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arXiv:quant-ph/0210078

Using unitary operations to preserve quantum states in the presence of relaxation

Benjamin Recht, Yael G. Maguire, Seth Lloyd, Isaac L. Chuang, Neil Gershenfeld

Published Oct 11, 2002• 3 citations
arXivHTMLPDF

Abstract

When a quantum system interacts with an external environment, it undergoes the loss of quantum correlation (decoherence) and the loss of energy (relaxation) and eventually all of the quantum information becomes classical. Here we show a general principle to use unitary operations to establish and preserve particular non-equilibrium states in arbitrary relaxing quantum systems. We elucidate these concepts with examples of state preservation in one-spin and two-spin entangled systems.

DOI
10.48550/arxiv.quant-ph/0210078

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Spingineering : quantum control in the presence of relaxation

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For certain problems quantum resources can exponentially increase computing power, but these quantum resources are very fragile in practice. When a quantum system interacts with an external environment, it undergoes decoherence the loss of quantum correlation and relaxation the loss of energy and eventually all of the quantum information is lost. Here we show a general principle of using unitary operators to suppress relaxation processes. Unitary operations do not cool a quantum system and seem an unlikely candidate for preventing irreversible thermodynamic heating processes, but surprisingly most decoherence processes can be corrected or ameliorated using open loop control with unitary controllers. We examine the different mechanisms of decoherence and relaxation on simple spin systems and discuss when the modes can be corrected. We show experimentally the feasibility of our correction schemes using nuclear magnetic resonance. We also demonstrate control of the nuclear spins over long time scales. Finally, we discuss the applications of unitary correction to higher dimensional systems and the potential applications to quantum information processing. Thesis Supervisor: Neil Gershenfeld Title: Associate Professor Spingineering: Quantum Control in the Presence of Relaxation by Benjamin Harris Recht Submitted to the Media Arts and Sciences department in partial fulfillment of the requirements for the degree of Master of Science in Media Arts and Sciences at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2002 Thesis Advisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neil Gershenfeld Associate Professor Department of Media Arts and Sciences Massachusetts Institute of Technology Thesis Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isaac Chuang Associate Professor Department of Media Arts and Sciences Massachusetts Institute of Technology Thesis Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Seth Lloyd Professor Department of Mechanical Engineering Massachusetts Institute of Technology Acknowledgments This thesis, being no different than any other, would have not been possible without some truly remarkable people. I would like to thank my advisor Neil Gershenfeld for his support and guidance through my first two years and my readers: Isaac Chuang for introducing me to and guiding me through the quantum world, and Seth Lloyd for always saying just the right thing at the right time. I owe a large thank you to Yael Maguire for inspiring this work and his eagerness to discuss my progress. Thanks to Matthias Steffen and Yael for making sure that my theorist hands never broke the ever fussy 500 MHz NMR spectrometer. I would also like to thank Aram Harrow and Ali Rahimi for invaluable discussions and suggestions. Thanks to Jason Taylor for bearing with me in the confines of our office and to the rest of the PHM crew: Matt, Rehmi, Ben, Rich, Ravi, Bernd, John-Paul, Dan, Caroline, Esa, Femi. Thanks to Linda Peterson and Pat Sokaloff for taking extra time to care about all of us lost and helpless graduate students. I also owe a great deal of thanks to Susan MurphyBottari, Mike Houlilhan, and Liz Hennessey for making the PHM machine run as smoothly as it does. I send much love and respect to the unlockedgroove crew and all of the people who frequent to the Appliance of Science for helping me maintain my sanity. Thank you Mom and Dad for 24 years of constant support and Marissa for never forgetting to tell me when I’m wrong. Thank you Lauren for being exactly who you are. Thanks to the MIT Media Laboratory and the Things That Think Consortium. This thesis would not have been possible without the constant support of Nicotine, Caffeine, Alcohol, and Cholesterol. Thank you for better living through chemistry.

•Jan 1, 2002•1 citation

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The subject of this thesis are various properties of quantum states that make them "non-classical" and their behaviour under unitary operations. In chapter 2 some basic concepts of quantum mechanics and quantum information are reviewed. In chapter 3 properties of quantum states analysed in the literature are identified. I chapter 4 relations between them are analysed: whether possessing of one property by a given state implies possessing another property by the same state. In chapter 5 the central notion of the thesis is introduced, namely possessing a given property in absolute vs. non-absolute way. A property is possessed by a given state in an absolute way if it is preserved under arbitrary unitary operation on this state. This will be applied notion to all properties listed in chapter 3. Two main questions asked here are as follows: Can a given property be possessed both in an absolute way and in a non-absolute way (by different classes of states)? If yes, what are necessary and sufficient conditions for a state possessing a given property in an absolute way?

