We investigate an original family of quantum distinguishability problems, where the goal is to perfectly distinguish between *M* quantum states that become identical under a completely decohering map. Similarly, we study distinguishability of *M* quantum channels that cannot be distinguished when one is restricted to decohered input and output states. The studied problems arise naturally in the presence of a superselection rule, allow one to quantify the amount of information that can be encoded in phase degrees of freedom (coherences), and are related to time-energy uncertainty relation. We present a collection of results on both necessary and sufficient conditions for the existence of *M* perfectly distinguishable states (channels) that are classically indistinguishable.

We adopt the perspective of similarity equivalence, in gate set tomography called the gauge, to analyze various properties of quantum operations belonging to a semigroup, Φ=*e**^{Lt}*,and therefore given through the Lindblad operator. We first observe that the non unital part of the channel decouples from the time evolution. Focusing on unital operations we restrict our attention to the single-qubit case, showing that the semigroup embedded inside the tetrahedron of Pauli channels is bounded by the surface composed of product probability vectors and includes the identity map together with the maximally depolarizing channel. Consequently, every member of the Pauli semigroup is unitarily equivalent to a unistochastic map, describing a coupling with one-qubit environment initially in the maximally mixed state, determined by a unitary matrix of order four.

Implementation of generalized quantum measurements is often experimentally demanding, as it requires performing a projective measurement on a system of interest extended by the ancilla. We report an alternative scheme for implementing generalized measurements that uses solely: (a) classical randomness and post-processing, (b) projective measurements on a relevant quantum system and (c) postselection on non-observing certain outcomes. The method implements arbitrary quantum measurement in d dimensional system with success probability 1/d. It is optimal in the sense that for any dimensionn d there exist measurements for which the success probability cannot be higher. We apply our results to bound the relative power of projective and generalised measurements for unambiguous state discrimination. Finally, we test our scheme experimentally on IBM quantum processor. Interestingly, due to noise involved in the implementation of entangling gates, the quality with which our scheme implements generalized qubit measurements is higher than the one obtained with the standard construction using the auxiliary system.

}, url = {https://arxiv.org/abs/1807.08449v1}, author = {Micha{\l} Oszmaniec and Filip B. Maciejewski and Zbigniew Pucha{\l}a} } @article {2551, title = {Almost all quantum channels are equidistant}, journal = {Journal of Mathematical Physics}, volume = {59}, year = {2018}, pages = {052201}, abstract = {In this work we analyze properties of generic quantum channels in the case of large system size. We use the random matrix theory and free probability to show that the distance between two independent random channels tends to a constant value as the dimension of the system grows larger. As a measure of the distance we use the diamond norm. In the case of a flat Hilbert-Schmidt distribution on quantum channels, we obtain that the distance converges to 1/2*+2/π*. Furthermore, we show that for a random state *ρ* acting on a bipartite Hilbert space $H_A \otimes H_B$, sampled from the Hilbert-Schmidt distribution, the reduced states $Tr_A\rho$ and $Tr_B \rho$ are arbitrarily close to the maximally mixed state. This implies that, for large dimensions, the state *ρ* may be interpreted as a Jamio{\l}kowski state of a unital map.

Is it always possible to explain random stochastic transitions between states of a finite-dimensional system as arising from the deterministic quantum evolution of the system? If not, then what is the minimal amount of randomness required by quantum theory to explain a given stochastic process? Here, we address this problem by studying possible coherifications of a quantum channel $\Phi$, i.e., we look for channels $\Phi^{\mathcal{C}}$ that induce the same classical transitions $T$, but are "more coherent". To quantify the coherence of a channel $\Phi$ we measure the coherence of the corresponding Jamio{\l}kowski state $J_{\Phi}$. We show that the classical transition matrix $T$ can be coherified to reversible unitary dynamics if and only if $T$ is unistochastic. Otherwise the Jamio{\l}kowski state $J_\Phi^{\mathcal{C}}$ of the optimally coherified channel is mixed, and the dynamics must necessarily be irreversible. To asses the extent to which an optimal process $\Phi^{\mathcal{C}}$ is indeterministic we find explicit bounds on the entropy and purity of $J_\Phi^{\mathcal{C}}$, and relate the latter to the unitarity of $\Phi^{\mathcal{C}}$. We also find optimal coherifications for several classes of channels, including all one-qubit channels. Finally, we provide a non-optimal coherification procedure that works for an arbitrary channel $\Phi$ and reduces its rank (the minimal number of required Kraus operators) from $d^2$ to $d$.

}, doi = {10.1088/1367-2630/aaaff3}, url = {https://doi.org/10.1088/1367-2630/aaaff3}, author = {Kamil Korzekwa and Stanis{\l}aw Czach{\'o}rski and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {2609, title = {Conditional entropic uncertainty relations for Tsallis entropies}, journal = {Quantum Information Processing}, volume = {17}, year = {2018}, pages = {193}, abstract = {The entropic uncertainty relations are a very active field of scientific inquiry. Their applications include quantum cryptography and studies of quantum phenomena such as correlations and non-locality. In this work we find state-independent entropic uncertainty relations in terms of the Tsallis entropies for states with a fixed amount of entanglement. Our main result is stated as Theorem. Taking the special case of von Neumann entropy and utilizing the concavity of conditional von Neumann entropies, we extend our result to mixed states. Finally we provide a lower bound on the amount of extractable key in a quantum cryptographic scenario.

