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Constructing all extreme instances of the set of completely positive trace-preserving (CPTP) maps, i.e., quantum channels, is a challenging valuable open problem in quantum information theory. Here we introduce a systematic approach that enables us to construct exactly those extreme channels that are covariant with respect to a finite discrete group or a compact connected Lie group. Innovative labeling of quantum channels by group representations enables us to identify the subset of group-covariant channels whose elements are group-covariant generalized-extreme channels. Furthermore, we exploit essentials of group representation theory to introduce equivalence classes for the labels and also partition the set of group-covariant channels. As a result we show that it is enough to construct one representative of each partition. We construct Kraus operators for group-covariant generalized-extreme channels by solving systems of linear and quadratic equations for all candidates satisfying the necessary condition for being group-covariant generalized-extreme channels. Deciding whether these constructed instances are extreme or quasi-extreme is accomplished by solving system of linear equations. We formalize the problem of constructing and classifying group-covariant generalized extreme channels, thereby yielding an algorithmic approach to solving, which we express as pseudocode. To illustrate the application and value of our method, we solve for explicit examples of group-covariant extreme channels. With unbounded computational resources to execute our algorithm, our method always delivers a description of an extreme channel for any finite-dimensional Hilbert-space and furthermore guarantees a description of a group-covariant extreme channel for any dimension and for any finite-discrete or compact connected Lie group if such an extreme channel exists.

The problem of dephasing channel discrimination is addressed for finite-dimensional systems. In particular, the optimization with respect to input states without energy constraint is solved analytically for qubit, qutrit and ququart. Additionally, it is shown that resorting to side entanglement assisted strategy is completely useless in this case.

We study the reduced dynamics of open quantum spin chains of arbitrary length $N$ with nearest neighbour $XX$ interactions, immersed within an external constant magnetic field along the $z$ direction, whose end spins are weakly coupled to heat baths at different temperatures, via energy preserving couplings. We find the analytic expression of the unique stationary state of the master equation obtained in the so-called global approach based on the spectralization of the full chain Hamiltonian. Hinging upon the explicit stationary state, we reveal the presence of sink and source terms in the spin-flow continuity equation and compare their behaviour with that of the stationary heat flow. Moreover, we also obtain analytic expressions for the steady state two-spin reduced density matrices and for their concurrence. We then set up an algorithm suited to compute the stationary bipartite entanglement along the chain and to study its dependence on the Hamiltonian parameters and on the bath temperatures.

Within the standard weak-coupling limit, the reduced dynamics of open quantum spin chains with their two end spins coupled to two distinct heat baths at different temperatures are mainly derived using the so-called \textit{global} and \textit{local} approaches, in which, respectively, the spin self-interaction is and is not taken into account. In order to compare the differences between the two regimes, we concentrate on an open three-site $XX$ spin-chain, provide systematic techniques to addressing the global and local asymptotic states and then compare the asymptotic spin-transport features by studying the spin flux through the middle site. %we examine the transport properties of an open three-site $XX$ spin-chain. %We analytically derive the exact steady state of the open chain in the global approach and its first order expansion with respect to the self-interaction in the local one. %The steady state transport properties through the middle spin is then studied %in order to compare the physical scenarios associated with the two regimes. %By analytical and numerical means, Basing on the analytical expressions of the stationary states in the two regimes, we find that the local approach misses important global effects emerging as spin sink and source terms that can only be due to non negligible inter-spin interactions. Moreover, we show that the local asympotic transport features cannot be recovered from the global ones by letting the inter-spin coupling vanish thus pointing to the existence of different coupling ranges where only one approach is physically tenable and possibly a region where the two descriptions may coexist.

We introduce a homology-based technique for the classification of multiqubit state vectors with genuine entanglement. In our approach, we associate state vectors to data sets by introducing a metric-like measure in terms of bipartite entanglement, and investigate the persistence of homologies at different scales. This leads to a novel classification of multiqubit entanglement. The relative occurrence frequency of various classes of entangled states is also shown.

We propose the usage of persistent homologies to characterize multipartite entanglement. On a multi-qubit data set we introduce metric-like measures defined in terms of bipartite entanglement and then we derive barcodes. We show that, depending on the distance, they are able to produce different classifications. In one case, it is possible to obtain the standard separability classes. In the other case, a new classification of entangled states of three and four qubits is provided.

