Ensembles of physical states and random quantum circuits on graphs (original) (raw)
Related papers
Quantum Entanglement in Random Physical States
Physical Review Letters, 2012
Most states in the Hilbert space are maximally entangled. This fact has proven useful to investigate-among other things-the foundations of statistical mechanics. Unfortunately, most states in the Hilbert space of a quantum many body system are not physically accessible. We define physical ensembles of states by acting on random factorized states by a circuit of length k of random and independent unitaries with local support. We study the typicality of entanglement by means of the purity of the reduced state. We find that for a time k = O(1) the typical purity obeys the area law. Thus, the upper bounds for area law are actually saturated in average, with a variance that goes to zero for large systems. Similarly, we prove that by means of local evolution a subsystem of linear dimensions L is typically entangled with a volume law when the time scales with the size of the subsystem. Moreover, we show that for large values of k the reduced state becomes very close to the completely mixed state.
Randomized graph states and their entanglement properties
Physical Review A, 2014
We introduce a class of mixed multi-qubit states, that corresponds to a randomized version of graph states. It is shown that unitary equivalences are lost by randomization using a rank argument. We study the entanglement features of these states by investigating both bipartite and genuine multipartite entanglement. Bipartite entanglement is studied via the concepts of connectedness and persistency, defined for graph states and strictly related to measurement based quantum computation. The presence of multipartite entanglement is characterized by witness operators which are subsequently adapted to study non-local properties through the violation of suitable Bell inequalities. Finally, we present results on the entanglement detection of particular randomized graph states, and we propose a method to further improve the detection of genuine multipartite entanglement.
Growth of graph states in quantum networks
Physical Review A, 2012
We propose a scheme to distribute graph states over quantum networks in the presence of noise in the channels and in the operations. The protocol can be implemented efficiently for large graph sates of arbitrary (complex) topology. We benchmark our scheme with two protocols where each connected component is prepared in a node belonging to the component and subsequently distributed via quantum repeaters to the remaining connected nodes. We show that the fidelity of the generated graphs can be written as the partition function of a classical Ising-type Hamiltonian. We give exact expressions of the fidelity of the linear cluster and results for its decay rate in random graphs with arbitrary (uncorrelated) degree distributions.
Universality in random quantum networks
Physical Review A, 2015
Networks constitute efficient tools for assessing universal features of complex systems. In physical contexts, classical as well as quantum, networks are used to describe a wide range of phenomena, such as phase transitions, intricate aspects of many-body quantum systems or even characteristic features of a future quantum internet. Random quantum networks and their associated directed graphs are employed for capturing statistically dominant features of complex quantum systems. Here, we develop an efficient iterative method capable of evaluating the probability of a graph being strongly connected. It is proven that random directed graphs with constant edge-establishing probability are typically strongly connected, i.e. any ordered pair of vertices is connected by a directed path. This typical topological property of directed random graphs is exploited to demonstrate universal features of the asymptotic evolution of large random qubit networks. These results are independent of our knowledge of the details of the network topology. These findings suggest that also other highly complex networks, such as a future quantum internet, may exhibit similar universal properties.
Random Unitary Matrices Associated to a Graph
Acta Physica Polonica A, 2013
We analyze composed quantum systems consisting of k subsystems, each described by states in the n-dimensional Hilbert space. Interaction between subsystems can be represented by a graph, with vertices corresponding to individual subsystems and edges denoting a generic interaction, modeled by random unitary matrices of order n 2. The global evolution operator is represented by a unitary matrix of size N = n k. We investigate statistical properties of such matrices and show that they display spectral properties characteristic to Haar random unitary matrices provided the corresponding graph is connected. Thus basing on random unitary matrices of a small size n 2 one can construct a fair approximation of large random unitary matrices of size n k. Graph-structured random unitary matrices investigated here allow one to define the corresponding structured ensembles of random pure states.
Area law for random graph states
Journal of Physics A: Mathematical and Theoretical, 2013
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 bi-partite, 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. Contents 1. Introduction 2. Random graph states 3. Exact area law for adapted partitions 4. One-vertex marginals 5. Defining the boundary surface for a general partition 6. A general area law for graph states 7. Rank of random graph states and a transport problem 8. Some results for different subsystem dimensions 9. Perspectives and open questions References
Physical Review Letters, 2008
We investigate the equidistribution of the eigenfunctions on quantum graphs in the high-energy limit. Our main result is an estimate of the deviations from equidistribution for large well-connected graphs. We use an exact field-theoretic expression in terms of a variant of the supersymmetric nonlinear σ-model. Our estimate is based on a saddle-point analysis of this expression and leads to a criterion for when equidistribution emerges asymptotically in the limit of large graphs. Our theory predicts a rate of convergence that is a significant refinement of previous estimates, long-assumed to be valid for quantum chaotic systems, agreeing with them in some situations but not all. We discuss specific examples for which the theory is tested numerically.
Entanglement features of random Hamiltonian dynamics
Physical Review B, 2018
We introduce the concept of entanglement features of unitary gates, as a collection of exponentiated entanglement entropies over all bipartitions of input and output channels. We obtained the general formula for time-dependent nth-Rényi entanglement features for unitary gates generated by random Hamiltonian. In particular, we propose an Ising formulation for the 2nd-Rényi entanglement features of random Hamiltonian dynamics, which admits a holographic tensor network interpretation. As a general description of entanglement properties, we show that the entanglement features can be applied to several dynamical measures of thermalization, including the out-of-time-order correlation and the entanglement growth after a quantum quench. We also analyze the Yoshida-Kitaev probabilistic protocol for random Hamiltonian dynamics.
Open-system dynamics of graph-state entanglement
Physical Review Letters, 2009
We consider graph states of an arbitrary number of particles undergoing generic decoherence. We present methods to obtain lower and upper bounds for the system's entanglement in terms of that of considerably smaller subsystems. For an important class of noisy channels, namely, the Pauli maps, these bounds coincide and thus provide the exact analytical expression for the entanglement evolution. All of the results apply also to (mixed) graph-diagonal states and hold true for any convex entanglement monotone. Since any state can be locally depolarized to some graph-diagonal state, our method provides a lower bound for the entanglement decay of any arbitrary state. Finally, this formalism also allows for the direct identification of the robustness under size scaling of graph states in the presence of decoherence, merely by inspection of their connectivities.