}, doi = {10.1007/s11128-018-1955-1}, url = {https://doi.org/10.1007/s11128-018-1955-1}, author = {Dariusz Kurzyk and {\L}ukasz Pawela and Zbigniew Pucha{\l}a} } @article {2606, title = {Gauge invariant information concerning quantum channels}, journal = {Quantum}, volume = {2}, year = {2018}, pages = {60}, abstract = {Motivated by the gate set tomography we study quantum channels from the perspective of information which is invariant with respect to the gauge realized through similarity of matrices representing channel superoperators. We thus use the complex spectrum of the superoperator to provide necessary conditions relevant for complete positivity of qubit channels and to express various metrics such as average gate fidelity.

}, doi = {10.22331/q-2018-04-11-60}, url = {https://doi.org/10.22331/q-2018-04-11-60}, author = {{\L}ukasz Rudnicki and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {2621, title = {Majorization uncertainty relations for mixed quantum states}, journal = {Journal of Physics A: Mathematical and Theoretical}, volume = {51}, year = {2018}, chapter = {175306}, abstract = {Majorization uncertainty relations are generalized for an arbitrary mixed quantum state *ρ* of a finite size *N*. In particular, a lower bound for the sum of two entropies characterizing probability distributions corresponding to measurements with respect to arbitrary two orthogonal bases is derived in terms of the spectrum of *ρ* and the entries of a unitary matrix *U* relating both bases. The obtained results can also be formulated for two measurements performed on a single subsystem of a bipartite system described by a pure state, and consequently expressed as uncertainty relation for the sum of conditional entropies.

We present an in-depth study of the problem of discrimination of von Neumann measurements in finite-dimensional Hilbert spaces. Specifically, we consider two scenarios: unambiguous and multiple-shot discrimination. In the first scenario we give the general expressions for the optimal discrimination probabilities with and without the assistance of entanglement. In the case of multiple-shot discrimination, we focus on discrimination of measurements with the assistance of entanglement. Interestingly, we prove that in this setting all pairs of distinct von Neumann measurements can be distinguished perfectly (i.e. with the unit success probability) using only a finite number of queries. We also show that in this scenario queering the measurements \emph{in parallel} gives the optimal strategy and hence any possible adaptive methods do not offer any advantage over the parallel scheme. Finally, we show that typical pairs of Haar-random von Neumann measurements can be perfectly distinguished with only two queries.

}, url = {https://arxiv.org/abs/1810.05122}, author = {Zbigniew Pucha{\l}a and {\L}ukasz Pawela and Aleksandra Krawiec and Ryszard Kukulski and Micha{\l} Oszmaniec} } @article {2683, title = {Strategies for optimal single-shot discrimination of quantum measurements}, journal = {Physical Review A}, volume = {98}, year = {2018}, chapter = {042103}, abstract = {In this work we study the problem of single-shot discrimination of von Neumann measurements. We associate each measurement with a measure-and-prepare channel. There are two possible approaches to this problem. The first one, which is simple, does not utilize entanglement. We focus only on discrimination of classical probability distribution, which are outputs of the channels. We find necessary and sufficient criterion for perfect discrimination in this case. A more advanced approach requires the usage and entanglement. We quantify the distance of the two measurements in terms of the diamond norm (called sometimes the completely bounded trace norm). We provide an exact expression for the optimal probability of correct distinction and relate it to the discrimination of unitary channels. We also state a necessary and sufficient condition for perfect discrimination and a semidefinite program which checks this condition. Our main result, however, is a cone program which calculates the distance of these measurements and hence provides an upper bound on the probability of their correct distinction. As a by-product the program also finds a strategy (input state) which achieves this bound. Finally, we provide a full description for the cases of Fourier matrices and mirror isometries.

}, doi = {10.1103/PhysRevA.98.042103}, url = {https://arxiv.org/abs/1804.05856}, author = {Zbigniew Pucha{\l}a and {\L}ukasz Pawela and Aleksandra Krawiec and Ryszard Kukulski} } @article {2663, title = {Unconditional Security of a K-State Quantum Key Distribution Protocol}, journal = {Quantum Information Processing}, volume = {17}, year = {2018}, pages = {228}, abstract = {Quantum key distribution protocols constitute an important part of quantum cryptography, where the security of sensitive information arises from the laws of physics. In this paper we introduce a new family of key distribution protocols and we compare its key with the well-known protocols such as BB84, PBC0 and generation rate to the well-known protocols such as BB84, PBC0 and R04. We also state the security analysis of these protocols based on the entanglement distillation and CSS codes techniques.

}, doi = {10.1007/s11128-018-1998-3}, url = {https://doi.org/10.1007/s11128-018-1998-3}, author = {Dariusz Kurzyk and {\L}ukasz Pawela and Zbigniew Pucha{\l}a} } @article {2655, title = {Unified approach to geometric and positive-map-based non-linear entanglement identifiers}, journal = {Phys. Rev. A }, volume = {97}, year = {2018}, pages = {042339}, abstract = {Detecting quantumness of correlations (especially entanglement) is a very hard task even in the simplest case, i.e., two-partite quantum systems. Here we provide an analysis of whether there exists a relation between two of the most popular types of entanglement identifiers: the first one based on positive maps and not directly applicable in the laboratory and the second one a geometric entanglement identifier which is based on specific Hermiticity-preserving maps. We show a profound relation between those two types of entanglement criteria. Hereunder we have proposed a general framework of nonlinear functional entanglement identifiers which allows us to construct experimentally friendly entanglement criteria.