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Theory and experiment have not fully resolved the apparent dichotomy , which has agonized physics for the past eighty years: on the one hand, the description of microsystems by quantum mechanics and, on the other, the description of macrosystems by classical dynamics or statistical mechanics. Derivations of the time-irreversible Liouville equation for an open quantum system, based on projecting out its environment, have narrowed the gap between the quantum and classical descriptions. Yet our `classical' intuition continues to be confronted by quantum-mechanical results like the Einstein--Podolsky--Rosen paradox that challenges the classical notion of locality, or the quantum Zeno effect which suggests that the isolation of a system is not the only way to preserve its quantum state. There are two key concepts in any discussion of such issues. 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Non-Markovian qubit dynamics in a thermal field bath: Relaxation, decoherence, and entanglement

Sanjiv Shresta, Charis Anastopoulos, Adrian A. Drǎgulescu +1 more

We study the non-Markovian dynamics of a qubit made up of a two-level atom interacting with an electromagnetic field (EMF) initially at finite temperature. Unlike most earlier studies where the bath is assumed to be fixed, we study the complete evolution of the combined qubit-EMF system, thus allowing for the coherent backaction from the bath on the qubit and the qubit on the bath in a self-consistent manner. In this way we can see the development of quantum correlations and entanglement between the system and its environment, and how that affects the decoherence and relaxation of the system. We find nonexponential decay for both the diagonal and nondiagonal matrix elements of the qubit's reduced density matrix in the pointer basis. The former shows the qubit relaxing to thermal equilibrium with the bath, while the latter shows the decoherence rate beginning at the usually predicted thermal rate, but changing to the zero-temperature value as the qubit and bath become entangled. The decoherence and relaxation rates are comparable, as in the zero-temperature case. On the entanglement of a qubit with the EMF we calculated the fidelity and the von Neumann entropy, which is a measure of the purity of the density matrix. The present more accurate non-Markovian calculations predict lower loss of fidelity and purity as compared with the Markovian results. Generally speaking, with the inclusion of quantum correlations between the qubit and its environment, the non-Markovian processes tend to slow down the drive of the system to equilibrium, prolonging the decoherence and better preserving the fidelity and purity of the system.

