of the graph represents a bi-partite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated to a given graph is presented. Our technique relies on Weingarten calculus and flow problems. We analyze statistical properties of spectra of such random density matrices and show for which cases they are described by the free Poissonian (Marchenko-Pastur) distribution. We derive a discrete family of generalized, Fuss-Catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues.

quant-ph↗

arXiv2010

Universality of spectra for interacting quantum chaotic systems

We analyze a model quantum dynamical system subjected to periodic interaction with an environment, which can describe quantum measurements. Under the condition of strong classical chaos and strong decoherence due to large coupling with the measurement device, the spectra of the evolution operator exhibit an universal behavior. A generic spectrum consists of a single eigenvalue equal to unity, which corresponds to the invariant state of the system, while all other eigenvalues are contained in a disk in the complex plane. Its radius depends on the number of the Kraus measurement operators, and determines the speed with which an arbitrary initial state converges to the unique invariant state. These spectral properties are characteristic of an ensemble of random quantum maps, which in turn can be described by an ensemble of real random Ginibre matrices. This will be proven in the limit of large dimension.

quant-ph↗

arXiv2010

Citation graph, weighted impact factors and performance indices

A scheme of evaluating an impact of a given scientific paper based on importance of papers quoting it is investigated. Introducing a weight of a given citation, dependent on the previous scientific achievements of the author of the citing paper, we define the weighting factor of a given scientist. Technically the weighting factors are defined by the components of the normalized leading eigenvector of the matrix describing the citation graph. The weighting factor of a given scientist, reflecting the scientific output of other researchers quoting his work, allows us to define weighted number of citation of a given paper, weighted impact factor of a journal and weighted Hirsch index of an individual scientist or of an entire scientific institution.

physics.soc-ph↗