}, doi = {10.1103/PhysRevA.97.042339}, url = {https://doi.org/10.1103/PhysRevA.97.042339}, author = {Marcin Markiewicz and Adrian Kolodziejski and Zbigniew Pucha{\l}a and Adam Rutkowski and Tomasz Tylec and Wieslaw Laskowski} } @article {2618, title = {Vertices cannot be hidden from quantum spatial search for almost all random graphs}, journal = {Quantum Information Processing}, volume = {17}, year = {2018}, pages = {81}, doi = {10.1007/s11128-018-1844-7}, url = {https://doi.org/10.1007/s11128-018-1844-7}, author = {Adam Glos and Aleksandra Krawiec and Ryszard Kukulski and Zbigniew Pucha{\l}a} } @article {2552, title = {Quantum noise generated by local random Hamiltonians}, journal = {Physical Review A}, volume = {95}, year = {2017}, chapter = {032333}, abstract = {We investigate the impact of a local random unitary noise on multipartite quantum states of arbitrary dimension. We follow the dynamical approach, in which the single-particle unitaries are generated by local random Hamiltonians. Assuming short evolution time we derive an upper bound on the fidelity between an initial and the final state transformed by this type of noise. This result is based on averaging the Tamm-Mandelstam bound and holds for a wide class of distributions of random Hamiltonians fulfilling specific symmetry conditions. It is showed that the sensitivity of a given pure quantum state to the discussed type of noise depends only on the properties of a single-particle and bipartite reduced states.

}, doi = {10.1103/PhysRevA.95.032333}, url = {https://doi.org/10.1103/PhysRevA.95.032333}, author = {Marcin Markiewicz and Zbigniew Pucha{\l}a and Anna de Rosier and Wies{\l}aw Laskowski and Karol {\.Z}yczkowski} } @article {iitisid_0518, title = {Symbolic integration with respect to the Haar measure on the unitary groups}, journal = {Bulletin of the Polish Academy of Sciences Technical Sciences}, volume = {65}, year = {2017}, note = {arXiv:1109.4244}, month = {02/2017}, chapter = {21}, abstract = {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.

}, doi = {10.1515/bpasts-2017-0003}, url = {https://doi.org/10.1515/bpasts-2017-0003}, author = {Zbigniew Pucha{\l}a and Jaroslaw Miszczak} } @article {iitisid_0689, title = {Asymptotic entropic uncertainty relations}, journal = {J. Math. Phys. }, volume = {57}, year = {2016}, note = {arXiv:1412.7065v1}, chapter = {032204}, abstract = {Entropic uncertainty relations are analyzed for the case of N-dimensional Hilbert space and two orthogonal measurements performed in two generic bases, related by a Haar random unitary matrix U. We derive estimations for the average norms of truncations of U of a given size, which allow us to study state-independent lower bounds for the sum of two entropies describing the measurements outcomes. In particular, we show that the Maassen{\textendash}Uffink bound asymptotically behaves as lnN-lnlnN-ln2, while the strong entropic majorization relation yields a nearly optimal bound, lnN-const. Analogous results are also obtained for a more general case of several orthogonal measurements performed in generic bases.

}, doi = {10.1063/1.4944425}, url = {http://dx.doi.org/10.1063/1.4944425}, author = {Rados{\l}aw Adamczak and Rafa{\l} Lata{\l}a and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {iitisid_0734, title = {Distinguishability of generic quantum states}, journal = {Physical Review A}, volume = {93}, year = {2016}, note = {arXiv:1507.05123}, chapter = {062112}, abstract = {Properties of random mixed states of order N distributed uniformly with respect to the Hilbert-Schmidt measure are investigated. We show that for large N, due to the concentration of measure, the trace distance between two random states tends to a fixed number D\ =1/4+1/π, which yields the Helstrom bound on their distinguishability. To arrive at this result we apply free random calculus and derive the symmetrized Marchenko{\textendash}Pastur distribution, which is shown to describe numerical data for the model of two coupled quantum kicked tops. Asymptotic values for the fidelity, Bures and transmission distances between two random states are obtained. Analogous results for quantum relative entropy and Chernoff quantity provide other bounds on the distinguishablity of both states in a multiple measurement setup due to the quantum Sanov theorem.

}, doi = {10.1103/PhysRevA.93.062112}, url = {https://doi.org/10.1103/PhysRevA.93.062112}, author = {Zbigniew Pucha{\l}a and {\L}ukasz Pawela and Karol {\.Z}yczkowski} } @article {iitisid_0690, title = {Quantifying channels output similarity with applications to quantum control}, journal = {Quantum Information Processing}, volume = {15}, year = {2016}, note = {arXiv:1412.8694}, chapter = {1455}, abstract = {In this work we aim at quantifying quantum channel output similarity. In order to achieve this, we introduce the notion of quantum channel superfidelity, which gives us an upper bound on the quantum channel fidelity. This quantity is expressed in a clear form using the Kraus representation of a quantum channel. As examples, we show potential applications of this quantity in the quantum control field.