To control and utilize quantum features in small scale for practical applications such as quantum transport, it is crucial to gain deep understanding of quantum characteristics of states such as coherence. Here by introducing a technique that simplifies solving the dynamical equation, we study the dynamics of coherence in a system of qubits interacting with each other through a common bath at non zero temperature. Our results demonstrate that depending on initial state, environment temperature affect coherence and excitation transfer in different ways. We show that when initial state is incoherent, as time goes on, coherence and probability of excitation transfer increase. But for coherent initial state, we find a critical value of temperature, below which system loses its coherence in time which diminishes the probability of excitation transfer. Hence in order to achieve higher value of coherence and also higher probability of excitation transfer, temperature of the bath should go beyond that critical value. Stationary coherence and probability of finding excited qubits in steady state, are discussed. We also elaborate on dependence of critical value of bath temperature on system size.

We introduce a model of a non-Gaussian quantum channel that stems from the composition of two physically relevant processes occurring in open quantum systems, namely, amplitude damping and dephasing. For it we find input states approaching zero output entropy while respecting the input energy constraint. These states fully exploit the infinite dimensionality of the Hilbert space. Upon truncation of the latter, the minimum output entropy remains finite, and optimal input states for such a case are conjectured thanks to numerical evidence.

We use the matrix product formalism to find exact ground states of two new spin-1 quantum chains with nearest neighbor interactions. One of the models, model I, describes a one-parameter family of quantum chains for which the ground state can be found exactly. In certain limit of the parameter, the Hamiltonian turns into the interesting case $H=\sum_i ({\bf S}_i\cdot {\bf S}_{i+1})^2$. The other model which we label as model II, corresponds to a family of solvable three-state vertex models on square two dimensional lattices. The ground state of this model is highly degenerate and the matrix product states is a generating state of such degenerate states. The simple structure of the matrix product state allows us to determine the properties of degenerate states which are otherwise difficult to determine. For both models we find exact expressions for correlation functions., 22 pages, references added, accepted for publication in European Physics Journal B

Three distant labs A, B and C, having no prior entanglement can establish a shared GHZ state, when one of them say A sends two particles to B and C for their local actions. The mediating particles remain separable from each other and from the particles of A, B and C. We prove that in this way, GHZ states are shared with a probability $\frac{1}{7}$. We also show how separable particles can be mediated to establish arbitrary $d-$ dimensional Bell states between distant labs. Our method is constructive and allows generaization of GHZ sharing between any number of parties and in any dimension. The proposed method may facilitate the construction of multi-node quantum networks and many other processes which use multi-partite entangled states., 5 pages, one figure, accepted for publication in Physical Review A

The stability of the topological order phase induced by the $Z_3$ Kitaev model, which is a candidate for fault-tolerant quantum computation, against the local order phase induced by the 3-State Potts model is studied. We show that the low energy sector of the Kitaev-Potts model is mapped to the Potts model in the presence of transverse magnetic field. Our study relies on two high-order series expansion based on continuous unitary transformations in the limits of small- and large-Potts couplings as well as mean-field approximation. Our analysis reveals that the topological phase of the $Z_3$ Kitaev model breaks down to the Potts model through a first order phase transition. We capture the phase transition by analysis of the ground state energy, one-quasiparticle gap and geometric measure of entanglement., Comment: 12 pages, 10 figures, Accepted for publication in Physical Review B

We consider two ensembles of qubit dissipating into two overlapping environments, that is with a certain number of qubit in common that dissipate into both environments. We then study the dynamics of bipartite entanglement between the two ensembles by excluding the common qubit. To get analytical solutions for an arbitrary number of qubit we consider initial states with a single excitation and show that the largest amount of entanglement can be created when excitations are initially located among side (non common) qubit. Moreover, the stationary entanglement exhibits a monotonic (resp. non-monotonic) scaling versus the number of common (resp. side) qubit.

