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Non Local Detection of quantum dynamical processes in spin chains

Saikat Sur, V. Subrahmanyam

Published Jun 24, 2016• 0 citations
OpenAlex

Abstract

We study the dynamics of a one dimensional quantum spin chain evolving from unentangled or entangled initial state. At a given instant of time a quantum dynamical process (ex. measurement) is performed on a single spin at one end of the chain. The aim is to detect if the process occurred or not from a measurement on a far away spin at a later time. From the dynamical evolution of the state from the epoch time it is possible to detect the occurrence of the dynamical process. The propagation of the signal for the dynamical process, and the speed of the signal are investigated for various spin models, viz. the Ising, Heisenberg, and the transverse-XY dynamics.

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We study the dynamics of a one dimensional quantum spin chain evolving from unentangled or entangled initial state. At a given instant of time a quantum dynamical process (ex. measurement) is performed on a single spin at one end of the chain. The aim is to detect if the process occurred or not from a measurement on a far away spin at a later time. From the dynamical evolution of the state from the epoch time it is possible to detect the occurrence of the dynamical process. The propagation of the signal for the dynamical process, and the speed of the signal are investigated for various spin models, viz. the Ising, Heisenberg, and the transverse-XY dynamics.
Abstract We study the dynamics of a one dimensional quantum spin chain evolving from unentangled or entangled initial state. At a given instant of time a quantum dynamical process (ex. quantum measurement operation) is performed on a single spin at one end of the chain, decohering the system. Through the further unitary evolution, a signal propagates in the spin chain, which can be detected from a measurement on a different spin at later times. From the dynamical unitary evolution of the decohered state from the epoch time, it is possible to detect the occurrence of the quantum operation. The propagation of the signal for the dynamical process, and the speed of the signal are investigated for various spin models, viz. using the Ising, Heisenberg and transverse- XY dynamics.
The evolution of entanglement in a one-dimensional Ising chain is numerically studied under various initial conditions. We analyze two problems concerning the dynamics of entanglement: (i) generation of the entanglement from the pseudopure separable state and (ii) transportation of the entanglement from one end of the chain to the other. The model investigated is a one-dimensional Ising spin-1/2 chain with nearest-neighbor interactions placed in an external magnetic field and irradiated by a weak resonant transverse field. The possibility of selective initialization of partially entangled states is considered. It was shown that, in spite of the use of a model with direct interactions between the nearest neighbors, entanglement between remote spins is generated.
Abstract We consider a local instantaneous quantum dynamical process (QDP) that disturbs the background unitary Hamiltonian dynamics of a spin chain. We consider both a non-unitary incoherent QDP and a coherent unitary QDP intervention of the dynamical evolution of the spin chain. To track the effect of QDP on the dynamics, we investigate the Loschmidt echo, which is quite sensitive to whether the background dynamics is integrable or not. For the integrable case, namely the Heisenberg model, the Loschmidt echo depends on the parameters corresponding to the QDP as well as the time of occurrence of QDP. The probability of reviving the system to its initial state is higher for non-unitary/incoherent QDPs occurring at large time intervals. In the case of a unitary/coherent QDP, some amount of the probability of reviving the state is lost. For the non-integrable background dynamics we consider a kicked Harper model. It exhibits a decaying behaviour when contrasted with integrable dynamics. The decay rate is slower when the corresponding classical Hamiltonian is non-chaotic.
We study the unitary time evolution of the entropy of entanglement of a one-dimensional system between the degrees of freedom in an interval of length and its complement, starting from a pure state which is not an eigenstate of the Hamiltonian. We use path integral methods of quantum field theory as well as explicit computations for the transverse Ising spin chain. In both cases, there is a maximum speed v of propagation of signals. In general the entanglement entropy increases linearly with time t up to , after which it saturates at a value proportional to , the coefficient depending on the initial state. This behaviour may be understood as a consequence of causality.
We study the dynamics of a Heisenberg XYXYXY spin chain with an unknown state coded into one qubit or a pair of entangled qubits, with the rest of the spins being in a polarized state. The time evolution involves magnon excitations, and through them the entanglement is transported across the channel. For a large number of qubits, explicit formulas for the concurrences, measures for two-qubit entanglements, and the fidelity for recovering the state some distance away are calculated as functions of time. Initial states with an entangled pair of qubits show better fidelity, which takes its first maximum value at earlier times, compared to initial states with no entangled pair. In particular initial states with a pair of qubits in an unknown state $\ensuremath{\alpha}\ensuremath{\uparrow}\ensuremath{\uparrow}+\ensuremath{\beta}\ensuremath{\downarrow}\ensuremath{\downarrow}$ are best suited for quantum-state transport.
We study the dynamics of the entanglement spectrum, that is the time evolution of the eigenvalues of the reduced density matrices after a bipartition of a one-dimensional spin chain. Starting from the ground state of an initial Hamiltonian, the state of the system is evolved in time with a new Hamiltonian. We consider both instantaneous and quasi adiabatic quenches of the system Hamiltonian across a quantum phase transition. We analyse the Ising model that can be exactly solved and the XXZ for which we employ the time-dependent density matrix renormalisation group algorithm. Our results show once more a connection between the Schmidt gap, i.e. the difference of the two largest eigenvalues of the reduced density matrix and order parameters, in this case the spontaneous magnetisation.