}, doi = {10.1007/s11128-015-1238-z}, url = {https://doi.org/10.1007/s11128-015-1238-z}, author = {{\L}ukasz Pawela and Zbigniew Pucha{\l}a} } @article {iitisid_0733, title = {Certainty relations, mutual entanglement and non-displacable manifolds}, journal = {Phys. Rev. A}, volume = {92}, year = {2015}, pages = {032109}, abstract = {{We derive explicit bounds for the average entropy characterizing measurements of a pure quantum state of size N in L orthogonal bases. Lower bounds lead to novel entropic uncertainty relations, while upper bounds allow us to formulate universal certainty relations. For L=2 the maximal average entropy saturates at logN as there exists a mutually coherent state, but certainty relations are shown to be nontrivial for L>=3 measurements. In the case of a prime power dimension

}, doi = {10.1103/PhysRevA.92.032109}, url = {https://doi.org/10.1103/PhysRevA.92.032109}, author = {Zbigniew Pucha{\l}a and {\L}. Rudnicki and K. Chabuda and M. Paraniak and Karol {\.Z}yczkowski} } @article {iitisid_0721, title = {Exploring boundaries of quantum convex structures: Special role of unitary processes}, journal = {Phys. Rev. A}, volume = {92}, year = {2015}, pages = {012304}, abstract = {We address the question of finding the most effective convex decompositions into boundary elements (so-called boundariness) for sets of quantum states, observables and channels. First we show that in general convex sets the boundariness essentially coincides with the question of the most distinguishable element, thus, providing an operational meaning for this concept. Unexpectedly, we discovered that for any interior point of the set of channels the optimal decomposition necessarily contains a unitary channel. In other words, for any given channel the best distinguishable one is some unitary channel. Further, we prove that boundariness is sub-multiplicative under composition of systems and explicitly evaluate its maximal value that is attained only for the most mixed elements of the considered convex structures.

}, doi = {10.1103/PhysRevA.92.012304}, url = {https://doi.org/10.1103/PhysRevA.92.012304}, author = {Zbigniew Pucha{\l}a and A. Jencova and M. Sedlak and M. Ziman} } @article {iitisid_0732, title = {Minimal Renyi-Ingarden-Urbanik entropy of multipartite quantum states}, journal = {Entropy}, volume = {17}, number = {7}, year = {2015}, pages = {5063{\textendash}5084}, abstract = {We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with d levels each. It can be described by the R{\textquoteright}enyi-Ingarden-Urbanik entropy Sq of a decomposition of the state in a product basis, minimized over all local unitary transformations. In the case q=0 this quantity becomes a function of the rank of the tensor representing the state, while in the limit q{\textrightarrow}$\infty$ the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system the entropy S1 coincides with the standard entanglement entropy. We analyze the distribution of the minimal entropy for random states of three and four-qubit systems. In the former case the distributions of 3-tangle is studied and some of its moments are evaluated, while in the latter case we analyze the distribution of the hyperdeterminant. The behavior of the maximum overlap of a three-qudit system with the closest separable state is also investigated in the asymptotic limit.

}, doi = {10.3390/e17075063}, url = {https://doi.org/10.3390/e17075063}, author = {M. Enr{\'\i}quez and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {iitisid_0621, title = {Quantum control robust with respect to coupling with an external environment}, journal = {Quantum Information Processing}, volume = {14}, number = {2}, year = {2015}, note = {arXiv:1306.6826}, pages = {437{\textendash}446}, doi = {10.1007/s11128-014-0879-7}, url = {https://doi.org/10.1007/s11128-014-0879-7}, author = {{\L}ukasz Pawela and Zbigniew Pucha{\l}a} } @article {iitisid_0686, title = {Real numerical shadow and generalized B-splines}, journal = {Linear Algebra and its Applications}, volume = {479}, year = {2015}, note = {arXiv:1409.4941}, pages = {12{\textendash}51}, abstract = {arXiv:1409.4941

}, doi = {10.1016/j.laa.2015.03.029}, url = {https://doi.org/10.1016/j.laa.2015.03.029}, author = {Charles F. Dunkl and Piotr Gawron and {\L}ukasz Pawela and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {2424, title = {Wybory samorz{\k a}dowe 2014 w poszukiwaniu anomalii statystycznych}, journal = {Electoral Studies}, volume = {30}, year = {2015}, pages = {534{\textendash}545}, author = {Piotr Gawron and {\L}ukasz Pawela and Zbigniew Pucha{\l}a and Szklarski, Jacek and Karol {\.Z}yczkowski} } @article {iitisid_0593, title = {Constructive entanglement test from triangle inequality}, journal = {J. Phys. A: Math. Theor.}, volume = {47}, number = {42}, year = {2014}, note = {arXiv:1211.2306}, pages = {424035}, abstract = {We derive a simple lower bound on the geometric measure of entanglement for mixed quantum states in the case of a general multipartite system. The main ingredient of the presented derivation is the triangle inequality applied to the root infidelity distance in the space of density matrices. The obtained bound leads to entanglement criteria with a straightforward interpretation. The proposed criteria provide an experimentally accessible, powerful entanglement test. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to {\textquoteright}50 years of Bell{\textquoteright}s theorem{\textquoteright}.

