Phase Covariant Qubit Dynamics and Divisibility

Cornell University - arXiv, 2022

We address the question of the existence of quantum channels that are divisible in two quantum channels but not in three, or more generally channels divisible in n but not in n + 1 parts. We show that for the qubit, those channels do not exist, whereas for general finite-dimensional quantum channels the same holds at least for full Kraus rank channels. To prove these results we introduce a novel decomposition of quantum channels which separates them in a boundary and Markovian part, and it holds for any finite dimension. Additionally, the introduced decomposition amounts to the well known connection between divisibility classes and implementation types of quantum dynamical maps, and can be used to implement quantum channels using smaller quantum registers.

Ufimskij matematičeskij žurnal, 2022

In this paper we study quantum dynamical mappings called also quantum processes. The set of values of such mapping is a one-parameter family of completely positive trace-preserving linear operators defined on a finite-dimensional Hilbert space. In quantum information theory such operators are referred to as quantum channels. An important concept for quantum dynamical mappings is their divisibility. There are different types of this concept. The present paper deals with so-called completely positive divisible quantum processes. For two such processes, which are bijective and satisfy a commutativity condition, we construct a compound quantum process. It is shown that this compound quantum process is also completely positive divisible. Endowing a set of quantum channels with the norm topology, we consider continuous quantum processes and continuous completely positive evolutions. The latter are defined as two-parameter families of quantum channels satisfying additional properties. We prove that a continuous bijective completely positive divisible quantum process generates a continuous completely positive evolution. In order to illustrate the considered concepts and the results on them, we provide examples of quantum dynamical mappings with values in the set of qubit channels. In particular, a completely positive divisible compound quantum process is constructed for two bijective commuting quantum processes. Geometric and physical interpretations of this compound quantum process are given.

Physical review letters, 2018

We analyze the relation between completely positive (CP) divisibility and the lack of information backflow for an arbitrary-not necessarily invertible-dynamical map. It is well known that CP divisibility always implies a lack of information backflow. Moreover, these two notions are equivalent for invertible maps. In this Letter, it is shown that for a map which is not invertible the lack of information backflow always implies the existence of a completely positive propagator which, however, needs not be trace preserving. Interestingly, for a wide class of image nonincreasing dynamical maps, this propagator becomes trace preserving as well, and hence, the lack of information backflow implies CP divisibility. This result sheds new light into the structure of the time-local generators giving rise to CP-divisible evolutions. We show that if the map is not invertible then positivity of dissipation/decoherence rates is no longer necessary for CP divisibility.

Physical Review A, 2010

For the prototypical example of the Ising chain in a transverse field, we study the impact of decoherence on the sweep through a second-order quantum phase transition. Apart from the advance in the general understanding of the dynamics of quantum phase transitions, these findings are relevant for adiabatic quantum algorithms due to the similarities between them. It turns out that (in contrast to first-order transitions studied previously) the impact of decoherence caused by a weak coupling to a rather general environment increases with system size (i.e., number of spins/qubits), which might limit the scalability of the system. PACS numbers: 03.67. Lx, 03.65.Yz, 75.10.Pq, 64.60.Ht. Recently, the dynamics of quantum phase transitions [1] attracted increasing interest, see, e.g., . In contrast to thermal transitions (usually driven by the competition between energy and entropy), they are characterized by a fundamental change of the ground state structure (e.g., from para-to ferro-magnetic) at the critical value of a variable external parameter (e.g., magnetic field). Quantum phase transitions are induced by quantum rather than thermal fluctuations and thus may occur at zero temperature. At the critical point, the energy levels become arbitrarily close and thus the response times diverge (in the continuum limit). Consequently, during the sweep trough such a phase transition by means of a time-dependent external parameter, small external perturbations or internal fluctuations become strongly amplified -leading to many interesting effects, see, e.g., . One of them is the anomalously high susceptibility to decoherence (see also ): Due to the convergence of the energy levels at the critical point, even low-energy modes of the environment may cause excitations and thus perturb the system. Here, we study the decoherence caused by a small coupling to a rather general reservoir for the quantum Ising chain in a transverse field, which is considered a prototypical example [1] for a second-order quantum phase transition (and further possesses the advantage of being analytically solvable).

