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arXiv:cond-mat/0301111 · Physical review. B, Condensed matter

Entanglement and decoherence in near-critical qubit chains

Dmitri V. Khveshchenko

Published Nov 19, 2003• 43 citations
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Abstract

We study the problem of environmentally induced decoherence in a near-critical one-dimensional system of $N\ensuremath{\gg}1$ coupled qubits. Using the Jordan-Wigner fermion representation of the qubit operators we identify the decoherence rates relevant for the two-qubit reduced density matrix. We find that a desirable onset of massive shared entanglement in the near-critical regime comes at the expense of decoherence, which also tends to increase as the system is tuned towards criticality. Our results reveal rather contradictory general requirements that future designs of a qubit-chain-based quantum information processor will need to satisfy.

DOI
10.1103/physrevb.68.193307

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We study the problem of environmentally induced decoherence in a near-critical one-dimensional system of N\ensuremath≫1N\ensuremath{\gg}1N\ensuremath≫1 coupled qubits. Using the Jordan-Wigner fermion representation of the qubit operators we identify the decoherence rates relevant for the two-qubit reduced density matrix. We find that a desirable onset of massive shared entanglement in the near-critical regime comes at the expense of decoherence, which also tends to increase as the system is tuned towards criticality. Our results reveal rather contradictory general requirements that future designs of a qubit-chain-based quantum information processor will need to satisfy.
We study the problem of environmentally-induced decoherence in a near-critical one-dimensional system of N>>1 coupled qubits. Using the Jordan-Wigner fermion representation of the qubit operators we identify the decoherence rates relevant for the two-qubit reduced density matrix. We find that a desirable onset of massive shared entanglement in the near-critical regime comes at the expense of decoherence which also tends to increase as the system is tuned towards criticality. Our results reveal rather contradictory general requirements that future designs of a qubit chain-based quantum information processor will need to satisfy. Comment: Latex, 5 pages, no figures
We study decoherence induced on a two-level system coupled to a one-dimensional quantum spin chain. We consider the cases where the dynamics of the chain is determined by the Ising, XYXYXY, or Heisenberg exchange Hamiltonian. This model of quantum baths can be of fundamental importance for the understanding of decoherence in open quantum systems, since it can be experimentally engineered by using atoms in optical lattices. As an example, here we show how to implement a pure dephasing model for a qubit system coupled to an interacting spin bath. We provide results that go beyond the case of a central spin coupled uniformly to all the spins of the bath, in particular showing what happens when the bath enters different phases, or becomes critical; we also study the dependence of the coherence loss on the number of bath spins to which the system is coupled and we describe a coupling-independent regime in which decoherence exhibits universal features, irrespective of the system-environment coupling strength. Finally, we establish a relation between decoherence and entanglement inside the bath. For the Ising and the XYXYXY models we are able to give an exact expression for the decay of coherences, while for the Heisenberg bath we resort to the numerical time-dependent density matrix renormalization group.
A relation between entanglement and criticality of spin chains is established. The entanglement we exploit is shared between auxiliary particles, which are isolated from each other, but are coupled to the same critical spin-1∕21∕21∕2 chain. We analytically evaluate the reduced density matrix, and numerically show the entanglement of the auxiliary particles in the proximity of the critical points of the spin chain. We find that the entanglement induced by the spin chain may reach 1, and it can signal very well the critical points of the chain. A physical understanding and experimental realization with trapped ions are presented.
A relation between entanglement and criticality of spin chains is established. The entanglement we exploit is shared between auxiliary particles, which are isolated from each other, but are coupled to the same critical spin-1/2 chain. We analytically evaluate the reduced density matrix, and numerically show the entanglement of the auxiliary particles in the proximity of the critical points of the spin chain. We find that the entanglement induced by a critical many-body system may reach one, and it can signal very well the critical points of the spin chain. A physical understanding and experimental realization with trapped ions are presented.
We consider a generalized central spin model, consisting of two central qubits and an environmental spin chain (with periodic boundary condition) to which these central qubits are locally and weakly connected either at the same site or at two different sites separated by a distance ddd. Our purpose is to study the subsequent temporal generation of entanglement, quantified by concurrence, when initially the qubits are in an unentangled state. In the equilibrium situation, we show that the concurrence survives for a larger value of ddd when the environmental spin chain is critical. Importantly, a common feature observed both in the equilibrium and the nonequilibrium situations while the latter is created by a sudden but global change of the environmental transverse field is that the two qubits become maximally entangled for the critical quenching. Following a nonequilibrium evolution of the spin chain, our study for $d\ensuremath{\ne}0$ indicates that there exists a threshold time above which concurrence attains a finite value. Additionally, we show that the number of independent decohering channels (DCs) is determined by ddd as well as the local difference of the transverse field of the two underlying Hamiltonians governing the time evolution; the concurrence can be enhanced by a higher number of independent channels. The qualitatively similar behavior displayed by the concurrence for critical and off-critical quenches, as reported here, is characterized by analyzing the nonequilibrium evolution of these channels. The concurrence is maximum when the decoherence factor or the echo associated with the most rapidly DC decays to zero; on the contrary, the condition when the concurrence vanishes is determined nontrivially by the associated decay of one of the intermediate DCs. Analyzing the reduced density of a single qubit, we also explain the observation that the dephasing rate is always slower than the unentanglement rate. We further establish that the maximally and minimally decohering channels show singular behavior which can be explained by invoking upon a quasiparticle picture.
Different approaches in quantifying environmentally-induced decoherence are considered. We identify a measure of decoherence, derived from the density matrix of the system of interest, that quantifies the environmentally induced error, i.e., deviation from the ideal isolated-system dynamics. This measure can be shown to have several useful features. Its behavior as a function of time has no dependence on the initial conditions, and is expected to be insensitive to the internal dynamical time scales of the system, thus only probing the decoherence-related time dependence. For a spin-boson model - a prototype of a qubit interacting with environment - we also demonstrate the property of additivity: in the regime of the onset of decoherence, the sum of the individual qubit error measures provides an estimate of the error for a several-qubit system, even if the qubits are entangled, as expected in quantum-computing applications. This makes it possible to estimate decoherence for several-qubits quantum computer gate designs for which explicit calculations are exceedingly difficult.