}, doi = {10.1088/1751-8113/47/42/424035}, url = {https://doi.org/10.1088/1751-8113/47/42/424035}, author = {{\L}. Rudnicki and Zbigniew Pucha{\l}a and Pawe{\l} Horodecki and Karol {\.Z}yczkowski} } @article {iitisid_0682, title = {Diagonal unitary entangling gates and contradiagonal quantum states}, journal = {Phys. Rev. A}, volume = {90}, year = {2014}, note = {arXiv:1407.1169 doi:10.1103/PhysRevA.90.032303}, pages = {032303}, abstract = {Nonlocal properties of an ensemble of diagonal random unitary matrices of order N2 are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as logN, in contrast to the logN2 behavior characteristic to random unitary gates. Entangling power of a diagonal gate U is related to the von Neumann entropy of an auxiliary quantum state ρ=AA{\textdagger}/N2, where the square matrix A is obtained by reshaping the vector of diagonal elements of U of length N2 into a square matrix of order N. This fact provides a motivation to study the ensemble of non-hermitian unimodular matrices A, with all entries of the same modulus and random phases and the ensemble of quantum states ρ, such that all their diagonal entries are equal to 1/N. Such a state is contradiagonal with respect to the computational basis, in sense that among all unitary equivalent states it maximizes the entropy copied to the environment due to the coarse graining process. The first four moments of the squared singular values of the unimodular ensemble are derived, based on which we conjecture a connection to a recently studied combinatorial object called the

}, doi = {10.1103/PhysRevA.90.032303}, url = {https://doi.org/10.1103/PhysRevA.90.032303}, author = {A. Lakshminarayan and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {iitisid_0558, title = {Quantum control with spectral constraints}, journal = {Quantum Information Processing.}, volume = {13}, year = {2014}, note = {arXiv:1204.6557}, pages = {227{\textendash}237}, abstract = {Various constraints concerning control fields can be imposed in the realistic implementations of quantum control systems. One of the most important is the restriction on the frequency spectrum of acceptable control parameters. It is important to consider the limitations of experimental equipment when trying to find appropriate control parameters. Therefore, in this paper we present a general method of obtaining a piecewise-constant controls, which are robust with respect to spectral constraints. We consider here a Heisenberg spin chain, however the method can be applied to a system with more general interactions. To model experimental restrictions we apply an ideal low-pass filter to numerically obtained control pulses. The usage of the proposed method has negligible impact on the control quality as opposed to the standard approach, which does not take into account spectral limitations.

}, doi = {10.1007/s11128-013-0644-3}, url = {https://doi.org/10.1007/s11128-013-0644-3}, author = {{\L}ukasz Pawela and Zbigniew Pucha{\l}a} } @article {iitisid_0656, title = {Strong Majorization Entropic Uncertainty Relations}, journal = {Phys. Rev. A}, volume = {89}, number = {5}, year = {2014}, note = {arXiv:1402.0129}, pages = {052115}, abstract = {We analyze entropic uncertainty relations in a finite dimensional Hilbert space and derive several strong bounds for the sum of two entropies obtained in projective measurements with respect to any two orthogonal bases. We improve the recent bounds by Coles and Piani [ArXiv:1307.4265], which are known to be stronger than the well known result of Maassen and Uffink. Furthermore, we find a novel bound based on majorization techniques, which also happens to be stronger than the recent results involving largest singular values of submatrices of the unitary matrix connecting both bases. The firsts set of new bounds give better results for unitary matrices close to the Fourier matrix, while the second one works better in the opposite sectors. Some results derived admit generalization to arbitrary mixed states and the bounds are increased by the von Neumann entropy of the measured state