Physics Letters A, 2004

We study dynamical semigroups of positive, but not completely positive maps on finitedimensional bipartite systems and analyze properties of their generators in relation to non-decomposability and bound-entanglement. An example of non-decomposable semigroup leading to a 4 × 4-dimensional bound-entangled density matrix is explicitly obtained.

2004

We propose an effective Hamiltonian approach to investigate decoherence of a quantum system in a non-Markovian reservoir, naturally imposing the complete positivity on the reduced dynamics of the system. The formalism is based on the notion of an effective reservoir, i.e., certain collective degrees of freedom in the reservoir that are responsible for the decoherence. As examples for completely positive decoherence, we present three typical decoherence processes for a qubit such as dephasing, depolarizing, and amplitude damping. The effects of the non-Markovian decoherence are compared to the Markovian decoherence.

arXiv: Quantum Physics, 2020

In this article, we study the behavior of complete complementarity relations under the action of some common noisy quantum channels (amplitude damping, phase damping, bit flip, bit-phase flip, phase flip, depolarizing, and correlated amplitude damping). By starting with an entangled bipartite pure quantum state, with the linear entropy being the quantifier of entanglement, we study how entanglement is redistributed and turned into general correlations between the degrees of freedom of the whole system. For instance, it's possible to express the entanglement entropy in terms of the multipartite quantum coherence, or in terms of the correlated quantum coherence, of the different partitions of the system. Besides, by considering the environment as part of the quantum system, such that the whole system can be regarded as a multipartite pure quantum system, the linear entropy is shown not just as a measure of mixedness of a particular subsystem, but as a correlation measure of the su...

Divisibility of dynamical maps is visualized by trajectories in the parameter space and analyzed within the framework of collision models. We introduce ultimate completely positive (CP) divisible processes, which lose CP divisibility under infinitesimal perturbations, and characterize Pauli dynamical semigroups exhibiting such a property. We construct collision models with factorized environment particles, which realize additivity and multiplicativity of generators of CP divisible maps. A mixture of dynamical maps is obtained with the help of correlated environment. Mixture of ultimate CP divisible processes is shown to result in a new class of eternal CP indivisible evolutions. We explicitly find collision models leading to weakly and essentially non-Markovian Pauli dynamical maps.

We investigate the Markovian and non-Markovian dynamics of Gaussian quantum channels, exploiting a recently introduced necessary and sufficient criterion and the ensuing measure of non-Markovianity based on the violation of the divisibility property of the dynamical map. We compare the paradigmatic instances of quantum Brownian motion (QBM) and pure damping channels, and for the former we find that the exact dynamical evolution is always non-Markovian in the finite-time as well as in the asymptotic regimes for any nonvanishing value of the non-Markovianity parameter. If one resorts to the rotating-wave approximated form of the QBM, that neglects the anomalous diffusion contribution to the system dynamics, we show that such an approximation fails to detect the non-Markovian nature of the dynamics. Finally, for the exact dynamics of the QBM in the asymptotic regime, we show that the quantifiers of non-Markovianity based on the distinguishability between quantum states fail to detect the non-Markovian nature of the dynamics.

New Journal of Physics, 2021

An evolution of a two-level system (qubit) interacting with a single-photon wave packet is analyzed. It is shown that a hierarchy of master equations gives rise to phase covariant qubit evolution. The temporal correlations in the input field induce nontrivial memory effects for the evolution of a qubit. It is shown that in the resonant case whenever time-local generator is regular (does not display singularities) the qubit evolution never displays information backflow. However, in general the generator might be highly singular leading to intricate non-Markovian effects. A detailed analysis of the exponential profile is provided which allows to illustrate all characteristic feature of the qubit evolution.