We discuss some recent results concerning the decoherence in controlled quantum open systems with... more We discuss some recent results concerning the decoherence in controlled quantum open systems within the mathematical setting corresponding to motion reversal experiments (the Loschmidt echo). We compare the case of randomly chosen sequence of unitary dynamical maps with the case of a constant dynamics corresponding to a classically chaotic evolution. The interplay between chaos and decoherence is illustrated by the new numerical results on the quantum Arnold cat map perturbed by a measurement process. Open problems related to the simple operational characterization of the decoherence strength are discussed.
Motivated by the recent interest in thermodynamics of micro- and mesoscopic quantum systems we st... more Motivated by the recent interest in thermodynamics of micro- and mesoscopic quantum systems we study the maximal amount of work that can be reversibly extracted from a quantum system used to store temporarily energy. Guided by the notion of passivity of a quantum state we show that entangling unitary controls extract in general more work than independent ones. In the limit of large number of copies one can reach the thermodynamical bound given by the variational principle for free energy.
We prove a no-go theorem for storing quantum information in equilibrium systems. Namely, quantum ... more We prove a no-go theorem for storing quantum information in equilibrium systems. Namely, quantum information cannot be stored in a system with time-independent Hamiltonian interacting with heat bath of temperature $T>0$ during time that grows with the number of used qubits. We prove it by showing, that storing quantum information for macroscopic time would imply existence of perpetuum mobile of the second kind. The general results are illustrated by the Kitaev model of quantum memory. In contrast, classical information can be stored in equilibrium states for arbitrary long times. We show how it is possible via phase-transition type phenomena. Our result shows that there is a fundamental difference between quantum and classical information in {\it physical} terms.
We analyse stability of the four-dimensional Kitaev model - a candidate for scalable quantum memo... more We analyse stability of the four-dimensional Kitaev model - a candidate for scalable quantum memory - in finite temperature within the weak coupling Markovian limit. It is shown that, below a critical temperature, certain topological qubit observables X and Z possess relaxation times exponentially long in the size of the system. Their construction involves polynomial in system's size algorithm which uses as an input the results of measurements performed on all individual spins. We also discuss the drawbacks of such candidate for quantum memory and mention the implications of the stability of qubit for statistical mechanics.
We develop a dynamical non-Markovian description of quantum computing in the weak-coupling limit,... more We develop a dynamical non-Markovian description of quantum computing in the weak-coupling limit, in the lowest-order approximation. We show that the long-range memory of the quantum reservoir (such as the 1/t4 one exhibited by electromagnetic vacuum) produces a strong interrelation between the structure of noise and the quantum algorithm, implying nonlocal attacks of noise. This shows that the implicit assumption of quantum error correction theory-independence of noise and self-dynamics-fails in long time regimes. We also use our approach to present pure decoherence and decoherence accompanied by dissipation in terms of the spectral density of the reservoir. The so-called dynamical decoupling method is discussed in this context. Finally, we propose a minimal decoherence model, in which the only source of decoherence is vacuum. We optimize the fidelity of quantum-information processing under the trade-off between the speed of the gate and the strength of decoherence.
There exists a large number of experimental and theoretical results supporting the picture of mac... more There exists a large number of experimental and theoretical results supporting the picture of macroscopic qubits implemented by nanoscopic Josephson junctions of three different types -- charge qubit, flux qubit and phase qubit. The standard unified description of such systems is based on the formal quantization of the phenomenological Kirchhoff equations for the corresponding circuits. In this paper a simplified version of the BCS theory for superconductors is used to derive microscopic models for all types of small Josephson junctions. For these models the state-dependent individual tunneling of Cooper pairs couples ground pair states with excited pair states what leads to a more complicated structure of the lowest lying energy levels. In particular, the highly degenerate levels emerge, which act as probability sinks for the qubit. These models allow also for the coupling to phonons as an efficient mechanism of relaxation for all types of junctions. The alternative formulas concerning basic spectral parameters of superconducting qubits are presented and compared with the experimental data. Finally, the question whether small Josephson junctions can be treated as macroscopic quantum systems is briefly discussed.