}, doi = {10.1103/PhysRevA.89.052115}, url = {https://doi.org/10.1103/PhysRevA.89.052115}, author = {{\L}. Rudnicki and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {iitisid_0595, title = {Analysis of patent activity in the field of quantum information processing}, journal = {International Journal of Quantum Information}, volume = {11}, number = {1}, year = {2013}, note = {arXiv:1212.2439}, pages = {1350007}, issn = {0219-7499}, author = {Ryszard Winiarczyk and Piotr Gawron and Jaroslaw Miszczak and {\L}ukasz Pawela and Zbigniew Pucha{\l}a} } @article {iitisid_0590, title = {Enhancing pseudo-telepathy in the Magic Square game}, journal = {PLOS ONE}, volume = {8}, year = {2013}, note = {arXiv:1211.1213}, pages = {e64694}, author = {{\L}ukasz Pawela and Piotr Gawron and Zbigniew Pucha{\l}a and Jan S{\l}adkowski} } @article {iitisid_0562, title = {Entropic trade-off relations for quantum operations}, journal = {Phys. Rev. A}, volume = {87}, number = {3}, year = {2013}, note = {arXiv:1206.2536 doi: 10.1103/PhysRevA.87.032308}, pages = {032308}, abstract = {Spectral properties of an arbitrary matrix can be characterized by the entropy of its rescaled singular values. Any quantum operation can be described by the associated dynamical matrix or by the corresponding superoperator. The entropy of the dynamical matrix describes the degree of decoherence introduced by the map, while the entropy of the superoperator characterizes the a priori knowledge of the receiver of the outcome of a quantum channel Φ. We prove that for any map acting on an N-dimensional quantum system the sum of both entropies is not smaller than lnN. For any bistochastic map this lower bound reads 2lnN. We investigate also the corresponding R{\'e}nyi entropies, providing an upper bound for their sum, and analyze the entanglement of the bi-partite quantum state associated with the channel.}, author = {Wojciech Roga and Zbigniew Pucha{\l}a and {\L}. Rudnicki and Karol {\.Z}yczkowski} } @article {iitisid_0537, title = {Increasing the security of the ping-pong protocol by using many mutually unbiased bases}, journal = {Quantum Information Processing}, volume = {12}, number = {1}, year = {2013}, note = {arXiv:1201.3230}, pages = {569{\textendash}576}, abstract = {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.}, author = {P. Zawadzki and Zbigniew Pucha{\l}a and Jaroslaw Miszczak} } @article {iitisid_0544, title = {Local controllability of quantum systems}, journal = {Quantum Information Processing}, volume = {12}, number = {1}, year = {2013}, note = {arXiv:1203.4056}, pages = {459{\textendash}466}, abstract = {We give a criterion that is sufficient for controllability of multipartite quantum systems. We generalize the graph infection criterion to the quantum systems that cannot be described with the use of a graph theory. We introduce the notation of hypergraphs and reformulate the infection property in this setting. The introduced criterion has a topological nature and therefore it is not connected to any particular experimental realization of quantum information processing.}, author = {Zbigniew Pucha{\l}a} } @article {iitisid_0617, title = {Majorization entropic uncertainty relations}, journal = {J. Phys. A: Math. Theor.}, volume = {46}, year = {2013}, note = {arXiv:1304.7755}, pages = {272002}, abstract = {Entropic uncertainty relations in a finite dimensional Hilbert space are investigated. Making use of the majorization technique we derive explicit lower bounds for the sum of Renyi entropies describing probability distributions associated with a given pure state expanded in eigenbases of two observables. The bounds obtained are expressed in terms of the largest singular values of submatrices of the unitary rotation matrix. For a generic unitary matrix of size N = 5 the bound obtained is stronger than the one of Maassen and Uffink (MU) with probability larger than 98\%, and this ratio increases with N. We show also that the bounds investigated are invariant for unitary matrices equivalent up to dephasing and permutation and derive a classical analogue of the MU uncertainty relation formulated for stochastic transition matrices.}, author = {Zbigniew Pucha{\l}a and {\L}. Rudnicki and Karol {\.Z}yczkowski} } @article {iitisid_0586, title = {A model for quantum queue}, journal = {International Journal of Quantum Information}, volume = {11}, number = {2}, year = {2013}, note = {arXiv:1210.8339}, pages = {1350023}, issn = {0219-7499}, author = {Piotr Gawron and Dariusz Kurzyk and Zbigniew Pucha{\l}a} } @article {iitisid_0589, title = {Collectibility for Mixed Quantum States}, journal = {Phys. Rev. A}, volume = {86}, number = {6}, year = {2012}, note = {arXiv:1211.0573 doi:10.1103/PhysRevA.86.062329}, pages = {062329}, abstract = {Bounds analogous to entropic uncertainty relations allow one to design practical tests to detect quantum entanglement by a collective measurement performed on several copies of the state analyzed. This approach, initially worked out for pure states only [ Phys. Rev. Lett. 107 150502 (2011)], is extended here for mixed quantum states. We define collectibility for any mixed states of a multipartite system. Deriving bounds for collectibility for positive partially transposed states of given purity provides insight into the structure of entangled quantum states. In the case of two qubits the application of complementary measurements and coincidence based detections leads to a test of entanglement of pseudopure states.}, author = {{\L}. Rudnicki and Zbigniew Pucha{\l}a and Pawe{\l} Horodecki and Karol {\.Z}yczkowski} } @article {iitisid_0522, title = {Notes on the Riccati operator equation in open quantum systems}, journal = {J. Math. Phys}, volume = {53}, year = {2012}, note = {arXiv:1111.4380 IF=1.291(2010); 5IF=1.210(2010);}, pages = {012106}, abstract = {Recent problem [B. Gardas, J. Math. Phys. {bf 52}, 042104 (2011)] concerning an antilinear solution of the Riccati equation is solved. We also exemplify that a simplification of the Riccati equation, even under reasonable assumptions, can lead to a not equivalent equation.