arXiv:quant-ph/0408084•Physical Review A•Feb 22, 2005•59 citations
When a quantum system interacts with an external environment, it undergoes the loss of quantum correlation (decoherence) and the loss of energy (relaxation) and eventually all of the quantum information becomes classical. Here we show a general principle to use unitary operations to establish and preserve particular non-equilibrium states in arbitrary relaxing quantum systems. We elucidate these concepts with examples of state preservation in one-spin and two-spin entangled systems.
For certain problems quantum resources can exponentially increase computing power, but these quantum resources are very fragile in practice. When a quantum system interacts with an external environment, it undergoes decoherence the loss of quantum correlation and relaxation the loss of energy and eventually all of the quantum information is lost. Here we show a general principle of using unitary operators to suppress relaxation processes. Unitary operations do not cool a quantum system and seem an unlikely candidate for preventing irreversible thermodynamic heating processes, but surprisingly most decoherence processes can be corrected or ameliorated using open loop control with unitary controllers. We examine the different mechanisms of decoherence and relaxation on simple spin systems and discuss when the modes can be corrected. We show experimentally the feasibility of our correction schemes using nuclear magnetic resonance. We also demonstrate control of the nuclear spins over long time scales. Finally, we discuss the applications of unitary correction to higher dimensional systems and the potential applications to quantum information processing. Thesis Supervisor: Neil Gershenfeld Title: Associate Professor Spingineering: Quantum Control in the Presence of Relaxation by Benjamin Harris Recht Submitted to the Media Arts and Sciences department in partial fulfillment of the requirements for the degree of Master of Science in Media Arts and Sciences at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2002 Thesis Advisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neil Gershenfeld Associate Professor Department of Media Arts and Sciences Massachusetts Institute of Technology Thesis Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isaac Chuang Associate Professor Department of Media Arts and Sciences Massachusetts Institute of Technology Thesis Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Seth Lloyd Professor Department of Mechanical Engineering Massachusetts Institute of Technology Acknowledgments This thesis, being no different than any other, would have not been possible without some truly remarkable people. I would like to thank my advisor Neil Gershenfeld for his support and guidance through my first two years and my readers: Isaac Chuang for introducing me to and guiding me through the quantum world, and Seth Lloyd for always saying just the right thing at the right time. I owe a large thank you to Yael Maguire for inspiring this work and his eagerness to discuss my progress. Thanks to Matthias Steffen and Yael for making sure that my theorist hands never broke the ever fussy 500 MHz NMR spectrometer. I would also like to thank Aram Harrow and Ali Rahimi for invaluable discussions and suggestions. Thanks to Jason Taylor for bearing with me in the confines of our office and to the rest of the PHM crew: Matt, Rehmi, Ben, Rich, Ravi, Bernd, John-Paul, Dan, Caroline, Esa, Femi. Thanks to Linda Peterson and Pat Sokaloff for taking extra time to care about all of us lost and helpless graduate students. I also owe a great deal of thanks to Susan MurphyBottari, Mike Houlilhan, and Liz Hennessey for making the PHM machine run as smoothly as it does. I send much love and respect to the unlockedgroove crew and all of the people who frequent to the Appliance of Science for helping me maintain my sanity. Thank you Mom and Dad for 24 years of constant support and Marissa for never forgetting to tell me when I’m wrong. Thank you Lauren for being exactly who you are. Thanks to the MIT Media Laboratory and the Things That Think Consortium. This thesis would not have been possible without the constant support of Nicotine, Caffeine, Alcohol, and Cholesterol. Thank you for better living through chemistry.
The subject of this thesis are various properties of quantum states that make them "non-classical" and their behaviour under unitary operations. In chapter 2 some basic concepts of quantum mechanics and quantum information are reviewed. In chapter 3 properties of quantum states analysed in the literature are identified. I chapter 4 relations between them are analysed: whether possessing of one property by a given state implies possessing another property by the same state. In chapter 5 the central notion of the thesis is introduced, namely possessing a given property in absolute vs. non-absolute way. A property is possessed by a given state in an absolute way if it is preserved under arbitrary unitary operation on this state. This will be applied notion to all properties listed in chapter 3. Two main questions asked here are as follows: Can a given property be possessed both in an absolute way and in a non-absolute way (by different classes of states)? If yes, what are necessary and sufficient conditions for a state possessing a given property in an absolute way?
Theory and experiment have not fully resolved the apparent dichotomy , which has agonized physics for the past eighty years: on the one hand, the description of microsystems by quantum mechanics and, on the other, the description of macrosystems by classical dynamics or statistical mechanics. Derivations of the time-irreversible Liouville equation for an open quantum system, based on projecting out its environment, have narrowed the gap between the quantum and classical descriptions. Yet our `classical' intuition continues to be confronted by quantum-mechanical results like the Einstein--Podolsky--Rosen paradox that challenges the classical notion of locality, or the quantum Zeno effect which suggests that the isolation of a system is not the only way to preserve its quantum state. There are two key concepts in any discussion of such issues. The first, which is responsible for the most salient nonclassical properties, is entanglement , that is partial or complete correlation or, more generally, inseparability of the elements comprising a quantum ensemble . Even after their interaction has ceased, this inseparability, originating from their past interaction, can affect the state of one element when another element is subject to a nonunitary action, such as its measurement, tracing- out, or thermalization. The second key concept is decoherence of open quantum systems, which is the consequence of their entanglement with their environment, a `meter' or a thermal `reservoir', followed by the tracing-out of the latter. Despite new insights into entanglement and decoherence, there are still no complete, unequivocal answers to the fundamental questions of the transition from quantal to classical behaviour: how do irreversibility and classicality emerge from unitarity as systems and their environments become increasingly complex ? At what stage does