We consider, within the algebraic formalism, the time dependence of fidelity for qubits encoded i... more We consider, within the algebraic formalism, the time dependence of fidelity for qubits encoded in an open physical system. We relate the decay of fidelity to the evolution of correlation functions and, in the particular case of a Markovian dynamics, to the spectral gap of the generator of the semigroup. The results are applicable to the analysis of models of quantum memories.
The purity, Tr( ρ2), measures how pure or mixed a quantum state ρ is. It is well known that quant... more The purity, Tr( ρ2), measures how pure or mixed a quantum state ρ is. It is well known that quantum dynamical semigroups that preserve the identity operator (which we refer to as unital) are strictly purity-decreasing transformations. Here, we provide an almost complete characterization of the class of strictly purity-decreasing quantum dynamical semigroups. We show that in the case of finite-dimensional Hilbert spaces, a dynamical semigroup is strictly purity-decreasing if and only if it is unital, while in the infinite dimensional case, unitality is only sufficient.
We revisit the notion of Kolmogorov-Sinai entropy for classical dynamical systems in terms of an ... more We revisit the notion of Kolmogorov-Sinai entropy for classical dynamical systems in terms of an algebraic formalism. This is the starting point for defining the entropy for general non-commutative systems. Hereby typical quantum tools are introduced in the statistical description of classical dynamical systems. We illustrate the power of these techniques by providing a simple, self-contained proof of the entropy formula for general automorphisms of n-dimensional tori.
The rate of temperature decrease of a cooled quantum bath is studied as its temperature is reduce... more The rate of temperature decrease of a cooled quantum bath is studied as its temperature is reduced to the absolute zero. The III-law of thermodynamics is then quantified dynamically by evaluating the characteristic exponent {\zeta} of the cooling process dT(t)/dt \sim -T^{\zeta} when approaching the absolute zero, T \rightarrow 0. A continuous model of a quantum refrigerator is employed consisting of a working medium composed either by two coupled harmonic oscillators or two coupled 2-level systems. The refrigerator is a nonlinear device merging three currents from three heat baths: a cold bath to be cooled, a hot bath as an entropy sink, and a driving bath which is the source of cooling power. A heat driven refrigerator (absorption refrigerator) is compared to a power driven refrigerator. When optimized both cases lead to the same exponent {\zeta}, showing a lack of dependence on the form of the working medium and the characteristics of the drivers. The characteristic exponent is therefore determined by the properties of the cold reservoir and its interaction with the system. Two generic heat baths models are considered, a bath composed of harmonic oscillators and a bath composed from ideal Bose/Fermi gas. The restrictions on the interaction Hamiltonian imposed by the III-law are discussed. In the appendix the theory of periodicaly driven open systems and its implication to thermodynamics is outlined.
Lecture Notes in Physics Berlin Springer Verlag, 2007
... to a member of the Editorial Board, or directly to the managing editor at Springer: Dr. Chris... more ... to a member of the Editorial Board, or directly to the managing editor at Springer: Dr. Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg/Germany christian. caron@ springer. com Page 4. Robert Alicki Karl Lendi Quantum ...
We prove continuity of quantum conditional information $S(\rho^{12}| \rho^2)$ with respect to the... more We prove continuity of quantum conditional information $S(\rho^{12}| \rho^2)$ with respect to the uniform convergence of states and obtain a bound which is independent of the dimension of the second party. This can, e.g., be used to prove the continuity of squashed entanglement.
This paper consists in two parts. First we set up a general scheme of local traps in an homogeneo... more This paper consists in two parts. First we set up a general scheme of local traps in an homogeneous deterministic quantum system. The current of particles caught by the trap is linked to the dynamical behaviour of the trap states. In this way, transport properties in an homogeneous system are related to spectral properties of a coherent dynamics. Next we apply the scheme to a system of Fermions in the one-particle approximation. We obtain in particular lower bounds for the dynamical entropy in terms of the current induced by the trap.
