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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