We study a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a finite-dimensional vector space. These states can be considered as generalized valence bond states, and they are dense in the set of all translation invariant states. We develop a complete theory of the ergodic decomposition of such states, including the decomposition into periodic "Neel ordered" states. The ergodic components have exponential decay of correlations. All states considered can be obtained as "local functions" of states of a special kind, so-called "purely generated states," which are shown to be ground states for suitably chosen finite range VBS interactions. We show that all these generalized VBS models have a spectral gap. Our theory does not require symmetry of the state with respect to a local gauge group. In particular we illustrate our results with a one-parameter family of examples which are not isotropic except for one special case. This isotropic model coincides with the one-dimensional antiferromagnet, recently studied by Affleck, Kennedy, Lieb, and Tasaki.
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 .
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