The Wigner-Weisskopf-type model developed by Alicki and Giraldi [J. Phys. B 44, 154020 (2011)], 1... more The Wigner-Weisskopf-type model developed by Alicki and Giraldi [J. Phys. B 44, 154020 (2011)], 10.1088/0953-4075/44/15/154020 is applied to the biological process of energy transfer from a large peripheral light harvesting antenna to the reaction center. This process is mediated by the Fenna-Matthews-Olson (FMO) photosynthetic complex with a remarkably high efficiency. The proposed model provides a simple resonance mechanism of this phenomenon employing exciton coherent motion and is described by analytical formulas. A coupling to the vibrational environment is a necessary component of this mechanism as well as a fine-tuning of the FMO complex Hamiltonian. The role of the relatively strong coupling to the energy sink in achieving the resonance condition and the absence of heating of the vibrational environment are emphasized.
The authors propose to describe the dynamics of unstable particles in relativistic quantum field ... more The authors propose to describe the dynamics of unstable particles in relativistic quantum field theory in terms of semigroups of transformations of the observables. This leads, in contrast to the usual Hilbert space level treatment, to a complete and consistent descriptions of the irreversible dynamics of decay processes. The scheme is explicitly worked out for the massive scalar quantum field and the evolution of the particle density and its higher moments is computed.
We comment that the model proposed in Phys. Rev. Lett. 108, 120603 (2012) violates the dynamical ... more We comment that the model proposed in Phys. Rev. Lett. 108, 120603 (2012) violates the dynamical version of the third law of thermodynamics. We discuses the different formulations of the third law of thermodynamics and suggest a possible reason for the violation.
It is often claimed, that from a quantum system of d levels, and entropy S and heat bath of tempe... more It is often claimed, that from a quantum system of d levels, and entropy S and heat bath of temperature T one can draw kT (ln d − S) amount of work. However, the usual arguments basing on Szilard engine, are not fully rigorous. Here we prove the formula within Hamiltonian description of drawing work from a quantum system and a heat bath, at the cost of entropy of the system. We base on the derivation of thermodynamical laws and quantities in [R. Alicki, J. Phys. A, 12, L103 (1979)] within weak coupling limit. Our result provides fully physical scenario for extracting thermodynamical work form quantum correlations [Oppenheim et al. Phys. Rev. Lett. 89, 180402 (2002)]. We also derive Landauer principle as a consequence of second law within the considered model.
We discuss some recent results concerning the decoherence in controlled quantum open systems with... more We discuss some recent results concerning the decoherence in controlled quantum open systems within the mathematical setting corresponding to motion reversal experiments (the Loschmidt echo). We compare the case of randomly chosen sequence of unitary dynamical maps with the case of a constant dynamics corresponding to a classically chaotic evolution. The interplay between chaos and decoherence is illustrated by the new numerical results on the quantum Arnold cat map perturbed by a measurement process. Open problems related to the simple operational characterization of the decoherence strength are discussed.
Motivated by the recent interest in thermodynamics of micro- and mesoscopic quantum systems we st... more Motivated by the recent interest in thermodynamics of micro- and mesoscopic quantum systems we study the maximal amount of work that can be reversibly extracted from a quantum system used to store temporarily energy. Guided by the notion of passivity of a quantum state we show that entangling unitary controls extract in general more work than independent ones. In the limit of large number of copies one can reach the thermodynamical bound given by the variational principle for free energy.
We prove a no-go theorem for storing quantum information in equilibrium systems. Namely, quantum ... more We prove a no-go theorem for storing quantum information in equilibrium systems. Namely, quantum information cannot be stored in a system with time-independent Hamiltonian interacting with heat bath of temperature $T>0$ during time that grows with the number of used qubits. We prove it by showing, that storing quantum information for macroscopic time would imply existence of perpetuum mobile of the second kind. The general results are illustrated by the Kitaev model of quantum memory. In contrast, classical information can be stored in equilibrium states for arbitrary long times. We show how it is possible via phase-transition type phenomena. Our result shows that there is a fundamental difference between quantum and classical information in {\it physical} terms.
We analyse stability of the four-dimensional Kitaev model - a candidate for scalable quantum memo... more We analyse stability of the four-dimensional Kitaev model - a candidate for scalable quantum memory - in finite temperature within the weak coupling Markovian limit. It is shown that, below a critical temperature, certain topological qubit observables X and Z possess relaxation times exponentially long in the size of the system. Their construction involves polynomial in system's size algorithm which uses as an input the results of measurements performed on all individual spins. We also discuss the drawbacks of such candidate for quantum memory and mention the implications of the stability of qubit for statistical mechanics.
