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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantum safe with classical key: physics of open systems from the point view of information theory

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

Published Jul 23, 2002• 0 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 ? 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.

DOI
10.1117/12.475875

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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.
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 $\ensuremath{\rho}$ 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.
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.
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 Collisional models are a category of microscopic framework designed to study open quantum systems. The framework involves a system sequentially interacting with a bath comprised of identically prepared units. In this regard, quantum homogenization is a process where the system state approaches the identically prepared state of bath unit in the asymptotic limit. Here, we study the homogenization process for a single qubit in the non-Markovian collisional model framework generated via additional bath-bath interaction. With partial swap operation as both system-bath and bath-bath unitary, we numerically demonstrate that homogenization is achieved irrespective of the initial states of the system or bath units. This is reminiscent of the Markovian scenario, where partial swap is the unique operation for a universal quantum homogenizer. On the other hand, we observe that the rate of homogenization is slower than its Markovian counter part. Interestingly, a different choice of bath-bath unitary speeds up the homogenization process but loses the universality, being dependent on the initial states of the bath units.
Here we investigate the role of quantum interference in the quantum homogenizer, whose convergence properties model a thermalization process. In the original quantum homogenizer protocol, a system qubit converges to the state of identical reservoir qubits through partial-swap interactions, that allow interference between reservoir qubits. We design an alternative, incoherent quantum homogenizer, where each system-reservoir interaction is moderated by a control qubit using a controlled-swap interaction. We show that our incoherent homogenizer satisfies the essential conditions for homogenization, being able to transform a qubit from any state to any other state to arbitrary accuracy, with negligible impact on the reservoir qubits' states. Our results show that the convergence properties of homogenization machines that are important for modeling thermalization are not dependent on coherence between qubits in the homogenization protocol. We then derive bounds on the resources required to re-use the homogenizers for performing state transformations. This demonstrates that both homogenizers are universal for any number of homogenizations, for an increased resource cost. Published by the American Physical Society 2024