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Physical Review A

Diluting quantum information: An analysis of information transfer in system-reservoir interactions

Mário Ziman, Peter Štelmachovič, Vladimír Bužek, Mark S. Hillery, Valerio Scarani, Nicolas Gisin

Published Mar 18, 2002• 173 citations
Publisher

Abstract

We design a universal quantum homogenizer, which is a quantum machine that takes as an input a system qubit initially in the state $\ensuremath{\rho}$ and a set of N reservoir qubits initially prepared in the same state $\ensuremath{\xi}.$ In the homogenizer the system qubit sequentially interacts with the reservoir qubits via the partial swap transformation. The homogenizer realizes, in the limit sense, the transformation such that at the output each qubit is in an arbitrarily small neighborhood of the state $\ensuremath{\xi}$ irrespective of the initial states of the system and the reservoir qubits. This means that the system qubit undergoes an evolution that has a fixed point, which is the reservoir state $\ensuremath{\xi}.$ We also study approximate homogenization when the reservoir is composed of a finite set of identically prepared qubits. The homogenizer allows us to understand various aspects of the dynamics of open systems interacting with environments in nonequilibrium states. In particular, the reversibility vs irreversibility of the dynamics of the open system is directly linked to specific (classical) information about the order in which the reservoir qubits interacted with the system qubit. This aspect of the homogenizer leads to a model of a quantum safe with a classical combination. We analyze in detail how entanglement between the reservoir and the system is created during the process of quantum homogenization. We show that the information about the initial state of the system qubit is stored in the entanglement between the homogenized qubits.

DOI
10.1103/physreva.65.042105

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We design a universal quantum homogenizer, which is a quantum machine that takes as an input a system qubit initially in the state $ρ$ and a set of N reservoir qubits initially prepared in the same state $ξ$. In the homogenizer the system qubit sequentially interacts with the reservoir qubits via the partial swap transformation. The homogenizer realizes, in the limit sense, the transformation such that at the output each qubit is in an arbitratily small neighbourhood of the state $ξ$ irrespective of the initial states of the system and the reservoir qubits. This means that the system qubit undergoes an evolution that has a fixed point, which is the reservoir state $ξ$. We also study approximate homogenization when the reservoir is composed of a finite set of identically prepared qubits. The homogenizer allows us to understand various aspects of the dynamics of open systems interacting with environments in non-equilibrium states. In particular, the reversibility vs or irreversibility of the dynamics of the open system is directly linked to specific (classical) information about the order in which the reservoir qubits interacted with the system qubit. This aspect of the homogenizer leads to a model of a quantum safe with a classical combination.We analyze in detail how entanglement between the reservoir and the system is created during the process of quantum homogenization. We show that the information about the initial state of the system qubit is stored in the entanglement between the homogenized qubits.

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We design a universal quantum homogenizer, which is a quantum machine that takes as an input a system qubit initially in the state ? and a set of N reservoir qubits initially prepared in the same state ?. In the homogenizer the system qubit sequentially interacts with the reservoir qubits via the partial swap transformation. The homogenizer realizes, in the limit sense, the transformation such that at the output each qubit is in an arbitratily small neighbourhood of the state ? irrespective of the initial states of the system and the reservoir qubits. This means that the system qubit undergoes an evolution that has a fixed point, which is the reservoir state ?. We also study approximate homogenization when the reservoir is composed of a finite set of identically prepared qubits. The homogenizer allows us to understand various aspects of the dynamics of open systems interacting with environments in non-equilibrium states. In particular, the reversibility vs or irreversibility of the dynamics of the open system is directly linked to specific (classical) information about the order in which the reservoir qubits interacted with the system qubit. This aspect of the homogenizer leads to a model of a quantum safe with a classical combination.We analyze in detail how entanglement between the reservoir and the system is created during the process of quantum homogenization. We show that the information about the initial state of the system qubit is stored in the entanglement between the homogenized qubits.