We develop a dynamical non-Markovian description of quantum computing in the weak-coupling limit,... more We develop a dynamical non-Markovian description of quantum computing in the weak-coupling limit, in the lowest-order approximation. We show that the long-range memory of the quantum reservoir (such as the 1/t4 one exhibited by electromagnetic vacuum) produces a strong interrelation between the structure of noise and the quantum algorithm, implying nonlocal attacks of noise. This shows that the implicit assumption of quantum error correction theory-independence of noise and self-dynamics-fails in long time regimes. We also use our approach to present pure decoherence and decoherence accompanied by dissipation in terms of the spectral density of the reservoir. The so-called dynamical decoupling method is discussed in this context. Finally, we propose a minimal decoherence model, in which the only source of decoherence is vacuum. We optimize the fidelity of quantum-information processing under the trade-off between the speed of the gate and the strength of decoherence.
There exists a large number of experimental and theoretical results supporting the picture of mac... more There exists a large number of experimental and theoretical results supporting the picture of macroscopic qubits implemented by nanoscopic Josephson junctions of three different types -- charge qubit, flux qubit and phase qubit. The standard unified description of such systems is based on the formal quantization of the phenomenological Kirchhoff equations for the corresponding circuits. In this paper a simplified version of the BCS theory for superconductors is used to derive microscopic models for all types of small Josephson junctions. For these models the state-dependent individual tunneling of Cooper pairs couples ground pair states with excited pair states what leads to a more complicated structure of the lowest lying energy levels. In particular, the highly degenerate levels emerge, which act as probability sinks for the qubit. These models allow also for the coupling to phonons as an efficient mechanism of relaxation for all types of junctions. The alternative formulas concerning basic spectral parameters of superconducting qubits are presented and compared with the experimental data. Finally, the question whether small Josephson junctions can be treated as macroscopic quantum systems is briefly discussed.
We consider, within the algebraic formalism, the time dependence of fidelity for qubits encoded i... more We consider, within the algebraic formalism, the time dependence of fidelity for qubits encoded in an open physical system. We relate the decay of fidelity to the evolution of correlation functions and, in the particular case of a Markovian dynamics, to the spectral gap of the generator of the semigroup. The results are applicable to the analysis of models of quantum memories.
The purity, Tr( ρ2), measures how pure or mixed a quantum state ρ is. It is well known that quant... more The purity, Tr( ρ2), measures how pure or mixed a quantum state ρ is. It is well known that quantum dynamical semigroups that preserve the identity operator (which we refer to as unital) are strictly purity-decreasing transformations. Here, we provide an almost complete characterization of the class of strictly purity-decreasing quantum dynamical semigroups. We show that in the case of finite-dimensional Hilbert spaces, a dynamical semigroup is strictly purity-decreasing if and only if it is unital, while in the infinite dimensional case, unitality is only sufficient.
We revisit the notion of Kolmogorov-Sinai entropy for classical dynamical systems in terms of an ... more We revisit the notion of Kolmogorov-Sinai entropy for classical dynamical systems in terms of an algebraic formalism. This is the starting point for defining the entropy for general non-commutative systems. Hereby typical quantum tools are introduced in the statistical description of classical dynamical systems. We illustrate the power of these techniques by providing a simple, self-contained proof of the entropy formula for general automorphisms of n-dimensional tori.
The rate of temperature decrease of a cooled quantum bath is studied as its temperature is reduce... more The rate of temperature decrease of a cooled quantum bath is studied as its temperature is reduced to the absolute zero. The III-law of thermodynamics is then quantified dynamically by evaluating the characteristic exponent {\zeta} of the cooling process dT(t)/dt \sim -T^{\zeta} when approaching the absolute zero, T \rightarrow 0. A continuous model of a quantum refrigerator is employed consisting of a working medium composed either by two coupled harmonic oscillators or two coupled 2-level systems. The refrigerator is a nonlinear device merging three currents from three heat baths: a cold bath to be cooled, a hot bath as an entropy sink, and a driving bath which is the source of cooling power. A heat driven refrigerator (absorption refrigerator) is compared to a power driven refrigerator. When optimized both cases lead to the same exponent {\zeta}, showing a lack of dependence on the form of the working medium and the characteristics of the drivers. The characteristic exponent is therefore determined by the properties of the cold reservoir and its interaction with the system. Two generic heat baths models are considered, a bath composed of harmonic oscillators and a bath composed from ideal Bose/Fermi gas. The restrictions on the interaction Hamiltonian imposed by the III-law are discussed. In the appendix the theory of periodicaly driven open systems and its implication to thermodynamics is outlined.
Lecture Notes in Physics Berlin Springer Verlag, 2007
... to a member of the Editorial Board, or directly to the managing editor at Springer: Dr. Chris... more ... to a member of the Editorial Board, or directly to the managing editor at Springer: Dr. Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg/Germany christian. caron@ springer. com Page 4. Robert Alicki Karl Lendi Quantum ...
We prove continuity of quantum conditional information $S(\rho^{12}| \rho^2)$ with respect to the... more We prove continuity of quantum conditional information $S(\rho^{12}| \rho^2)$ with respect to the uniform convergence of states and obtain a bound which is independent of the dimension of the second party. This can, e.g., be used to prove the continuity of squashed entanglement.
This paper consists in two parts. First we set up a general scheme of local traps in an homogeneo... more This paper consists in two parts. First we set up a general scheme of local traps in an homogeneous deterministic quantum system. The current of particles caught by the trap is linked to the dynamical behaviour of the trap states. In this way, transport properties in an homogeneous system are related to spectral properties of a coherent dynamics. Next we apply the scheme to a system of Fermions in the one-particle approximation. We obtain in particular lower bounds for the dynamical entropy in terms of the current induced by the trap.
The Wigner-Weisskopf-type model developed by Alicki and Giraldi [J. Phys. B 44, 154020 (2011)], 1... more The Wigner-Weisskopf-type model developed by Alicki and Giraldi [J. Phys. B 44, 154020 (2011)], 10.1088/0953-4075/44/15/154020 is applied to the biological process of energy transfer from a large peripheral light harvesting antenna to the reaction center. This process is mediated by the Fenna-Matthews-Olson (FMO) photosynthetic complex with a remarkably high efficiency. The proposed model provides a simple resonance mechanism of this phenomenon employing exciton coherent motion and is described by analytical formulas. A coupling to the vibrational environment is a necessary component of this mechanism as well as a fine-tuning of the FMO complex Hamiltonian. The role of the relatively strong coupling to the energy sink in achieving the resonance condition and the absence of heating of the vibrational environment are emphasized.
The authors propose to describe the dynamics of unstable particles in relativistic quantum field ... more The authors propose to describe the dynamics of unstable particles in relativistic quantum field theory in terms of semigroups of transformations of the observables. This leads, in contrast to the usual Hilbert space level treatment, to a complete and consistent descriptions of the irreversible dynamics of decay processes. The scheme is explicitly worked out for the massive scalar quantum field and the evolution of the particle density and its higher moments is computed.
We comment that the model proposed in Phys. Rev. Lett. 108, 120603 (2012) violates the dynamical ... more We comment that the model proposed in Phys. Rev. Lett. 108, 120603 (2012) violates the dynamical version of the third law of thermodynamics. We discuses the different formulations of the third law of thermodynamics and suggest a possible reason for the violation.
It is often claimed, that from a quantum system of d levels, and entropy S and heat bath of tempe... more It is often claimed, that from a quantum system of d levels, and entropy S and heat bath of temperature T one can draw kT (ln d − S) amount of work. However, the usual arguments basing on Szilard engine, are not fully rigorous. Here we prove the formula within Hamiltonian description of drawing work from a quantum system and a heat bath, at the cost of entropy of the system. We base on the derivation of thermodynamical laws and quantities in [R. Alicki, J. Phys. A, 12, L103 (1979)] within weak coupling limit. Our result provides fully physical scenario for extracting thermodynamical work form quantum correlations [Oppenheim et al. Phys. Rev. Lett. 89, 180402 (2002)]. We also derive Landauer principle as a consequence of second law within the considered model.
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