arXiv subjects
Publications and source records attributed to Jarosław Adam Miszczak.
We present IntU package for Mathematica computer algebra system. The presented package performs a symbolic integration of polynomial functions over the unitary group with respect to unique normalized Haar measure. We describe a number of special cases which can be used to optimize the calculation speed for some classes of integrals. We also provide some examples of usage of the presented package.
We introduce a family of quantum walks on cycles parametrized by their liveliness, defined by the ability to execute a long-range move. We investigate the behaviour of the probability distribution and time-averaged probability distribution. We show that the liveliness parameter, controlling the magnitude of the additional long-range move, has a direct impact on the periodicity of the limiting distribution. We also show that the introduced model provides a method for network exploration which is robust against trapping.
The existence of an undetectable eavesdropping of dense coded information has been already demonstrated by Pavičić for the quantum direct communication based on the ping-pong paradigm. However, a) the explicit scheme of the circuit is only given and no design rules are provided, b) the existence of losses is implicitly assumed, c) the attack has been formulated against qubit based protocol only and it is not clear whether it can be adapted to higher dimensional systems. These deficiencies are removed in the presented contribution. A new generic eavesdropping scheme built on a firm theoretical background is proposed. In contrast to the previous approach, it does not refer to the properties of the vacuum state, so it is fully consistent with the absence of losses assumption. Moreover, the scheme applies to the communication paradigm based on signal particles of any dimensionality. It is also shown that some well known attacks are special cases of the proposed scheme.
We provide a complete work-flow, based on the language of quantum information theory, suitable for processing data for the purpose of pattern recognition. The main advantage of the introduced scheme is that it can be easily implemented and applied to process real-world data using modest computation resources. At the same time it can be used to investigate the difference in the pattern recognition resulting from the utilization of the tensor product structure of the space of quantum states. We illustrate this difference by providing a simple example based on the classification of 2D data.
We develop a model which can be used to analyse the scenario of exploring quantum network with a distracted sense of direction. Using this model we analyse the behaviour of quantum mobile agents operating with non-adaptive and adaptive strategies which can be employed in this scenario. We introduce the notion of node visiting suitable for analysing quantum superpositions of states by distinguishing between visiting and attaining a position. We show that without a proper model of adaptiveness, it is not possible for the party representing the distraction in the sense of direction, to obtain the results analogous to the classical case. Moreover, with additional control resources the total number of attained positions is maintained if the number of visited positions is strictly limited.
We introduce a method of analyzing entanglement enhanced quantum games on regular lattices of agents. Our method is valid for setups with periodic and non-periodic boundary conditions. To demonstrate our approach we study two different types games, namely the prisoner's dilemma game and a cooperative Parrondo's game. In both cases we obtain results showing, that entanglement is a crucial resource necessary for the agents to achieve positive capital gain.
We study the model of quantum walks on cycles enriched by the addition of 1-step memory. We provide a formula for the probability distribution and the time-averaged limiting probability distribution of the introduced quantum walk. Using the obtained results, we discuss the properties of the introduced model and the difference in comparison to the memoryless model.
We develop a framework which aims to simplify the analysis of quantum states and quantum operations by harnessing the potential of function programming paradigm. We show that the introduced framework allows a seamless manipulation of quantum channels, in particular to convert between different representations of quantum channels, and thus that the use of functional programming concepts facilitates the manipulation of abstract objects used in the language of quantum theory. For the purpose of our presentation we will use Mathematica computer algebra system. This choice is motivated twofold. First, it offers a rich programming language based on the functional paradigm. Second, this programming language is combined with powerful symbolic and numeric manipulation capabilities.
We introduce a general approach for the analysis of a quantum direct communication protocol. The method is based on the investigation of the superoperator acting on a joint system of the communicating parties and the eavesdropper. The introduced method is more versatile than the approaches used so far as it permits to incorporate different noise models in a unified way. Moreover, it make use of a well grounded theory of quantum discrimination for the purpose of estimating the eavesdropper's information gain.
This paper provides an analysis of patent activity in the field of quantum information processing. Data from the PatentScope database from the years 1993-2011 was used. In order to predict the future trends in the number of filed patents time series models were used.
The restricted numerical range $W_R(A)$ of an operator $A$ acting on a $D$-dimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset $R$ of the of set of pure quantum states of dimension $D$. One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of $A$ -- a normalized probability distribution on the complex plane supported in $W_R(A)$. Its value at point $z \in {\mathbbm C}$ is equal to the probability that the expectation value $< ψ|A|ψ>$ is equal to $z$, where $|ψ>$ represents a random quantum state in subset $R$ distributed according to the natural measure on this set, induced by the unitarily invariant Fubini--Study measure. Studying restricted shadows of operators of a fixed size $D=N_A N_B$ we analyse the geometry of sets of separable and maximally entangled states of the $N_A \times N_B$ composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow we study the dynamics of quantum entanglement. A similar analysis extended for operators on $D=2^3$ dimensional Hilbert space allows us to investigate the structure of the orbits of $GHZ$ and $W$ quantum states of a three--qubit system.
In this paper we propose an extended version of the ping-pong protocol and study its security. The proposed protocol incorporates the usage of mutually unbiased bases in the control mode. We show that, by increasing the number of bases, it is possible to improve the security of this protocol. We also provide the upper bounds on eavesdropping average non-detection probability and propose a control mode modification that increases the attack detection probability.
We study the probability measure on the space of density matrices induced by the metric defined by using superfidelity. We give the formula for the probability density of eigenvalues. We also study some statistical properties of the set of density matrices equipped with the introduced measure and provide a method for generating density matrices according to the introduced measure.
We present the application of Principal Component Analysis for data acquired during the design of a natural gesture interface. We investigate the concept of an eigengesture for motion capture hand gesture data and present the visualisation of principal components obtained in the course of conducted experiments. We also show the influence of dimensionality reduction on reconstructed gesture data quality.
We describe an informal methodology for developing on-line applications, which is, to some extent, complementary to the Web 2.0 aspects of web development. The presented methodology is suitable for developing low-cost and non-cost web sites targeted at medium-sized communities. We present basic building blocks used in the described strategy. To achieve a better understanding of the discussed concepts we comment on their application during the realization of two web projects. We focus on the role of community-driven development, which is crucial for projects of the discussed type.
We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.
Numerical range of a Hermitian operator X is defined as the set of all possible expectation values of this observable among a normalized quantum state. We analyze a modification of this definition in which the expectation value is taken among a certain subset of the set of all quantum states. One considers for instance the set of real states, the set of product states, separable states, or the set of maximally entangled states. We show exemplary applications of these algebraic tools in the theory of quantum information: analysis of k-positive maps and entanglement witnesses, as well as study of the minimal output entropy of a quantum channel. Product numerical range of a unitary operator is used to solve the problem of local distinguishability of a family of two unitary gates.
We present two measures of distance between quantum processes based on the superfidelity, introduced recently to provide an upper bound for quantum fidelity. We show that the introduced measures partially fulfill the requirements for distance measure between quantum processes. We also argue that they can be especially useful as diagnostic measures to get preliminary knowledge about imperfections in an experimental setup. In particular we provide quantum circuit which can be used to measure the superfidelity between quantum processes. As the behavior of the superfidelity between quantum processes is crucial for the properties of the introduced measures, we study its behavior for several families of quantum channels. We calculate superfidelity between arbitrary one-qubit channels using affine parametrization and superfidelity between generalized Pauli channels in arbitrary dimensions. Statistical behavior of the proposed quantities for the ensembles of quantum operations in low dimensions indicates that the proposed measures can be indeed used to distinguish quantum processes.