Recent zbMATH articles in MSC 30H20https://zbmath.org/atom/cc/30H202021-05-28T16:06:00+00:00WerkzeugThe resolvent of the Nelson Hamiltonian improves positivity.https://zbmath.org/1459.810742021-05-28T16:06:00+00:00"Lampart, Jonas"https://zbmath.org/authors/?q=ai:lampart.jonasSummary: We give a new proof that the resolvent of the renormalised Nelson Hamiltonian at fixed total momentum \(P\) improves positivity in the (momentum) Fock-representation, for every \(P\). The argument is based on an explicit representation of the renormalised operator and its domain using interior boundary conditions, which allows us to avoid the intermediate steps of regularisation and renormalisation used in other proofs of this result.A common parametrization for finite mode Gaussian states, their symmetries, and associated contractions with some applications.https://zbmath.org/1459.811412021-05-28T16:06:00+00:00"John, Tiju Cherian"https://zbmath.org/authors/?q=ai:john.tiju-cherian"Parthasarathy, K. R."https://zbmath.org/authors/?q=ai:parthasarathy.k-r|parthasarathy.kalyanapuram-rangachariSummary: Let \({\Gamma}(\mathcal{H})\) be the boson Fock space over a finite dimensional Hilbert space \(\mathcal{H} \). It is shown that every Gaussian symmetry admits a Klauder-Bargmann integral representation in terms of coherent states. Furthermore, Gaussian states, Gaussian symmetries, and second quantization contractions belong to a weakly closed self-adjoint semigroup \(\mathcal{E}_2(\mathcal{H})\) of bounded operators in \({\Gamma}(\mathcal{H})\). This yields a common parametrization for these operators. It is shown that the new parametrization for Gaussian states is a fruitful alternative to the customary parametrization by position-momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries: (i) every Gaussian state \(\rho\) admits a factorization \(\rho = Z_1^\dagger Z_1\), where \(Z_1\) is an element of \(\mathcal{E}_2(\mathcal{H})\) and has the form \(Z_1 = \sqrt{c} {\Gamma}(P) \exp \left\{\sum_{r = 1}^n \lambda_r a_r + \sum_{r, s = 1}^n \alpha_{r s} a_r a_s\right\}\) on the dense linear manifold generated by all exponential vectors, where \(c\) is a positive scalar, \( \Gamma (P)\) is the second quantization of a positive contractive operator \(P\) in \(\mathcal{H}, a_{r}, 1 \leq r \leq n\), are the annihilation operators corresponding to the \(n\) different modes in \({\Gamma}(\mathcal{H}), \lambda_r \in \mathbb{C} \), and \([ \alpha_{rs} ]\) is a symmetric matrix in \(M_n(\mathbb{C})\); (ii) an explicit particle basis expansion of an arbitrary mean zero pure Gaussian state vector along with a density matrix formula for a general Gaussian state in terms of its \(\mathcal{E}_2(\mathcal{H})\)-parameters; (iii) a class of examples of pure \(n\)-mode Gaussian states that are completely entangled; (iv) tomography of an unknown Gaussian state in \({\Gamma}(\mathbb{C}^n)\) by the estimation of its \(\mathcal{E}_2(\mathbb{C}^n)\) parameters using \(O(n^2)\) measurements with a finite number of outcomes.
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