The expected return time to the original state is a key concept characterizing systems obeying both classical or quantum dynamics. We consider iterated open quantum dynamical systems in finite-dimensional Hilbert spaces, a broad class of systems that includes classical Markov chains and unitary discrete-time quantum walks on networks. Starting from a pure state, the time evolution is induced by repeated applications of a superoperator (quantum channel) in each time step followed by a measurement to detect whether the system has returned to the original state. We prove that if the superoperator is unital in the part of the Hilbert space explored by the system, then the expectation value of the return time is an integer, equal to the dimension of this relevant Hilbert space. We illustrate our results on partially coherent quantum walks on finite graphs. Our work shows that the expected return time is a quantitative measure of the size of the part of the Hilbert space available to the system when the dynamics is started from a certain state.
The expected return time to the original state is a key concept characterizing systems obeying both classical or quantum dynamics. We consider iterated open quantum dynamical systems in finite-dimensional Hilbert spaces, a broad class of systems that includes classical Markov chains and unitary discrete-time quantum walks on networks. Starting from a pure state, the time evolution is induced by repeated applications of a superoperator (quantum channel) in each time step followed by a measurement to detect whether the system has returned to the original state. We prove that if the superoperator is unital in the part of the Hilbert space explored by the system, then the expectation value of the return time is an integer, equal to the dimension of this relevant Hilbert space. We illustrate our results on partially coherent quantum walks on finite graphs. Our work shows that the expected return time is a quantitative measure of the size of the part of the Hilbert space available to the system when the dynamics is started from a certain state.
We consider recurrence to the initial state after repeated actions of a quantum channel. After each iteration a projective measurement is applied to check recurrence. The corresponding return time is known to be an integer for the special case of unital channels, including unitary channels. We prove that for a more general class of quantum channels the expected return time can be given as the inverse of the weight of the initial state in the steady state. This statement is a generalization of the Kac lemma for classical Markov chains.
Abstract Randomly repeated measurements during the evolution of a closed quantum system create a sequence of probabilities for the first detection of a certain quantum state. The related discrete monitored evolution for the return of the quantum system to its initial state is investigated. We found that the mean number of measurements (MNM) until the first detection is an integer, namely the dimensionality of the accessible Hilbert space. Moreover, the mean first detected return (FDR) time is equal to the average time step between successive measurements times the MNM. Thus, the mean FDR time scales linearly with the dimensionality of the accessible Hilbert space. The main goal of this work is to explain the quantization of the mean return time in terms of a quantized Berry phase.
The evolution of an isolated quantum system inevitably exhibits recurrence: the state returns to the vicinity of its initial condition after finite time. Despite its fundamental nature, a rigorous quantitative understanding of recurrence has been lacking. We establish upper bounds on the recurrence time, trec≲texit(ε)(1/ε)d, where d is the Hilbert-space dimension, ε the neighborhood size, and texit(ε) the escape time from this neighborhood. For pure states evolving under a Hamiltonian H, estimating texit is equivalent to an inverse quantum speed limit problem: finding upper bounds on the time a time-evolved state ψt needs to depart from the ε-vicinity of the initial state ψ0. We provide a partial solution, showing that under mild assumptions texit(ε)≈ε/Δ(H2), with Δ(H2) the Hamiltonian variance in ψ0. We show that our upper bound on trec is generically saturated for random Hamiltonians. Finally, we analyze the impact of coherence of the initial state in the eigenbasis of H on recurrence behavior.
While the expected time of first visit to a subspace under some quantum evolution is a relevant statistic, it is also important to confront this information with the associated variance, so that one may have a better understanding of how representative such means are. In this work, we study these matters in the context of quantum channels acting on finite-dimensional Hilbert spaces, and we provide practical methods for calculating the above data with respect to goal subspaces of interest. By considering an assumption which is strictly weaker than irreducibility, we are also able to consider unitary quantum walks. Our approach is based on the monitored evolution of positive, trace-preserving operators given by quantum Markov chains as described by Gudder. Then, given any channel and some goal subspace of interest, one may induce an absorbing chain in a natural way so that the monitored evolution of the original map can be examined.
A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. Hitting times for discrete quantum walks on graphs give an average time before the walk reaches an ending condition. We derive an expression for hitting time using superoperators, and numerically evaluate it for the walk on the hypercube for various coins and decoherence models. The hitting time for a classical random walk on a connected graph will always be finite. We show that, by contrast, quantum walks can have infinite hitting times for some initial states. We seek criteria to determine if a given walk on a graph will have infinite hitting times, and find a sufficient condition, which for discrete time quantum walks is that the degeneracy of the evolution operator be greater than the degree of the graph. The phenomenon of infinite hitting times is in general a consequence of the symmetry of the graph and its automorphism group.
Symmetries of a graph, given by its automorphism group, can be inherited by the evolution operator. Using the irreducible representations of the automorphism group, we derive conditions such that quantum walks defined on this graph must have infinite hitting times for some initial states. Symmetry can cause the walk to also be confined to a subspace of the original Hilbert space for cartain initial states. We show that a quantum walk confined to the subspace corresponding to this symmetry group can be seen as a different quantum walk on a smaller quotient graph. We give an explicit construction of the quotient graph for any subgroup H of the automorphism group. The automorphisms of the quotient graph which are inherited from the original graph are the original automorphism group modulo the subgroup H used to construct it. We conjecture that the existence of a small quotient graph with finite hitting times is necessary for a walk to exhibit a quantum speed-up. Finally, we use symmetry and the theory of decoherence-free subspaces to determine when the subspace of the quotient graph is a decoherence-free subspace of the dynamics.
We make use of matrix representations of completely positive maps in order to study open quantum dynamics on graphs, with emphasis on quantum walks and the associated trajectories obtained via a monitoring of the position. We discuss the discrete and continuous time setting and consider the statistics of particles located on a vertex with some internal degree of freedom. Other classes of semigroup generators, such as graph-induced generators are also considered. In the case of primitive quantum channels we present expressions of the mean hitting time for a particle to reach a vertex in terms of a quantum version of the fundamental matrix, thus extending recent results in the setting of open quantum random walks. An open quantum version of Kac's Lemma for the expected return time is discussed. By considering appropriate block matrix expressions we are able to make computations in terms of matrix-valued transition rates, in analogy with the notion of rates from classical continuous-time Markov chains. Also inspired by the classical case a recurrence criterion for the continuous time open walk setting is presented.