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arXiv:1104.0485 · Physical Review A

Maximization of thermal entanglement of arbitrarily interacting two qubits

Tomotaka Kuwahara, Naomichi Hatano

Published Jun 10, 2011• 7 citations
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Abstract

We investigate the thermal entanglement of interacting two qubits. We maximize it by tuning a local Hamiltonian under a given interaction Hamiltonian. We prove that the optimizing local Hamiltonian takes a simple form which does not depend on the temperature and that the corresponding optimized thermal entanglement decays as $1/(T\mathrm{ln}T)$ at high temperatures. We also find that at low temperatures the thermal entanglement is maximum without any local Hamiltonians and that the second derivative of the maximized thermal entanglement changes discontinuously at the boundary between the high- and low-temperature phases.

DOI
10.1103/physreva.83.062311

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We investigate the thermal entanglement of interacting two qubits. We maximize it by tuning a local Hamiltonian under a given interaction Hamiltonian. We prove that the optimizing local Hamiltonian takes a simple form which does not depend on the temperature and that the corresponding optimized thermal entanglement decays as 1/(TlnT)1/(T\mathrm{ln}T)1/(TlnT) at high temperatures. We also find that at low temperatures the thermal entanglement is maximum without any local Hamiltonians and that the second derivative of the maximized thermal entanglement changes discontinuously at the boundary between the high- and low-temperature phases.
In the present thesis, we report several general properties of the enhancement of the entanglement by external fields. First, we investigate the thermal entanglement of interacting two qubits. We maximize it by tuning a local Hamiltonian under a given interaction Hamiltonian. We prove that the optimizing local Hamiltonian takes a simple form which dose not depend on the temperature and that the corresponding optimized thermal entanglement decays as 1/(T log T ) at high temperatures. We also find that at low temperatures the thermal entanglement is maximum without any local Hamiltonians and that the second derivative of the maximized thermal entanglement changes discontinuously at the boundary between the highand low-temperature phases. Second, we investigate the maximized entanglement of indirectly interacting two spins, that is, through other spins. We present a necessary condition for the indirect interaction to give a non-zero maximized entanglement between the focused spins. We also prove that if the focused spins are separated by two spins, there is a critical temperature above which the maximized entanglement between the focused spins vanishes. Then, we numerically calculate the maximized entanglement between the end spins of three-spin chains and four-spin chains. We discover that the maximizing local fields on the end spins have asymmetric forms. In the three-spin chains, we attribute the entanglement enhancement to the asymmetry of the local fields qualitatively and quantitatively in terms of the magnons. In XX and XY four-spin chains, we find that the critical temperature shows qualitatively different behavior depending on the conservation of the angular momentum in the z direction.
We investigate the response of entanglement to temperature variations in interacting two-qubit thermal states. For a general two-qubit interaction Hamiltonian, we derive exact expressions for the thermal concurrence, its first and second derivatives with respect to inverse temperature, and the thermal quantum Fisher information. We show that the rate of change of thermal entanglement is bounded by the thermal quantum Fisher information. We further derive a bound relating entanglement curvature and thermal quantum Fisher information, and show that temperature uncertainty induces a loss of entanglement bounded by the same quantity that determines thermometric sensitivity. These results establish thermal quantum Fisher information as a fundamental constraint on the response and robustness of entanglement in interacting two-qubit thermal states.
The entanglement of two qubits, each defined as an effective two-level, spin 1/2 system, is investigated for the case that the qubits interact via a Heisenberg XY interaction and are subject to decoherence due to population relaxation and thermal effects. For zero temperature, the time-dependent concurrence is studied analytically and numerically for some typical initial states, including a separable (unentangled) initial state. An analytical formula for non-zero steady state concurrence is found for any initial state, and optimal parameter values for maximizing steady state concurrence are given. The steady state concurrence is found analytically to remain non-zero for low, finite temperatures. We also identify the contributions of global and local coherence to the steady state entanglement.
We have found that for a wide range of two-qubit Hamiltonians the canonical-ensemble thermal state is entangled in two distinct temperature regions. In most cases the ground state is entangled; however we have also found an example where the ground state is separable and there are still two regions. This demonstrates that the qualitative behavior of entanglement with temperature can be much more complicated than might otherwise have been expected; it is not simply determined by the entanglement of the ground state, even for the simple case of two qubits. Furthermore, we prove a finite bound on the number of possible entangled regions for two qubits, thus showing that arbitrarily many transitions from entanglement to separability are not possible. We also show that only one entangled region is possible for the special case of Hamiltonians without magnetic fields.
We derive the entangled thermal mixed states by using the formalism of phase states for a finite-dimensional algebra of a multi-qubit system in contact with an independent thermal environment of absolute temperature [Formula: see text]. Thermal entangled states describing the multi-qubit system in equilibrium with the thermal bath are a special kind of mixed states that exhibit genuine multipartite correlation. We define the unitary phase operators for a multipartite system of non-interacting qubits. Entangled density matrices are derived for qubits interacting via an Hermitian Hamiltonian of Heisenberg type [Formula: see text]. By assuming that the noisy interaction of the entangled qubit ensemble with the bath is governed by a local Hamiltonian [Formula: see text], we show that the entangled phase states can be decohered. When the multi-qubits entangled system reaches the equilibrium with the thermal bath, the decohered mixed states are identified with entangled thermal states. The thermal mixed states for bipartite and multipartite systems are explicitly expressed and their bipartite entanglement properties are investigated.
The stationary entanglement and Bell violation of two qubits collectively interacting with a common thermal reservoir is investigated, and the analytical relations among average thermal photon number of the reservoir, entanglement and Bell violation of two qubits are given in detail. The initial fraction of Bell singlet state determines whether the common thermal reservoir can enhance the entanglement of two qubits or not in this case. Moreover, the stationary maximally entangled mixed state can emerge in the collective decay of two qubits in a thermal reservoir at zero-temperature only if the fraction of Bell singlet state in the initial state is not smaller than 2/3. Furthermore, it is shown that the collective decay can induce the entanglement of two qutrits initially in the maximally mixed state, and induce distillable entanglement from bound entangled states.