We investigate the correlation properties of separable two qubit states with maximally mixed marginals. These states are divided to two sets with the same geometric quantum correlation. However a closer scrutiny of these states reveals a profound difference between their quantum correlations as measured by more probing measures. Although these two sets of states are prepared by the same type of quantum operations acting on classically correlated states with equal classical correlations, the amount of final quantum correlation is different. We investigate this difference and trace it back to the hidden classical correlation which exists in their preparation process. We also compare these states with regard to their usefulness for entanglement distribution and their robustness against noise., Accepted for Publication in Physical Review A

We study the Kitaev-Ising model, where ferromagnetic Ising interactions are added to the Kitaev model on a lattice. This model has two phases which are characterized by topological and ferromagnetic order. Transitions between these two kinds of order are then studied on a quasi-one-dimensional system, on a ladder, and on a two-dimensional periodic lattice, a torus. By exactly mapping the quasi-one-dimensional case to an anisotropic $XY$ chain we show that the transition occurs at zero $\ensuremath{\lambda}$, where $\ensuremath{\lambda}$ is the strength of the ferromagnetic coupling. In the two-dimensional case the model is mapped to a two-dimensional Ising model in transverse field, where it shows a transition at a finite value of $\ensuremath{\lambda}$. A mean-field treatment reveals the qualitative character of the transition and an approximate value for the transition point. Furthermore with perturbative calculation, we show that the expectation value of Wilson loops behaves as expected in the topological and ferromagnetic phases.

We provide an analytical investigation of the entanglement dynamics for a system composed of an arbitrary number of qubits dissipating into a common environment. Specifically, we consider product initial states with a given number of excitations whose evolution remains confined on low-dimensional subspaces of the operators space. We then find for which pairs of qubits entanglement can be generated and can persist at a steady state. Finally, we determine the stationary distribution of entanglement as well as its scaling versus the total number of qubits in the system.

We consider a model of bosonic memory channel, which induces correlations among the transmitted signals. The application of suitable unitary transformations at encoding and decoding stages allows the complete removal of correlations, mapping the memory channel into a memoryless one. However, such transformations, being global over an arbitrary large number of bosonic modes, are not realistically implementable. We then introduce a family of efficiently realizable transformations which can be used to partially remove correlations among errors, and we quantify the reduction of the gap with memoryless channels., 6 pages, 6 figures, comments welcome

We investigate the possibility of chaining qubits by letting pairs of nearest neighbours qubits dissipating into common environments. We then study entanglement dynamics within the chain and show that steady state entanglement can be achieved.

We characterize the completely positive trace-preserving maps on qutrits (qutrit channels) according to their covariance and symmetry properties. Both discrete and continuous groups are considered. It is shown how each symmetry group restricts arbitrariness in the parameters of the channel to a very small set. Although the explicit examples are related to qutrit channels, the formalism is sufficiently general to be applied to qudit channels.

We investigate the possibility of correcting errors occurring on a multipartite system through a feedback mechanism that acquires information from partial access to the environment. A partial control scheme of this kind might be useful when dealing with correlated errors. In fact, in such a case, it could be enough to gather local information to decide what kind of global recovery to perform. Then, we apply this scheme to the depolarizing and correlated errors, and quantify its performance by means of the entanglement fidelity.

We introduce a quantum channel describing photon losses depending on the degree of polarization mixedness. This can be regarded as a model of quantum channel with correlated errors between discrete and continuous degrees of freedom. We consider classical information over a continuous alphabet encoded on weak coherent states as well as classical information over a discrete alphabet encoded on single photons using dual rail representation. In both cases we study the one-shot capacity of the channel and its behaviour in terms of correlation between losses and polarization mixedness.

The notion of forgetfulness, used in discrete quantum memory channels, is slightly weakened in order to be applied to the case of continuous channels. This is done in the context of quantum memory channels with Markovian noise. As a case study, we apply the notion of weak-forgetfulness to a bosonic memory channel with additive noise. A suitable encoding and decoding unitary transformation allows us to unravel the effects of the memory, hence the channel capacities can be computed using known results from the memoryless setting., Comment: 6 pages, 2 figures, comments are welcome. Minor corrections and acknoledgment added

We investigate the question of optimal input ensembles for memory channels and construct a rather large class of Pauli channels with correlated noise which can be studied analytically with regard to the entanglement of their optimal input ensembles. In a more detailed study of a subclass of these channels, the complete phase diagram of the two-qubit channel, which shows three distinct phases is obtained. While increasing the correlation generally changes the optimal state from separable to maximally entangled states, this is done via an intermediate region where both separable and maximally entangled states are optimal. A more concrete model, based on random rotations of the error operators which mimic the behavior of this subclass of channels is also presented., Comment: 13 pages, Latex