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Decoherence in Quantum Networks

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The title of the present dissertation Decoherence in Quantum Network sounds very general and all-inclusive. Indeed it embraces two topics (decoherence and quantum network) from the area of Quantum Mechanics each of which is described in all respects by a huge literature developed in the last three decades [...]. Quantum decoherence, as the name lets it mean, is the mechanism that makes a quantum system loose its coherence properties, and with them the capability of giving rise to interference phenomena or to other interesting quantum effects [...]. The key idea promoted by decoherence is the insight that realistic quantum systems are never isolated, but are immersed in the surrounding environment and interact continuously with it [...]. As an example one may consider a two-level quantum system (i.e. a quantum bit, usually shortened with a terminology from information science to \\qubit" ) in contact with a wide environment. Hence, quantum systems are open systems, and continuously interact or exchange information with an external environment whose degrees of freedom are too numerous to be monitored. The resulting correlation between the system and the environment spoils quantum coherence and brings about the transition from a pure quantum state to a mixture of quantum state resulting a classical state. To describe decoherence different kind of approaches can be used (for example Master equation, random matrix, etc). A quantum network typically consists of a number of quantum objects (e.g., atoms, ions, quantum dots, cavities, etc.), to be referred to hereafter as the sites or the nodes of the network. They can interact and the interactions (or their correlations) will be usually described by the edges of a graph. Quantum networks can address different information processing tasks. For instance a quantum state can be transferred from qubit to qubit down a chain solely due to the interactions, that is according to the laws of quantum physics [...]. Quantum networks offer us new opportunities and phenomena as compared to classical networks. An extension to large scale of the idea of a quantum network could lead to a futurible quantum internet [...]. The study of networks has traditionally been the territory of graph theory [...], also with the advent of their quantum versions. Within simple quantum network model information processing is usually described by assuming perfect control of the underlying graph. However, this is not much realistic since randomness is often present and it leads to decoherence effects [...]. In contrast, the conservation of coherence is essential for any quantum information process [...], hence there is a persistent interest in decoherence effects in quantum networks, which motivate us to study models for describing such noisy effects. We consider a simple model of quantum network, employing qubits (spin-1/2 particles) attached to the nodes of an underlying graph and we study the simplest task, namely information storage (on a single and two qubits), when the graph randomly changes in time. Actually we randomly add edges to an initially disconnected graph according to the Gilbert model characterized by a weighting parameter ex [...] and in an identically and independent way at each time step. We find that by increasing ex the dynamics of relevant quantities like fidelity, entropy or concurrence, gradually transforms from damped to damped oscillatory and finally to purely oscillatory. That leads to the paper [see, Information dissipation in random quantum networks, by U. Farooq and S. Mancini, OSID 21(3), 1450004, 2014]. We also study a system composed by pairs of qubits attached to each node of a linear chain, a model that stems from quantum dot arrays. Here we use the approach of evolution with a stochastic Hamiltonian to describe the noisy effects. We then evaluate the effect of two most common disorders, namely exchange coupling and hyperfine interaction fluctuations, in adiabatic preparation of ground state in such model. We show that the adiabatic ground state preparation is highly robust against these disorders making the chain a good analog simulator. Moreover, we also study the adiabatic information transfer, using singlet-triplet states, across the chain. In contrast to ground state preparation the transfer mechanism is highly affected by disorders. This suggested that for communication tasks across such chains adiabatic evolution is not as effective and quantum quenches would be preferable. That leads to the paper [see, Adiabatic many-body state preparation and information transfer in quantum dot arrays, by U. Farooq, A. Bayat, S. Mancini and S. Bose Phys. Rev. B 91, 134303, 2015]. The present work is organized as follows. In chapter 1, we shall give a survey of the various types of approach which can be employed to analyse the dynamics of open quantum systems that leads to decoherence effects. In chapter 2, we shall give a general description about quantum network and its possible applications. In chapter 3, we shall discuss the problem of quantum state transfer in qubit network and shall give a brief overview of some scheme that enable nearly prefect state transfer. In chapter 4, we shall discuss singlet-triplet networks, that is networks having on each site a pair of (generally entangled) qubit. Then within this framework we propose a model stemming from quantum dot array. There we shall address the problem of ground state preparation and state transfer. Finally we shall describe the inherent entanglement of the ground state of strongly correlated systems can be exploited for both classical and quantum communications. In chapter 5, we shall propose a decoherence model for qubit networks based on edges representing XY interactions randomly added to a disconnected graph accordingly to a suitable probability distribution. In this way we shall describe dissipation of information initially localize in single or two qubits all over the network. In chapter 6 we shall model the noisy effects in the quantum dot array introduced in chapter 4 and investigate their consequences on the preparation of ground state and quantum state transfer mechanism. Finally we shall draw conclusions.