}, author = {Bart{\l}omiej Gardas and Zbigniew Pucha{\l}a} } @article {iitisid_0513, title = {Qubit flip game on a Heisenberg spin chain}, journal = {Quantum Information Processing}, volume = {11}, number = {6}, year = {2012}, note = {arXiv:1108.0642}, pages = {1571{\textendash}1583}, abstract = {We study a quantum version of a penny flip game played using control parameters of the Hamiltonian in the Heisenberg model. Moreover, we extend this game by introducing auxiliary spins which can be used to alter the behaviour of the system. We show that a player aware of the complex structure of the system used to implement the game can use this knowledge to gain higher mean payoff.}, author = {Jaroslaw Miszczak and Piotr Gawron and Zbigniew Pucha{\l}a} } @article {iitisid_0536, title = {Restricted numerical shadow and geometry of quantum entanglement}, journal = {J. Phys. A: Math. Theor.}, volume = {45}, year = {2012}, note = {arXiv:1201.2524}, pages = {415309}, abstract = {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 {\textendash} a normalized probability distribution on the complex plane supported in W\_R(A). Its value at point z in C is equal to the probability that the expectation value is equal to z, where |psi> represents a random quantum state in subset R distributed according to the natural measure on this set, induced by the unitarily invariant Fubini{\textendash}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 x 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{\textendash}qubit system.}, author = {Zbigniew Pucha{\l}a and Jaroslaw Miszczak and Piotr Gawron and Charles F. Dunkl and J.A. Holbrook and Karol {\.Z}yczkowski} } @inbook {iitisid_0512, title = {Eigengestures for natural human computer interface}, booktitle = {Man-Machine Interactions 2}, year = {2011}, note = {arXiv:1105.1293}, pages = {49{\textendash}56}, publisher = {Springer}, organization = {Springer}, address = {Berlin / Heidelberg}, abstract = {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.}, isbn = {978-3-642-23168-1}, author = {Piotr Gawron and Przemys{\l}aw G{\l}omb and Jaroslaw Miszczak and Zbigniew Pucha{\l}a}, editor = {Tadeusz Czach{\'o}rski and Stanis{\l}aw Kozielski and Urszula Sta{\'n}czyk} } @article {iitisid_0466, title = {Experimentally feasible measures of distance between quantum operations}, journal = {Quantum Information Processing}, volume = {10}, number = {1}, year = {2011}, note = {arXiv:0911.0567 IF=2.085(2010);}, pages = {1{\textendash}12}, abstract = {We present two measures of distance between quantum processes which can be measured directly in laboratory without resorting to process tomography. The measures are 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. We also provide a physical interpretation of the introduced metrics based on the continuity of channel capacity.}, issn = {1570-0755}, author = {Zbigniew Pucha{\l}a and Jaroslaw Miszczak and Piotr Gawron and Bart{\l}omiej Gardas} } @article {iitisid_0509, title = {Numerical shadow and geometry of quantum states}, journal = {J. Phys. A: Math. Theor.}, volume = {44}, number = {33}, year = {2011}, note = {arXiv:1104.2760 IF=1.641(2010);}, pages = {335301}, abstract = {The totality of normalised density matrices of order N forms a convex set Q\_N in R^(N^2-1). Working with the flat geometry induced by the Hilbert-Schmidt distance we consider images of orthogonal projections of Q\_N onto a two-plane and show that they are similar to the numerical ranges of matrices of order N. For a matrix A of a order N one defines its numerical shadow as a probability distribution supported on its numerical range W(A), induced by the unitarily invariant Fubini-Study measure on the complex projective manifold CP^(N-1). We define generalized, mixed-states shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.}, issn = {1751-8113}, author = {Charles F. Dunkl and Piotr Gawron and J.A. Holbrook and Jaroslaw Miszczak and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {iitisid_0504, title = {Numerical shadows: measures and densities on the numerical range}, journal = {Linear Algebra Appl.}, volume = {434}, year = {2011}, note = {arXiv:1010.4189 IF=1.005(2010);}, pages = {2042{\textendash}2080}, abstract = {For any operator M acting on an N-dimensional Hilbert space H\_N we introduce its numerical shadow, which is a probability measure on the complex plane supported by the numerical range of M. The shadow of M at point z is defined as the probability that the inner product (Mu,u) is equal to z, where u stands for a random complex vector from H\_N, satisfying ||u||=1. In the case of N=2 the numerical shadow of a non-normal operator can be interpreted as a shadow of a hollow sphere projected on a plane. A similar interpretation is provided also for higher dimensions. For a hermitian M its numerical shadow forms a probability distribution on the real axis which is shown to be a one dimensional B-spline. In the case of a normal M the numerical shadow corresponds to a shadow of a transparent solid simplex in R^{N-1} onto the complex plane. Numerical shadow is found explicitly for Jordan matrices J\_N, direct sums of matrices and in all cases where the shadow is rotation invariant. Results concerning the moments of shadow measures play an important role. A general technique to study numerical shadow via the Cartesian decomposition is described, and a link of the numerical shadow of an operator to its higher-rank numerical range is emphasized.}, issn = {0024-3795}, author = {Charles F. Dunkl and Piotr Gawron and J.A. Holbrook and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {iitisid_0514, title = {Probability measure generated by the superfidelity}, journal = {J. Phys. A: Math. Theor.}, volume = {44}, year = {2011}, note = {arXiv:1107.2792 IF=1.641(2010);}, pages = {405301}, abstract = {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.}, author = {Zbigniew Pucha{\l}a and Jaroslaw Miszczak} } @article {iitisid_0473, title = {Product numerical range in a space with tensor product structure}, journal = {Linear Algebra Appl.}, volume = {434}, number = {1}, year = {2011}, note = {arXiv:1008.3482 IF=1.005(2010);}, pages = {327{\textendash}342}, abstract = {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.}, issn = {0024-3795}, author = {Zbigniew Pucha{\l}a and Piotr Gawron and Jaroslaw Miszczak and {\L}. Skowronek and Man-Duen Choi and Karol {\.Z}yczkowski} } @article {iitisid_0505, title = {Stationary states of two-level open quantum systems}, journal = {J. Phys. A: Math. Theor.