On Kadison-Schwarz type quantum quadratic operators on \bm_2(\mathbb{C})

In the present paper we study description of Kadison-Schwarz type quantum quadratic operators acting from \bm_2(\mathbb{C}) into \bm_2(\mathbb{C})\o\bm_2(\mathbb{C}). Note that such kind of operator is a generalization of quantum convolution. By means of such a description we provide an example of q.q.o. which is not a Kadision-Schwartz operator. Moreover, we study dynamics of an associated nonlinear (i.e. quadratic) operators acting on the state space of \bm_2(\mathbb{C}).Comment: 14 pages. arXiv admin note: text overlap with arXiv:0902.450

Entropy production rates of bistochastic strictly contractive quantum channels on a matrix algebra

We derive, for a bistochastic strictly contractive quantum channel on a matrix algebra, a relation between the contraction rate and the rate of entropy production. We also sketch some applications of our result to the statistical physics of irreversible processes and to quantum information processing.Comment: 7 pages; revised version submitted to J. Phys.

On pure quasi quantum quadratic operators of M_2(C)

In the present paper we study quasi quantum quadratic operators (q.q.o) acting on the algebra of $2\times 2$ matrices $M_2(C)$. It is known that a channel is called pure if it sends pure states to pure ones. In this papers, we introduce a weaker condition, called $q$-purity, than purity of the channel. To study $q$-pure channels, we concentrate ourselves to quasi q.q.o. acting on $M_2(C)$. We describe all trace-preserving quasi q.q.o. on $M_2(C)$, which allowed us to prove that if a trace-preserving symmetric quasi q.q.o. such that the corresponding quadratic operator is linear, then its $q$-purity implies its positivity. If a symmetric quasi q.q.o. has a Haar state $\tau$, then its corresponding quadratic operator is nonlinear, and it is proved that such $q$-pure symmetric quasi q.q.o. cannot be positive. We think that such a result will allow to check whether a given mapping from $M_2(C)$ to M_2(C)\o M_2(C) is pure or not. On the other hand, our study is related to construction of pure quantum nonlinear channels. Moreover, it is also considered that nonlinear dynamics associated with quasi pure q.q.o. may have differen kind of dynamics, i.e. it may behave chaotically or trivially, respectively.Comment: 14 page

Role of Partial Transpose and Generalized Choi maps in Quantum Dynamical Semigroups involving Separable and Entangled States

Power symmetric matrices defned and studied by R. Sinkhorn (1981) and their generalization by R.B. Bapat, S.K. Jain and K. Manjunatha Prasad (1999) have been utilized to give positive block matrices with trace one possessing positive partial transpose, the so-called PPT states. Another method to construct such PPT states is given, it uses the form of a matrix unitarily equivalent to its transpose obtained by S.R. Garcia and J.E. Tener (2012). Evolvement or suppression of separability or entanglement of various levels for a quantum dynamical semigroup of completely positive maps has been studied using Choi-Jamiolkowsky matrix of such maps and the famous Horodecki's criteria (1996). A Trichotomy Theorem has been proved, and examples have been given that depend mainly on generalized Choi maps and clearly distinguish the levels of entanglement breaking.Comment: A few corrections and changes in view of discussion with Matthias Christand

FLOW OF QUANTUM GENETIC LOTKA-VOLTERRA ALGEBRAS ON M2(ℂ)

In this thesis, a class of flow quantum Lotka-Volterra genetic algebras (FQLVG-A) is investigated and its structure is studied. Moreover, the necessary and sufficient conditions for the associativity and alternatively of FQGLV-A are derived. In addition, idempotent elements in FQGLV-A are found. Also, derivations of a class of FQLVG-A are described. Also, the automorphisms of a class of FQLVG-A and their positivity are examined

Geometry of Quantum States from Symmetric Informationally Complete Probabilities

It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system correspond to complex vectors of unit length. These vectors can be combined to create more general states expressed in terms of positive semidefinite matrices of unit trace called density operators. A density operator tells us everything we know about a quantum system. In particular, it specifies a unique probability for any measurement outcome. Thus, to fully appreciate quantum mechanics as a statistical model for physical phenomena, it is necessary to understand the basic properties of its set of states. Studying the convex geometry of quantum states provides important clues as to why the theory is expressed most naturally in terms of complex amplitudes. At the very least, it gives us a new perspective into thinking about structure of quantum mechanics. This thesis is concerned with the structure of quantum state space obtained from the geometry of the convex set of probability distributions for a special class of measurements called symmetric informationally complete (SIC) measurements. In this context, quantum mechanics is seen as a particular restriction of a regular simplex, where the state space is postulated to carry a symmetric set of states called SICs, which are associated with equiangular lines in a complex vector space. The analysis applies specifically to 3-dimensional quantum systems or qutrits, which is the simplest nontrivial case to consider according to Gleason's theorem. It includes a full characterization of qutrit SICs and includes specific proposals for implementing them using linear optics. The infinitely many qutrit SICs are classified into inequivalent families according to the Clifford group, where equivalence is defined by geometrically invariant numbers called triple products. The multiplication of SIC projectors is also used to define structure coefficients, which are convenient for elucidating some additional structure possessed by SICs, such as the Lie algebra associated with the operator basis defined by SICs, and a linear dependency structure inherited from the Weyl-Heisenberg symmetry. After describing the general one-to-one correspondence between density operators and SIC probabilities, many interesting features of the set of qutrits are described, including an elegant formula for its pure states, which reveals a permutation symmetry related to the structure of a finite affine plane, the exact rotational equivalence of different SIC probability spaces, the shape of qutrit state space defined by the radial distance of the boundary from the maximally mixed state, and a comparison of the 2-dimensional cross-sections of SIC probabilities to known results. Towards the end, the representation of quantum states in terms of SICs is used to develop a method for reconstructing quantum theory from the postulate of maximal consistency, and a procedure for building up qutrit state space from a finite set of points corresponding to a Hesse configuration in Hilbert space is sketched briefly