•May 11, 2015•0 citations
We design a universal quantum homogenizer, which is a quantum machine that takes as an input a system qubit initially in the state \ensuremathρ\ensuremath{\rho}\ensuremathρ and a set of N reservoir qubits initially prepared in the same state \ensuremathξ.\ensuremath{\xi}.\ensuremathξ. In the homogenizer the system qubit sequentially interacts with the reservoir qubits via the partial swap transformation. The homogenizer realizes, in the limit sense, the transformation such that at the output each qubit is in an arbitrarily small neighborhood of the state \ensuremathξ\ensuremath{\xi}\ensuremathξ irrespective of the initial states of the system and the reservoir qubits. This means that the system qubit undergoes an evolution that has a fixed point, which is the reservoir state \ensuremathξ.\ensuremath{\xi}.\ensuremathξ. We also study approximate homogenization when the reservoir is composed of a finite set of identically prepared qubits. The homogenizer allows us to understand various aspects of the dynamics of open systems interacting with environments in nonequilibrium states. In particular, the reversibility vs irreversibility of the dynamics of the open system is directly linked to specific (classical) information about the order in which the reservoir qubits interacted with the system qubit. This aspect of the homogenizer leads to a model of a quantum safe with a classical combination. We analyze in detail how entanglement between the reservoir and the system is created during the process of quantum homogenization. We show that the information about the initial state of the system qubit is stored in the entanglement between the homogenized qubits.
We design a universal quantum homogenizer, which is a quantum machine that takes as an input a system qubit initially in the state ρρρ and a set of N reservoir qubits initially prepared in the same state ξξξ. In the homogenizer the system qubit sequentially interacts with the reservoir qubits via the partial swap transformation. The homogenizer realizes, in the limit sense, the transformation such that at the output each qubit is in an arbitratily small neighbourhood of the state ξξξ irrespective of the initial states of the system and the reservoir qubits. This means that the system qubit undergoes an evolution that has a fixed point, which is the reservoir state ξξξ. We also study approximate homogenization when the reservoir is composed of a finite set of identically prepared qubits. The homogenizer allows us to understand various aspects of the dynamics of open systems interacting with environments in non-equilibrium states. In particular, the reversibility vs or irreversibility of the dynamics of the open system is directly linked to specific (classical) information about the order in which the reservoir qubits interacted with the system qubit. This aspect of the homogenizer leads to a model of a quantum safe with a classical combination.We analyze in detail how entanglement between the reservoir and the system is created during the process of quantum homogenization. We show that the information about the initial state of the system qubit is stored in the entanglement between the homogenized qubits.
We design a universal quantum homogenizer, which is a quantum machine that takes as an input a system qubit initially in the state ? and a set of N reservoir qubits initially prepared in the same state ?. In the homogenizer the system qubit sequentially interacts with the reservoir qubits via the partial swap transformation. The homogenizer realizes, in the limit sense, the transformation such that at the output each qubit is in an arbitratily small neighbourhood of the state ? irrespective of the initial states of the system and the reservoir qubits. This means that the system qubit undergoes an evolution that has a fixed point, which is the reservoir state ?. We also study approximate homogenization when the reservoir is composed of a finite set of identically prepared qubits. The homogenizer allows us to understand various aspects of the dynamics of open systems interacting with environments in non-equilibrium states. In particular, the reversibility vs or irreversibility of the dynamics of the open system is directly linked to specific (classical) information about the order in which the reservoir qubits interacted with the system qubit. This aspect of the homogenizer leads to a model of a quantum safe with a classical combination.We analyze in detail how entanglement between the reservoir and the system is created during the process of quantum homogenization. We show that the information about the initial state of the system qubit is stored in the entanglement between the homogenized qubits.
