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Interdisciplinary Information Sciences

Non-Equilibrium Thermo Field Dynamics and Its Application to Error-Correction for Spatially Correlated Quantum Errors

Toshihico Arimitsu

Published Jan 1, 2009• 1 citation
Publisher

Abstract

The framework of the canonical operator formalism for dissipative and/or stochastic quantum systems named non-equilibrium thermo field dynamics (NETFD) is presented in a self-contained manner. As an application of the formalism, a model of noisy-channel which is under the influence of spatially correlated quantum Brownian motion is investigated within the framework of NETFD. It is shown that errors due to spatially correlated noises can be corrected by the quantum error-correction code and error-correction procedure prepared for those for independent noises.

DOI
10.4036/iis.2009.441

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The framework of the canonical operator formalism for dissipative and/or stochastic quantum systems named non-equilibrium thermo field dynamics (NETFD) is presented in a self-contained manner. As an application of the formalism, a model of noisy-channel which is under the influence of spatially correlated quantum Brownian motion is investigated within the framework of NETFD. It is shown that errors due to spatially correlated noises can be corrected by the quantum error-correction code and error-correction procedure prepared for those for independent noises.
It is shown that errors on qubits caused by spatially correlated noise of quantum Brownian motion can be corrected with a stabilizer code (a quantum error correction code) and recovery operations prepared for uncorrelated noise without modifications. The analysis is made by means of the quantum stochastic Liouville equation approach which has been developed within the canonical operator formalism for dissipative systems called non-equilibrium thermo field dynamics. This approach yields a transparent procedure to derive completely positive maps describing errors on qubits under given statistics of noise.
A fundamental framework for the construction of a quantum field theory for open systems, which will be called non-equilibrium thermo field dynamics (NETFD), is built upon two concepts. One is the thermal state in the thermal-Liouville space, by which the quasi-particle superoperator is introduced generally from the thermal state condition. The other is the coarse graining which is realized by projecting out some partial space from the complete thermal-Liouville space, and by which the dissipation is introduced. Most properties of the usual quantum field theory (e.g., the operator formalism, the time-ordered formulation of the Green’s functions, the Feynman diagram method in real time, etc.) are preserved in NETFD. The entropy for the nonequilibrium state can be also introduced in NETFD.
A unified framework of stochastic differential equations for quantum systems, formulated within Non-Equilibrium Thermo Field Dynamics (NETFD), is applied to a model of non-linear damped oscillator. The quantum stochastic Liouville equation and the quantum Langevin equations (both of Ito and Stratonovich type), which are consistent with the corresponding master equation, are written down explicitly in the case of a non-conventional treatment. This solves Kubo's third problem: how one can obtain the correlations of random force operators for the Langevin equation compatible with the master equation derived by the non-conventional treatment of the damping theory.