One-way coupling often occurs in multi-dimensional models in finance. In this paper, we present a dimension reduction technique for Monte Carlo (MC) methods, referred to as drMC, that exploits this structure for pricing plain-vanilla European options under an N-dimensional one-way coupled model, where N is arbitrary. The dimension reduction also often produces a significant variance reduction.The drMC method is a dimension reduction technique built upon (i) the conditional MC technique applied to one of the factors which does not depend on any other factors in the model, and (ii) the derivation of a closed-form solution to the conditional partial differential equation (PDE) that arises via Fourier transforms. In the drMC approach, the option price can be computed simply by taking the expectation of this closed-form solution. Hence, the approach results in a powerful dimension reduction from N to one, which often results in a significant variance reduction as well, since the variance associated with the other (N−1) factors in the original model are completely removed from the drMC simulation. Moreover, under the drMC framework, hedging parameters, or Greeks, can be computed in a much more efficient way than in traditional MC techniques. A variance reduction analysis of the method is presented and numerical results illustrating the method’s efficiency are provided.

In this article, we illustrate the scaling properties of a family of solutions for N attractive bosonic atoms in the limit of large N. These solutions represent the quantized dynamics of solitonic degrees of freedom in atomic droplets. In dimensions lower than two, or d = 2 - epsilon, we demonstrate that the number of isotropic droplet states scales as N-3/2/epsilon(1/2), and for epsilon = 0, or d = 2, scales as N-2. The ground-state energies scale as N2/epsilon+1 in d = 2 - epsilon, and when d = 2, scale as an exponential function of N. We obtain the universal energy spectra and the generalized Tjon relation; their scaling properties are uniquely determined by the asymptotic freedom of quantum bosonic fields at short distances, a distinct feature in low dimensions. We also investigate the effect of quantum loop corrections that arise from various virtual processes and show that the resultant lifetime for a wide range of excited states scales as (NE1-epsilon/2)-E-epsilon/2.

In the absence of experimental constraints, optimal measurement schemes for quantum state tomography are well understood. We consider the scenario where the experimenter doesn't have arbitrary freedom to construct their measurement set, and may therefore not be able to implement a known optimal scheme. We introduce a simple procedure for minimizing the uncertainty in the reconstructed quantum state for an arbitrary tomographic scheme. We do this by defining a figure of merit based on the equally weighted variance of the measurement statistics. This figure of merit is straightforwardly based on the singular value decomposition of the measurement matrix, making it well suited for optimization.

We propose a technique for performing quantum state tomography of photonic polarization-encoded multiqubit states. Our method uses a single rotating wave plate, a polarizing beam splitter, and two photon-counting detectors per photon mode. As the wave plate rotates, the photon counters measure a pseudocontinuous signal which is then Fourier transformed. The density matrix of the state is reconstructed using the relationship between the Fourier coefficients of the signal and the Stokes' parameters that represent the state. The experimental complexity, i.e., different wave plate rotation frequencies, scales linearly with the number of qubits.