}, volume = {44}, number = {21}, year = {2011}, note = {arXiv:1006.3328 IF=1.641(2010);}, pages = {215306}, abstract = {A problem of finding stationary states of open quantum systems is addressed. We focus our attention on a generic type of open system: a qubit coupled to its environment. We apply the theory of block operator matrices and find stationary states of two-level open quantum systems under certain conditions applied on both the qubit and the surrounding.}, author = {Bart{\l}omiej Gardas and Zbigniew Pucha{\l}a} } @article {iitisid_0476, title = {Restricted numerical range: A versatile tool in the theory of quantum information}, journal = {J. Math. Phys.}, volume = {51}, number = {10}, year = {2010}, note = {arXiv:0905.3646 IF=1.291(2010);}, pages = {102204}, issn = {00222488}, author = {Piotr Gawron and Zbigniew Pucha{\l}a and Jaroslaw Miszczak and {\L}. Skowronek and Karol {\.Z}yczkowski} } @article {iitisid_0431, title = {Bound on trace distance based on superfidelity}, journal = {Phys. Rev. A}, volume = {79}, year = {2009}, note = {arXiv:0811.2323 IF=2.866(2009);}, pages = {024302}, abstract = {We provide a bound for the trace distance between two quantum states. The lower bound is based on the superfidelity, which provides the upper bound on quantum fidelity. One of the advantages of the presented bound is that it can be estimated using a simple measurement procedure. We also compare this bound with the one provided in terms of fidelity.}, author = {Zbigniew Pucha{\l}a and Jaroslaw Miszczak} } @article {iitisid_0421, title = {Sub- and super-fidelity as bounds for quantum fidelity}, journal = {Quantum Information \& Computation}, volume = {9}, number = {1\&2}, year = {2009}, note = {arXiv:0805.2037 IF=2.980(209); IF5=2.402(2010);}, month = {1}, pages = {0103{\textendash}0130}, abstract = {We derive several bounds on fidelity between quantum states. In particular we show that fidelity is bounded from above by a simple to compute quantity we call super{\textendash}fidelity. It is analogous to another quantity called sub{\textendash}fidelity. For any two states of a two{\textendash}dimensional quantum system ($N=2$) all three quantities coincide. We demonstrate that sub{\textendash} and super{\textendash}fidelity are concave functions. We also show that super{\textendash}fidelity is super{\textendash}multiplicative while sub{\textendash}fidelity is sub{\textendash}multiplicative and design feasible schemes to measure these quantities in an experiment. Super{\textendash}fidelity can be used to define a distance between quantum states. With respect to this metric the set of quantum states forms a part of a $N^2-1$ dimensional hypersphere.}, issn = {1533-7146}, author = {Jaroslaw Miszczak and Zbigniew Pucha{\l}a and Pawe{\l} Horodecki and A. Uhlmann and Karol {\.Z}yczkowski} } @article {iitisid_0425, title = {The exact asymptotic of the collision time tail distribution for independent Brownian particles with different drifts}, journal = {Probability Theory and Related Fields}, volume = {3-4}, year = {2008}, note = {arXiv:0704.0215 IF=1.569(2008);}, month = {11}, pages = {595{\textendash}617}, author = {Zbigniew Pucha{\l}a and T. Rolski} } @article {iitisid_0402, title = {Quantum state discrimination: A geometric approach}, journal = {Phys. Rev. A}, volume = {77}, number = {4}, year = {2008}, note = {arXiv:0711.4286 IF=2.908(2008);}, pages = {042111}, abstract = {We analyze the problem of finding sets of quantum states that can be deterministically discriminated. From a geometric point of view, this problem is equivalent to that of embedding a simplex of points whose distances are maximal with respect to the Bures distance (or trace distance). We derive upper and lower bounds for the trace distance and for the fidelity between two quantum states, which imply bounds for the Bures distance between the unitary orbits of both states. We thus show that, when analyzing minimal and maximal distances between states of fixed spectra, it is sufficient to consider diagonal states only. Hence when optimal discrimination is considered, given freedom up to unitary orbits, it is sufficient to consider diagonal states. This is illustrated geometrically in terms of Weyl chambers.}, issn = {1094-1622}, author = {D. Markham and Jaroslaw Miszczak and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {iitisid_0396, title = {Context selection for efficient bit modeling of contourlet transform coefficients}, journal = {Theoretical and Applied Informatics}, volume = {19}, number = {2}, year = {2007}, pages = {135{\textendash}146}, abstract = {Contourlet transform is named as one possible successor to the wavelet transform as a tool for representation of image information. It has better properties in representation of edge information, which make it suitable for designing image compression systems. It is not clear, however, how to exploit the statistical structure of the coefficients generated by this transform in effective compression algorithm. In this work, authors propose a general, nonparametric method to achieve context map, thus allowing for efficient application of EBCOT/JPEG 2000 bit modeling and encoding approach to contourlet transformed images. The article presents the derivation of the method and experiments performed on a publicly available image set. Keywords: contourlet transform, coefficient bit modeling, context estimation}, issn = {ISSN 1896{\textendash}5334}, author = {Przemys{\l}aw G{\l}omb and Zbigniew Pucha{\l}a and Arkadiusz Sochan} } @article {iitisid_0390, title = {The exact asymptotic of the time to collision}, journal = {Electronic Journal of Probability}, volume = {10}, number = {40}, year = {2005}, note = {IF=0.676(2006);}, month = {11}, pages = {1359{\textendash}1380}, abstract = {Abstract In this note we consider the time of the collision $tau$ for $n$ independent copies of Markov processes $X^1_t,. . .,X^n_t$, each starting from $x_i$,where $x_1 t) = t^{-n(n-1)/4}(Ch(x)+o(1)),$ where $C$ is known and $h(x)$ is the Vandermonde determinant. From the proof one can see that the result also holds for $X_t$ being the Brownian motion or the Poisson process. An application to skew standard Young tableaux is given.}, author = {Zbigniew Pucha{\l}a and T. Rolski} } @article {iitisid_0391, title = {Proof of Grabiner{\textquoteright}s theorem on non-colliding particles}, journal = {Probability and Mathematical Statistics}, volume = {25}, number = {1}, year = {2005}, pages = {129{\textendash}132}, abstract = {A detail proof of Grabiner{\textquoteright}s theorem [1] on the exact asymptotics of the time to collision for n independent Brownian motions is given.}, author = {Zbigniew Pucha{\l}a} }