We introduce a simple representation for irreducible spherical tensor operators of the rotation group of arbitrary integer or half integer rank and use these tensor operators to construct matrix product states corresponding to all the variety of valence-bond states proposed in the Affleck-Kennedy-Lieb-Tasaki (AKLT) construction. These include the fully dimerized states of arbitrary spins, with uniform or alternating patterns of spins, which are ground states of Hamiltonians with nearest and next-nearest neighbor interactions, and the partially dimerized or AKLT/VBS (Valence Bond Solid) states, which are constructed from them by projection. The latter states are translation-invariant ground states of Hamiltonians with nearest-neighbor interactions., Comment: 24 pages, references added, the version which appears in the journal

We consider a one-parameter family of matrix-product states of spin-1 particles on a periodic chain and study in detail the entanglement properties of such a state. In particular, we calculate exactly the entanglement of one site with the rest of the chain, and the entanglement of two distant sites with each other, and show that the derivative of both these properties diverge when the parameter $g$ of the states passes through a critical point. Such a point can be called a point of quantum phase transition, since at this point the character of the matrix-product state, which is the ground state of a Hamiltonian, changes discontinuously. We also study the finite size effects and show how the entanglement depends on the size of the chain. This later part is relevant to the field of quantum computation where the problem of initial state preparation in finite arrays of qubits or qutrits is important. It is also shown that the entanglement of two sites have scaling behavior near the critical point.

We investigate the problem of enhancement of mutual information by encoding classical data into entangled input states of arbitrary length and show that while there is a threshold memory or correlation parameter beyond which entangled states outperform the separable states, resulting in a higher mutual information, this memory threshold increases toward unity as the length of the string increases. These observations imply that encoding classical data into entangled states may not enhance the classical capacity of quantum channels., 14 pages, 8 figures, latex, accepted for publication in Physical Review A

For the space of two identical systems of arbitrary dimensions, we introduce a continuous family of bases with the following properties: i) the bases are orthonormal, ii) in each basis, all the states have the same values of entanglement, and iii) they continuously interpolate between the product basis and the maximally entangled basis. The states thus constructed may find applications in many areas related to quantum information science including quantum cryptography, optimal Bell tests and investigation of enhancement of channel capacity due to entanglement., 10 pages, 2 figures, 1 table, Accepted for publication in Phys. Rev. A

We analyze a recent suggestion \cite{kauffman1,kauffman2} on a possible relation between topological and quantum mechanical entanglements. We show that a one to one correspondence does not exist, neither between topologically linked diagrams and entangled states, nor between braid operators and quantum entanglers. We also add a new dimension to the question of entangling properties of unitary operators in general., RevTex, 7 eps figures, to be published in Phys. Lett. A (2004)

Given two arbitrary pure states $ |��>$ and $ |��>$ of qubits or higher level states, we provide arguments in favor of states of the form $ \frac{1}{\sqrt{2}}(|��> |��> + i |��> |��>) $ instead of symmetric or anti-symmetric states, as natural candidates for optimally entangled states constructed from these states. We show that such states firstly have on the average a high value of concurrence, secondly can be constructed by a universal unitary operator independent of the input states. We also show that these states are the only ones which can be produced with perfect fidelity by any quantum operation designed for intertwining two pure states with a relative phase. A probabilistic method is proposed for producing any pre-determined relative phase into the combination of any two arbitrary states., 6 pages, 1 figure

We investigate the quantum information reclaim from the environment after amplitude damping has occurred. In particular we address the question of optimal measurement on the environment to perform the best possible correction on two and three dimensional quantum systems. Depending on the dimension we show that the entanglement fidelity (the measure quantifying the correction performance) is or is not the same for all possible measurements and uncover the optimal measurement leading to the maximum entanglement fidelity.

We study the quantum capacity of continuous variable dephasing channel, which is a notable example of non-Gaussian quantum channel. We prove that a single letter formula applies. We then consider input energy restriction and show that by increasing it, the capacity saturates to a finite value. The optimal input state is found to be diagonal in the Fock basis and with a distribution that is a discrete version of a Gaussian. Relations between its mean/variance and dephasing rate/input energy are put forward. We also show that quantum capacity decays exponentially for large values of dephasing rates.