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arXiv:1112.5889 · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Pure Gaussian state generation via dissipation: a quantum stochastic differential equation approach

Naoki Yamamoto

Published Oct 22, 2012• 33 citations
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Abstract

Recently, the complete characterization of a general Gaussian dissipative system having a unique pure steady state was obtained. This result provides a clear guideline for engineering an environment such that the dissipative system has a desired pure steady state such as a cluster state. In this paper, we describe the system in terms of a quantum stochastic differential equation (QSDE) so that the environment channels can be explicitly dealt with. Then, a physical meaning of that characterization, which cannot be seen without the QSDE representation, is clarified; more specifically, the nullifier dynamics of any Gaussian system generating a unique pure steady state is passive. In addition, again based on the QSDE framework, we provide a general and practical method to implement a desired dissipative Gaussian system, which has a structure of quantum state transfer.

DOI
10.1098/rsta.2011.0529

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Highly multipartite entangled states play an important role in various quantum computing tasks. We investigate the dissipative generation of a complex entanglement structure as in a cluster state through engineered Markovian dynamics in the spin systems coupled via Ising interactions. Using the Lindblad master equation, we design a projection based dissipative channel that drives the system toward a unique pure steady state corresponding to the desired cluster state. This is done by removing the contribution of the orthogonal states. By explicitly constructing the Liouvillian superoperator in the full 2N2^N2N-dimensional Hilbert space, we compute the steady-state density matrix, the Liouvillian spectral gap, entanglement witness and the fidelity with respect to the ideal cluster state. The results demonstrate that the cluster state emerges as the steady state when the engineered Liouvillian dissipation dominates over the local Ising interaction between spins. Moreover, we find that the fidelity and Liouvillian spectral gap is relatively insensitive to the system size once the saturation dissipation has been achieved that scales linearly with the qubit number. This analysis illustrates a physically realizable path towards steady-state entanglement generation in the spin systems using engineered dissipation.
This paper introduces Quantum-Inspired Stochastic Differential Equations (QSDEs), a novel approach to modeling stochastic systems by leveraging core concepts from quantum mechanics. Traditional Stochastic Differential Equations (SDEs) often rely on Gaussian noise, which can be a significant limitation when accurately representing complex, real-world phenomena. QSDEs propose a fundamental shift by incorporating superposition and entanglement principles to model noise, moving beyond Gaussian distributions. The core mechanism involves introducing a "quantum noise" term, denoted as *η(t, x, P)*, where *η* is a superposition of probability distributions dynamically adjusted based on the system's state *x*. This allows for a more nuanced representation of uncertainty, potentially leading to vastly improved simulations across diverse fields. We outline the theoretical framework, explore the key mathematical formulations, and discuss the potential impact of QSDEs on areas such as financial modeling, physics simulations, and control systems. The central claim of this work is that classical SDEs can be significantly enhanced by incorporating quantum mechanical principles.
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