The asymptotic dynamics of many-qubit quantum systems is investigated under iteratively and randomly applied unitary transformations. For a one-parameter family of unitary transformations, which entangle pairs of qubits, two main theorems are proved. They characterize completely the dependence of the resulting asymptotic dynamics on the topology of the interaction graph that encodes all possible qubit couplings. These theorems exhibit clearly which aspects of an interaction graph are relevant and which ones are irrelevant to the asymptotic dynamics. On the basis of these theorems, the local entropy transport between an open quantum system and its environment are explored for strong non-Markovian couplings and for different sizes of the environment and different interaction topologies. It is shown that although the randomly applied unitary entanglement operations cannot decrease the overall entropy of such a qubit network, a local entropy decrease or 'cooling' of subsystems is possible for special classes of interaction topologies.
The asymptotic dynamics of many-qubit quantum systems is investigated under iteratively and randomly applied unitary transformations. For a one-parameter family of unitary transformations, which entangle pairs of qubits, two main theorems are proved. They characterize completely the dependence of the resulting asymptotic dynamics on the topology of the interaction graph that encodes all possible qubit couplings. These theorems exhibit clearly which aspects of an interaction graph are relevant and which ones are irrelevant to the asymptotic dynamics. On the basis of these theorems, the local entropy transport between an open quantum system and its environment are explored for strong non-Markovian couplings and for different sizes of the environment and different interaction topologies. It is shown that although the randomly applied unitary entanglement operations cannot decrease the overall entropy of such a qubit network, a local entropy decrease or 'cooling' of subsystems is possible for special classes of interaction topologies.
We investigate the asymptotic dynamics of quantum networks under repeated applications of random unitary operations. It is shown that in the asymptotic limit of large numbers of iterations this dynamics is generally governed by a typically low dimensional attractor space. This space is determined completely by the unitary operations involved and it is independent of the probabilities with which these unitary operations are applied. Based on this general feature analytical results are presented for the asymptotic dynamics of arbitrarily large cyclic qubit networks whose nodes are coupled by randomly applied controlled-NOT operations.
Systems evolving under the influence of competing two- and three-body interactions are of particular interest in exploring the stability of the equilibrium states of a strongly interacting many-body system. We present a solvable model based on qubit networks, which allows us to investigate the intricate influence of these couplings on the possible asymptotic equilibrium states. We study the asymptotic evolution of finite qubit networks under two- and three-qubit interactions. As representatives of three-qubit interactions we choose controlled unitary interactions (cu-interactions) with one and two control qubits. It is shown that networks with purely three-qubit interactions exhibit different asymptotic dynamics depending on whether we deal with interactions controlled by one or two qubits. However, when we allow three-qubit interactions next to two-qubit interactions, the asymptotics is dictated by two-qubit interactions only. Finally, we prove that the simultaneous presence of two types of three-qubit interactions results in the asymptotic dynamics characteristic for two-qubit cu-interactions.
Abstract We analyze the asymptotic dynamics of quantum systems resulting from large numbers of iterations of random unitary operations. Although, in general, these quantum operations cannot be diagonalized it is shown that their resulting asymptotic dynamics is described by a diagonalizable superoperator. We prove that this asymptotic dynamics takes place in a typically low dimensional attractor space which is independent of the probability distribution of the unitary operations applied. This vector space is spanned by all eigenvectors of the unitary operations involved which are associated with eigenvalues of unit modulus. Implications for possible asymptotic dynamics of iterated random unitary operations are presented and exemplified in an example involving random controlled-not operations acting on two qubits.
The question, whether an open system dynamics is Markovian or non-Markovian can be answered by studying the direction of the information flow in the dynamics. In Markovian dynamics, information must always flow from the system to the environment. If the environment is interacting with only one of the subsystems of a bipartite system, the dynamics of the entanglement in the bipartite system can be used to identify the direction of information flow. Here we study the dynamics of a two-level system interacting with an environment, which is also a heat bath, and consists of a large number of two-level quantum systems. Our model can be seen as a close approximation to the 'spin bath' model at low temperatures. We analyze the Markovian nature of the dynamics, as we change the coupling between the system and the environment. We find the Kraus operators of the dynamics for certain classes of couplings. We show that any form of time-independent or time-polynomial coupling gives rise to non-Markovianity. Also, we witness non-Markovianity for certain parameter values of time-exponential coupling. Moreover, we study the transition from non-Markovian to Markovian dynamics as we change the value of coupling strength.
Motivated by recent advances in quantum technologies, we investigate entanglement spreading and excitation transport within quantum complex networks. For this purpose, we adopt a description in which network nodes represent sites of the physical system (e.g. qubits), and links represent pairs of sites coupled by the Hamiltonian [Bentsen et al., Phys Rev Lett (2019)]. We consider a random tunable-range model [Gori et al., Phys Rev E (2017)] which, in terms of only two parameters, encompasses a large variety of graphs and reduces, in limiting cases, to well-studied benchmarks. Then, it is equipped with a paradigmatic Hamiltonian consisting of hopping terms on the network links. Building on known results [Gori et al., Phys Rev E (2017)], we first study the classical case, providing new semi-analytical estimates. We then address quantum dynamics: For the single-particle case, we analyse the asymptotic entanglement entropy between two halves of the system, computed by Exact Diagonalization. Surprisingly, we observe a new localization region for intermediate ranges of the network links. To understand this phenomenon, we introduce an approximate one-dimensional model and try to relate dynamics to network measures. The analysis is extended to half-filling using fermionic Gaussian states, obtaining results close to the single-particle ones that raise new interesting questions for future work. It is concluded that this model exhibits non-trivial behaviour, especially in the region where unexpected localization occurs. We infer that this is determined by the interplay between network features and quantum effects, possibly induced by structural disorder.
Random pure states of multi-partite quantum systems, associated with arbitrary graphs, are investigated. Each vertex of the graph represents a generic interaction between subsystems, described by a random unitary matrix distributed according to the Haar measure, while each edge of the graph represents a bipartite, maximally entangled state. For any splitting of the graph into two parts we consider the corresponding partition of the quantum system and compute the average entropy of entanglement. First, in the special case where the partition does not cross any vertex of the graph, we show that the area law is satisfied exactly. In the general case, we show that the entropy of entanglement obeys an area law on average, this time with a correction term that depends on the topologies of the graph and of the partition. The results obtained are applied to the problem of distribution of quantum entanglement in a quantum network with prescribed topology.