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arXiv:1611.07530 · Physical Review A

Purification in rapid-repeated-interaction systems

Daniel Grimmer, Robert B. Mann, Eduardo Martín-Martínez

Published Apr 10, 2017• 8 citations
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Abstract

We investigate the open dynamics of a quantum system which is rapidly repeatedly updated by a quantum channel. Specifically, we analyze when this dynamics can purify the system. We develop a necessary and sufficient condition for such purification effects to occur and characterize their strength. We thoroughly analyze the specific scenario of a quantum system undergoing rapid unitary interactions with a sequence of ancillary quantum systems. We find that, while the purification effects are generally present, in order for these effects to be strong compared to the decoherence effects the interaction Hamiltonian must have a minimum degree of complexity. Specifically, a tensor product interaction ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}}_{\text{S}}\ensuremath{\bigotimes}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{R}}_{\text{A}}$ as well as many common light-matter interactions cannot purify efficiently.

DOI
10.1103/physreva.95.042114

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We investigate the open dynamics of a quantum system which is rapidly repeatedly updated by a quantum channel. Specifically, we analyze when this dynamics can purify the system. We develop a necessary and sufficient condition for such purification effects to occur and characterize their strength. We thoroughly analyze the specific scenario of a quantum system undergoing rapid unitary interactions with a sequence of ancillary quantum systems. We find that, while the purification effects are generally present, in order for these effects to be strong compared to the decoherence effects the interaction Hamiltonian must have a minimum degree of complexity. Specifically, a tensor product interaction Q\ifmmode^\elseˆ\fiS\ensuremath⨂R\ifmmode^\elseˆ\fiA{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}}_{\text{S}}\ensuremath{\bigotimes}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{R}}_{\text{A}}Q\ifmmode^\elseˆ\fi​S​\ensuremath⨂R\ifmmode^\elseˆ\fi​A​ as well as many common light-matter interactions cannot purify efficiently.
We investigate the emergent open dynamics of a quantum system that undergoes rapid repeated unitary interactions with a sequence of ancillary systems. We study in detail how decoherence appears as a subleading effect when a quantum system is bombarded by a quick succession of ancillas. In the most general case, these ancillas are (a) taken from an ensemble of quantum systems of different dimensions, (b) prepared in different states, and (c) interacting with the system through different Hamiltonians. We derive an upper bound on decoherence rates in this regime, and show how a rich variety of phenomena in open dynamics (such as projection, thermalization, purification, and dephasing) can emerge out of our general model of repeated interaction. Furthermore, we show a fundamental link between the strength of the leading-order dissipation and the intrinsic ``unpredictability'' in the system-ancilla interaction. We also discuss how these results encompass and extend results obtained with other earlier models of repeated interaction.
We present a novel procedure to purify quantum states, i.e., purification through Zeno-like measurements. By simply repeating one and the same measurement on a quantum system, one can purify another system in interaction with the former. The conditions for the (efficient) purification are specified on a rather general setting, and the framework of the method possesses wide applicability. It is explicitly demonstrated on a specific setup that the purification becomes very efficient by tuning relevant parameters.
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.
We explore the effect of decoherence on many-body mixed states from a purification perspective. Here, quantum channels map to unitary transformations acting on a purified state defined within an extended Hilbert space. First, we exploit this approach to analyze the phenomenon of spontaneous strong-to-weak symmetry breaking (sw-SSB) triggered by strongly symmetric local quantum channels. We find that sw-SSB for mixed states relates to proximity to a symmetry-protected topological (SPT) order in the purified state. Remarkably, the measurement-induced long-range order in the purified SPT state, as characterized by the R\'enyi-2 correlator, mirrors the ensuing long-range order in the mixed state due to sw-SSB. Moreover, we establish a 1-to-1 correspondence between various sw-SSB order parameters for the mixed state, and strange correlators in the purification, which signify the SPT order. This purification perspective is further extended to explore intrinsic mixed-state topological order and decoherent symmetry-protected topological phases.
We introduce a continuous-time model of many-body quantum dynamics based on infinitesimal random unitary operations, combined with projective measurements. We consider purification dynamics in this model, where the system is initialized in a mixed state, which then purifies over time as a result of the measurements. By mapping our model to a family of effective 1D quantum Hamiltonians, we are able to derive analytic expressions that capture how the entropy of the system decays in time. Our results confirm the existence of two distinct dynamical phases, where purification occurs over a timescale that is exponential versus constant in system size. We compare our analytic expressions for this microscopic model to results derived from field theories that are expected to capture such measurement-induced phase transitions and find quantitative agreement between the two.
Abstract Bang–bang dynamical decoupling protects an open quantum system from decoherence due to its interaction with the surrounding bath/environment. In its standard form, this is achieved by strongly kicking the system with cycles of unitary operations, which average out the interaction Hamiltonian. In this paper, we generalize the notion of dynamical decoupling to repeated kicks with a quantum channel, which is applied to the bath. We derive necessary and sufficient conditions on the employed quantum channel and find that bath dynamical decoupling works if and only if the kick is ergodic. Furthermore, we study in which circumstances completely positive trace-preserving (CPTP) kicks on a mono-partite quantum system induce quantum Zeno dynamics with its Hamiltonian cancelled out. This does not require the ergodicity of the kicks, and the absence of decoherence-free subsystems is both necessary and sufficient. While the standard unitary dynamical decoupling is essentially the same as the quantum Zeno dynamics, our investigation implies that this is no longer true in the case of CPTP kicks. To derive our results, we prove some spectral properties of ergodic quantum channels, that might be of independent interest. Our approach establishes an enhanced and unified mathematical understanding of several recent experimental demonstrations and might form the basis of new dynamical decoupling schemes that harness environmental noise degrees of freedom.