system--meter entanglement give rise to a classical readout of the meter? Is there an upper limit on the size or complexity of systems displaying entanglement? Major developments have opened new vistas into controlled entanglement , which is the resource of quantum information processing. However, these developments have mainly focused on ensembles of simple two- and three-level systems that are thoroughly isolated from their environments. Treatments of coherence in quantum computing have mostly assumed that only a single or a few the elements of the quantum ensemble may simultaneously undergo an uncontrolled intervention---a quantum error. Decoherence-control protocols for more general types of errors are still incomplete. In order to resolve the outstanding issues of the quantum--classical transition, and study the control of entanglement and decoherence without the foregoing restrictions, we must venture into the domain of Quantum Complex Systems (QUACS), either consisting of a large number of inseparable elements or having many coupled degrees of freedom. Our conviction is that fundamental understanding and manipulation of entanglement within QUACS or their entanglement with the environment or a meter, call for the creation of a new conceptual framework or paradigm, that would encompass phenomena common to cold atoms in laser fields, large molecules, Josephson junctions, quantum gases and solids, with the view of employing these systems for quantum information processing and computing. Progress within this paradigm should allow us to answer the questions: does entanglement play an essential role in the evolution of large collections of complex systems? What are the size and complexity limits of systems and ensembles still controllable by an external intervention? What are the most appropriate decoherence protection schemes and control algorithms? This special issue addresses the challenge of understanding in depth and manipulating the basic quantum properties of optical, atomic, molecular and condensed-matter QUACS, and large ensembles thereof. In view of the interdisciplinary character of this issue, we find it expedient to look at the presented articles not only according to the physical objects they describe, but also from a unifying standpoint. Several unifying themes may be discerned: memory effects in dephasing and relaxation; protection of quantum information by its distribution throughout the ensemble; control of decoherence and entanglement by time-dependent interventions or measurements; manifestations of entanglement in scattering off statistical ensembles; measurment-induced dynamics. The concerted discussion of the topics outlined above should help us advance the new paradigm that addresses our abilities to diagnose and manipulate the entangled states of complex quantum objects and their robustness against decoherence. These abilities are required for quantum information (QI) applications in matter--wave interferometry in molecular, semiconducting or superconducting systems. On the fundamental level, this book may help establish the notion of dynamical information exchange between quantum systems and chart in detail the route from unitarity to classicality. Further developments within the outlined paradigm should yield novel, advantageous QI processing schemes and high-sensitivity interferometers, owing to decoherence suppression and effective control of many degrees of freedom. In the long run, these strategies may prove to be the first step towards the (hitherto unattempted) use of entanglement in nanotechnology, metrology and chemistry, with potentially remarkable novel applications.
We address the interaction between two quantum systems (AAA and BBB) that is mediated by their common linear environment. If the environment is out of equilibrium, the resulting interaction violates Onsager relations and cannot be described by a Hamiltonian. In simple terms, the action of system AAA on system BBB does not necessarily produce a back action. We derive general quantum equations describing the situation, and we analyze in detail their classical correspondence. Changing the properties of the environment, one can easily change and engineer the resulting interaction. It is tempting to use this for quantum manipulation of the systems. However, the resulting quantum gate is not always unitary and may induce a loss of quantum coherence. For a relevant example, we consider systems AAA and BBB to be spins of arbitrary values and arrange the interaction to realize an analog of the two-qubit cnot gate. The direction of spin AAA controls the rotation of spin BBB while spin AAA is not rotated experiencing no back action from spin BBB. We solve the quantum dynamics equations and analyze the purity of the resulting density matrix. The resulting purity essentially depends on the initial states of the systems. We attempt to find a universal characteristic of the purity, optimizing it for the worst choice of initial states. For both spins sA=sB=1/2{s}_{A}={s}_{B}=1/2sA​=sB​=1/2, the optimized purity is bounded by 1/2 irrespective of the details of the gate. We also study in detail the semiclassical limit of large spins. In this case, the optimized purity is bounded by ${(1+\ensuremath{\pi}/2)}^{\ensuremath{-}1}\ensuremath{\approx}0.39$. This is much better than the typical purity of a large spin state $\ensuremath{\sim}{s}^{\ensuremath{-}1}$. We conclude that although the quantum manipulation without back action inevitably causes decoherence of the quantum states, the actual purity of the resulting state can be optimized and made relatively high.
We study the non-Markovian dynamics of a qubit made up of a two-level atom interacting with an electromagnetic field (EMF) initially at finite temperature. Unlike most earlier studies where the bath is assumed to be fixed, we study the complete evolution of the combined qubit-EMF system, thus allowing for the coherent backaction from the bath on the qubit and the qubit on the bath in a self-consistent manner. In this way we can see the development of quantum correlations and entanglement between the system and its environment, and how that affects the decoherence and relaxation of the system. We find nonexponential decay for both the diagonal and nondiagonal matrix elements of the qubit's reduced density matrix in the pointer basis. The former shows the qubit relaxing to thermal equilibrium with the bath, while the latter shows the decoherence rate beginning at the usually predicted thermal rate, but changing to the zero-temperature value as the qubit and bath become entangled. The decoherence and relaxation rates are comparable, as in the zero-temperature case. On the entanglement of a qubit with the EMF we calculated the fidelity and the von Neumann entropy, which is a measure of the purity of the density matrix. The present more accurate non-Markovian calculations predict lower loss of fidelity and purity as compared with the Markovian results. Generally speaking, with the inclusion of quantum correlations between the qubit and its environment, the non-Markovian processes tend to slow down the drive of the system to equilibrium, prolonging the decoherence and better preserving the fidelity and purity of the system.