Recently Ziman et al. [Phys. Rev. A 65, 042105 (2002)] have introduced a concept of a universal quantum homogenizer which is a quantum machine that takes as input a given (system) qubit initially in an arbitrary state $\ensuremath{\rho}$ and a set of N reservoir qubits initially prepared in the state $\ensuremath{\xi}.$ The homogenizer realizes, in the limit sense, the transformation such that at the output each qubit is in an arbitrarily small neighborhood of the state $\ensuremath{\xi}$ irrespective of the initial states of the system and the reservoir qubits. In this paper we generalize the concept of quantum homogenization for qudits, that is, for d-dimensional quantum systems. We prove that the partial-swap operation induces a contractive map with the fixed point which is the original state of the reservoir. We propose an optical realization of the quantum homogenization for Gaussian states. We prove that an incoming state of a photon field is homogenized in an array of beam splitters. Using Simon's criterion, we study entanglement between outgoing beams from beam splitters. We derive an inseparability condition for a pair of output beams as a function of the degree of squeezing in input beams.
Abstract Recently Ziman et al. [3] have introduced a concept of a universal quantum homogenizer which is a quantum machine that takes as an input a system qubit initially in an arbitrary state ρ and a set of N reservoir qubits initially prepared in the same state ξ. The homogenizer realizes, in the limit sense, the transformation such that at the output each qubit is in an arbitrarily small neighbourhood of the state ξ irrespective of the initial states of the system and the reservoir qubits. In this paper we generalize the concept of quantum homogenization for qudits, that is, for d‐dimensional quantum systems. We prove that the partial swap operation induces a contractive map with the fixed point which is the original state of the reservoir. Finally we propose an optical realization of the quantum homogenization.
We study how entanglement between an open system and a reservoir is established. The system is considered to be a qubit, while the reservoir is modelled as a collection of qubits. The system and the reservoir qubits interact via a sequence of partial-swap operations. This processes is called quantum homogenization since at the output the system as well as all reservoir qubits are in states that are, in a limit sense, equal to the original state of the reservoir qubits. We show that in this process the Coffman–Kundu–Wootters inequalities are saturated. This means that no intrinsic multi-partite entanglement is created.
The title of the present dissertation Decoherence in Quantum Network sounds very general and all-inclusive. Indeed it embraces two topics (decoherence and quantum network) from the area of Quantum Mechanics each of which is described in all respects by a huge literature developed in the last three decades [...]. Quantum decoherence, as the name lets it mean, is the mechanism that makes a quantum system loose its coherence properties, and with them the capability of giving rise to interference phenomena or to other interesting quantum effects [...]. The key idea promoted by decoherence is the insight that realistic quantum systems are never isolated, but are immersed in the surrounding environment and interact continuously with it [...]. As an example one may consider a two-level quantum system (i.e. a quantum bit, usually shortened with a terminology from information science to \\qubit" ) in contact with a wide environment. Hence, quantum systems are open systems, and continuously interact or exchange information with an external environment whose degrees of freedom are too numerous to be monitored. The resulting correlation between the system and the environment spoils quantum coherence and brings about the transition from a pure quantum state to a mixture of quantum state resulting a classical state. To describe decoherence different kind of approaches can be used (for example Master equation, random matrix, etc). A quantum network typically consists of a number of quantum objects (e.g., atoms, ions, quantum dots, cavities, etc.), to be referred to hereafter as the sites or the nodes of the network. They can interact and the interactions (or their correlations) will be usually described by the edges of a graph. Quantum networks can address different information processing tasks. For instance a quantum state can be transferred from qubit to qubit down a chain solely due to the interactions, that is according to the laws of quantum physics [...]