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arXiv:quant-ph/0609141 · Physical Review A

Resonant purification of mixed states for closed and open quantum systems

Raffaele Romano

Published Feb 20, 2007• 9 citations
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Abstract

Pure states are fundamental for the implementation of quantum technologies, and several methods for the purification of the state of a quantum system $S$ have been developed in the past years. In this work we describe a mechanism leading to purification of mixed states, based on the interaction of $S$ with an auxiliary system $P$. Considering two-level systems and assuming a particular interaction between them, we study how the dynamical parameters of the system $P$ affect the purification of $S$. By using analytical and numerical tools, we show that the purification process exhibits a resonant behavior in both the cases of system isolated from the external environment or not.

DOI
10.1103/physreva.75.024301

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Pure states are fundamental for the implementation of quantum technologies, and several methods for the purification of the state of a quantum system SSS have been developed in the past years. In this work we describe a mechanism leading to purification of mixed states, based on the interaction of SSS with an auxiliary system PPP. Considering two-level systems and assuming a particular interaction between them, we study how the dynamical parameters of the system PPP affect the purification of SSS. By using analytical and numerical tools, we show that the purification process exhibits a resonant behavior in both the cases of system isolated from the external environment or not.
Pure states are fundamental for the implementation of quantum technologies. Several methods for the purification of the state of a quantum system S have been developed in the past years. Considering two-level systems and assuming a particular interaction between the system S and probe P, we propose a control strategy that purifies an arbitrary mixed state to an eigenstate. An interaction control is designed to improve the quality of the purification of mixed state. The Lyapunov-based control method is used to design the part of interaction control. Compared numerical simulation experiments demonstrate superiority of the purification strategy proposed: the purification of an arbitrary mixed state can be realized in the energy difference between S and P being not only zero case, but also in the nonzero case.
A quantum system put in interaction with another one that is repeatedly measured is subject to a nonunitary dynamics, through which it is possible to extract subspaces. This key idea has been exploited to propose schemes aimed at the generation of pure quantum states (purification). All such schemes have so far been considered in the ideal situations of isolated systems. In this paper, we analyze the influence of non-negligible interactions with environment during the extraction process, with the scope of investigating the possibility of purifying the state of a system in spite of the sources of dissipation. A general framework is presented and a paradigmatic example consisting of two interacting spins immersed in a bosonic bath is studied. The effectiveness of the purification scheme is discussed in terms of purity for different values of the relevant parameters and in connection with the bath temperature.
It is known that an arbitrary quantum state can always be expressed in a pure state by preparing an appropriate ancilla system, which is called purification. The purification can always be performed when considering only a quantity not dependent on the quantum state of the ancilla system. Otherwise, the possibility of purification cannot be elucidated. In this study, we focus on the cases where a quantum system S and the ancilla system interact with each other, and investigate whether purification, which correctly reproduces the time evolution of the system S, is possible. Accordingly, when the ancilla system is a macrosystem, it can be shown that this purification is possible for a specific initial state.
Accurate modeling of quantum systems interacting with environments requires addressing non-unitary dynamics, which significantly complicates computational approaches. In this work, we enhance an open quantum system (OQS) theory using density-matrix purification, enabling a unitary description of dynamics by entangling the system with an environment of equal dimension. We first establish the connection between density-matrix purification and conventional OQS methods. We then demonstrate the standalone applicability of purification theory by deriving system-environment interactions from fundamental design principles. Using model systems, we show that the purification approach extends beyond the complete positivity condition and effectively models both Markovian and non-Markovian dynamics. Finally, we implement density-matrix purification on a quantum simulator, illustrating its capability to map non-unitary OQS dynamics onto a unitary framework suitable for quantum computers.
Pure quantum states are a fundamental resource for quantum information processing, but unavoidable noise degrades their purity and limits the performance of quantum tasks. Quantum state purification attempts to restore purity by jointly processing several noisy copies of an unknown state, typically in a probabilistic way. Whether purification is possible, and how much it can achieve, depends strongly on the set of operations that are physically allowed. In this work, we study universal purification under \textit{energy-preserving operations}, which cannot exchange energy with an external environment. We identify exactly when the energy-preserving constraint forbids any universal purification advantage, and we provide concrete examples in which an advantage available to unrestricted operations disappears under energy conservation. When energy-preserving purification remains possible, we determine its optimal fidelity and success probability and construct the corresponding protocols. The framework includes unrestricted purification as a special case and extends naturally to purification assisted by an external energy resource.
Purification of mixed states in Quantum Mechanics, by which we mean the transformation into pure states, has been viewed as an {\it Operation} in the sense of Kraus et al and explicit {\it Kraus Operators} \cite{kra1,kra2,kra3} have been constructed for two seperate purification protocols. The first one, initially due to Schrodinger \cite{sch} and subsequently elaborated by Sudarshan et al \cite{sudar}, is based on the {\it preservation of probabilities}. We have constructed a second protocol here based on {\it optimization of fidelities}. Both purification protocols have been implemented on a single qubit in an attempt to improve the fidelity of the purified post measurement state of the qubit with the initial pure state. We have considered both {\it complete} and {\it partial} measurements and have established bounds and inequalities for various fidelities. We show that our purification protocol leads to better state reconstruction, most explicitly so, when partial measurements are made.