Numerical shadow and geometry of quantum states

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.Comment: 19 pages, 5 figure

Numerical simulations of mixed states quantum computation

We describe quantum-octave package of functions useful for simulations of quantum algorithms and protocols. Presented package allows to perform simulations with mixed states. We present numerical implementation of important quantum mechanical operations - partial trace and partial transpose. Those operations are used as building blocks of algorithms for analysis of entanglement and quantum error correction codes. Simulation of Shor's algorithm is presented as an example of package capabilities.Comment: 6 pages, 4 figures, presented at Foundations of Quantum Information, 16th-19th April 2004, Camerino, Ital

Singular value decomposition and matrix reorderings in quantum information theory

We review Schmidt and Kraus decompositions in the form of singular value decomposition using operations of reshaping, vectorization and reshuffling. We use the introduced notation to analyse the correspondence between quantum states and operations with the help of Jamiolkowski isomorphism. The presented matrix reorderings allow us to obtain simple formulae for the composition of quantum channels and partial operations used in quantum information theory. To provide examples of the discussed operations we utilize a package for the Mathematica computing system implementing basic functions used in the calculations related to quantum information theory.Comment: 11 pages, no figures, see http://zksi.iitis.pl/wiki/projects:mathematica-qi for related softwar

Restricted numerical shadow and geometry of quantum entanglement

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 $$ 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--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.Comment: 33 pages, 8 figures, IOP styl

Fidelity approach to quantum phase transitions

We review briefly the quantum fidelity approach to quantum phase transitions in a pedagogical manner. We try to relate all established but scattered results on the leading term of the fidelity into a systematic theoretical framework, which might provide an alternative paradigm for understanding quantum critical phenomena. The definition of the fidelity and the scaling behavior of its leading term, as well as their explicit applications to the one-dimensional transverse-field Ising model and the Lipkin-Meshkov-Glick model, are introduced at the graduate-student level. In addition, we survey also other types of fidelity approach, such as the fidelity per site, reduced fidelity, thermal-state fidelity, operator fidelity, etc; as well as relevant works on the fidelity approach to quantum phase transitions occurring in various many-body systems.Comment: 41 pages, 31 figures. We apologize if we omit acknowledging your relevant works. Do tell. An updated version with clearer figures can be found at: http://www.phy.cuhk.edu.hk/~sjgu/fidelitynote.pd

Alternative fidelity measure for quantum states

We propose an alternative fidelity measure (namely, a measure of the degree of similarity) between quantum states and benchmark it against a number of properties of the standard Uhlmann-Jozsa fidelity. This measure is a simple function of the linear entropy and the Hilbert-Schmidt inner product between the given states and is thus, in comparison, not as computationally demanding. It also features several remarkable properties such as being jointly concave and satisfying all of "Jozsa's axioms". The trade-off, however, is that it is supermultiplicative and does not behave monotonically under quantum operations. In addition, new metrics for the space of density matrices are identified and the joint concavity of the Uhlmann-Jozsa fidelity for qubit states is established.Comment: 12 pages, 3 figures. v2 includes minor changes, new references and new numerical results (Sec. IV

Immunogenicity and safety of a quadrivalent high-dose inactivated influenza vaccine compared with a standard-dose quadrivalent influenza vaccine in healthy people aged 60 years or older: a randomized Phase III trial

A quadrivalent high-dose inactivated influenza vaccine (IIV4-HD) is licensed for adults 6565 y of age based on immunogenicity and efficacy studies. However, IIV4-HD has not been evaluated in adults aged 60\u201364 y. This study compared immunogenicity and safety of IIV4-HD with a standard-dose quadrivalent influenza vaccine (IIV4-SD) in adults aged 6560 y. This Phase III, randomized, modified double-blind, active-controlled study enrolled 1,528 participants aged 6560 y, randomized 1:1 to a single injection of IIV4-HD or IIV4-SD. Hemagglutination inhibition (HAI) geometric mean titers (GMTs) were measured at baseline and D 28 and seroconversion assessed. Safety was described for 180 d after vaccination. The primary immunogenicity objective was superiority of IIV4-HD versus IIV4-SD, for all four influenza strains 28 d post vaccination in participants aged 60\u201364 and 6565 y. IIV4-HD induced a superior immune response versus IIV4-SD in terms of GMTs in participants aged 60\u201364 y and those aged 6565 y for all four influenza strains. IIV4-HD induced higher GMTs in those aged 60\u201364 y than those aged 6565 y. Seroconversion rates were higher for IIV4-HD versus IIV4-SD in each age-group for all influenza strains. Both vaccines were well tolerated in participants 6560 y of age, with no safety concerns identified. More solicited reactions were reported with IIV4-HD than with IIV4-SD. IIV4-HD provided superior immunogenicity versus IIV4-SD and was well tolerated in adults aged 6560 y. IIV4-HD is assumed to offer improved protection against influenza compared with IIV4-SD in adults aged 6560 y, as was previously assessed for adults aged 6565 y

Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa