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We describe a scalable stochastic method for the experimental measurement of generalized fidelities characterizing the accuracy of the implementation of a coherent quantum transformation. The method is based on the motion reversal of random unitary operators. In the simplest case our method enables direct estimation of the average gate fidelity. The more general fidelities are characterized by a universal exponential rate of fidelity loss. In all cases the measurable fidelity decrease is directly related to the strength of the noise affecting the implementation -quantified by the trace of the superoperator describing the non-unitary dynamics. While the scalability of our stochastic protocol makes it most relevant in large Hilbert spaces (when quantum process tomography is infeasible), our method should be immediately useful for evaluating the degree of control that is achievable in any prototype quantum processing device. By varying over different experimental arrangements and error-correction strategies additional information about the noise can be determined.

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 temporarily store 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 a large number of copies one can reach the thermodynamical bound given by the variational principle for the free energy.

Introduction 1 2 Basic tools for quantum mechanics 4 2.1 Hilbert spaces and operators 5 2.1.1 Vector spaces 5 2.1.2 Banach and Hilbert spaces 7 2.1.3 Geometrical properties of Hilbert spaces 9 2.1.4 Orthonormal bases 10 2.1.5 Subspaces and projectors-11 2.1.6 Linear maps between Banach spaces 13 2.1.7 Linear functionals and Dirac notation 14 2.1.8 Adjoints of bounded operators 18 2.1.9 Hermitian, unitary and normal operators 19 2.1.10 Partial isometries and polar decomposition 21 2.1.11 Spectra of operators 22 2.1.12 Unbounded operators 24 2.2 Measures 25 2.2.1 Measures and integration 25 2.2.2 Distributions 29 2.2.3 Hilbert spaces of functions 30 2.2.4 Spectral measures 31 2.3 Probability in quantum mechanics '34 2.3.1 Pure states. .35 2.3.2 Mixed states, density matrices 37 2.4 Observables in quantum mechanics 38 2.4.1 Compact operators 38 2.4.2 Weyl quantization 40 2.5 Composed systems 44 2.5.1 Direct sums 45 2.5.2 Tensor products 46 2.5.3 Observables and states of composite systems 48

We prove continuity of quantum conditional information S(ρ 12 | ρ 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.A, generally mixed, state of a bipartite system is given by a density matrix ρ 12 on a Hilbert space H 12 = H 1 ⊗ H 2 . We shall, in order to avoid technical complications, restrict our attention to finite dimensional systems and not distinguish between the density matrix ρ 12 and its associated expectation functional a → ρ 12 (a) := Tr ρ 12 a, a a linear operator on H 12 .

In traditional thermodynamics the Carnot cycle yields the ideal performance bound of heat engines and refrigerators. We propose and analyze a minimal model of a heat machine that can play a similar role in quantum regimes. The minimal model consists of a single two-level system with periodically modulated energy splitting that is permanently, weakly, coupled to two spectrally separated heat baths at different temperatures. The equation of motion allows us to compute the stationary power and heat currents in the machine consistent with the second law of thermodynamics. This dual-purpose machine can act as either an engine or a refrigerator (heat pump) depending on the modulation rate. In both modes of operation, the maximal Carnot efficiency is reached at zero power. We study the conditions for finite-time optimal performance for several variants of the model. Possible realizations of the model are discussed.

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 ζ of the cooling process dT (t) dt ∼ −T ζ when approaching the absolute zero, T → 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 ζ, 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.