Interest is in evaluating, by Markov chain Monte Carlo (MCMC) simulation, the expected value of a function with respect to a, possibly unnormalized, probability distribution. A general purpose variance reduction technique for the MCMC estimator, based on the zero-variance principle introduced in the physics literature, is proposed. Conditions for asymptotic unbiasedness of the zero-variance estimator are derived. A central limit theorem is also proved under regularity conditions. The potential of the idea is illustrated with real applications to probit, logit and GARCH Bayesian models. For all these models, a central limit theorem and unbiasedness for the zero-variance estimator are proved (see the supplementary material available on-line).

Heisenberg and Schrödinger uncertainty principles give lower bounds for the product of variances 2000 Mathematics Subject Classification. Primary 62B10, 94A17; Secondary 46L30, 46L60.

An inequality relating the Wigner-Yanase information and the SLD-quantum Fisher information was established by Luo (Proc. Amer. Math. Soc., 132, pp. 885-890, 2004). In this paper, we generalize Luo's inequality to any regular quantum Fisher information. Moreover, we show that this general inequality can be derived from the Kubo-Ando inequality, which states that any matrix mean is greater than the harmonic mean and smaller than the arithmetic mean

Let A1,...,A(N) be complex self-adjoint matrices and let rho be a density matrix. The Robertson uncertainty principle det{Cov rho(A(h), A(j))} >= det{ -1/2Tr(rho[A(h), A(j)])} gives a bound for the quantum generalized variance in terms of the commutators [A(h), A(j)]. The right side matrix is antisymmetric and therefore the bound is trivial (equal to zero) in the odd case N = 2m + 1. Let f be an arbitrary normalized symmetric operator monotone function and let (.,.)(rho,f) be the associated quantum Fisher information. We have conjectured the inequality det{Cov rho(A(h), A(j))} >= det {f(0)/2 < i[rho, A(h)], i[rho, Aj]>rho,f} that gives a non-trivial bound for any N is an element of N using the commutators [rho, A(h)]. In the present paper the conjecture is proved by mean of the Kubo-Ando mean inequalit

Let A1, ..., AN be complex self-adjoint matrices and let ρ be a density matrix. The Robertson uncertainty principleff gives a bound for the quantum generalized covariance in terms of the commutators [A h , Aj]. The right side matrix is antisymmetric and therefore the bound is trivial (equal to zero) in the odd caseLet f be an arbitrary normalized symmetric operator monotone function and let •, • ρ,f be the associated quantum Fisher information. In this paper we conjecture the inequalityff that gives a non-trivial bound for any N ∈ N using the commutators i[ρ, A h ]. The inequality has been proved in the cases N = 1, 2 by the joint efforts of many authors (see the Introduction). In this paper we prove the (real) case N = 3.

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