. Quantum networks offer us new opportunities and phenomena as compared to classical networks. An extension to large scale of the idea of a quantum network could lead to a futurible quantum internet [...]. The study of networks has traditionally been the territory of graph theory [...], also with the advent of their quantum versions. Within simple quantum network model information processing is usually described by assuming perfect control of the underlying graph. However, this is not much realistic since randomness is often present and it leads to decoherence effects [...]. In contrast, the conservation of coherence is essential for any quantum information process [...], hence there is a persistent interest in decoherence effects in quantum networks, which motivate us to study models for describing such noisy effects. We consider a simple model of quantum network, employing qubits (spin-1/2 particles) attached to the nodes of an underlying graph and we study the simplest task, namely information storage (on a single and two qubits), when the graph randomly changes in time. Actually we randomly add edges to an initially disconnected graph according to the Gilbert model characterized by a weighting parameter ex [...] and in an identically and independent way at each time step. We find that by increasing ex the dynamics of relevant quantities like fidelity, entropy or concurrence, gradually transforms from damped to damped oscillatory and finally to purely oscillatory. That leads to the paper [see, Information dissipation in random quantum networks, by U. Farooq and S. Mancini, OSID 21(3), 1450004, 2014]. We also study a system composed by pairs of qubits attached to each node of a linear chain, a model that stems from quantum dot arrays. Here we use the approach of evolution with a stochastic Hamiltonian to describe the noisy effects. We then evaluate the effect of two most common disorders, namely exchange coupling and hyperfine interaction fluctuations, in adiabatic preparation of ground state in such model. We show that the adiabatic ground state preparation is highly robust against these disorders making the chain a good analog simulator. Moreover, we also study the adiabatic information transfer, using singlet-triplet states, across the chain. In contrast to ground state preparation the transfer mechanism is highly affected by disorders. This suggested that for communication tasks across such chains adiabatic evolution is not as effective and quantum quenches would be preferable. That leads to the paper [see, Adiabatic many-body state preparation and information transfer in quantum dot arrays, by U. Farooq, A. Bayat, S. Mancini and S. Bose Phys. Rev. B 91, 134303, 2015]. The present work is organized as follows. In chapter 1, we shall give a survey of the various types of approach which can be employed to analyse the dynamics of open quantum systems that leads to decoherence effects. In chapter 2, we shall give a general description about quantum network and its possible applications. In chapter 3, we shall discuss the problem of quantum state transfer in qubit network and shall give a brief overview of some scheme that enable nearly prefect state transfer. In chapter 4, we shall discuss singlet-triplet networks, that is networks having on each site a pair of (generally entangled) qubit. Then within this framework we propose a model stemming from quantum dot array. There we shall address the problem of ground state preparation and state transfer. Finally we shall describe the inherent entanglement of the ground state of strongly correlated systems can be exploited for both classical and quantum communications. In chapter 5, we shall propose a decoherence model for qubit networks based on edges representing XY interactions randomly added to a disconnected graph accordingly to a suitable probability distribution. In this way we shall describe dissipation of information initially localize in single or two qubits all over the network. In chapter 6 we shall model the noisy effects in the quantum dot array introduced in chapter 4 and investigate their consequences on the preparation of ground state and quantum state transfer mechanism. Finally we shall draw conclusions.