Quantum Statistics of Optical Parameteric Oscillation

R. Graham

Barry C. Sanders, Peter Høyer, Mark Adcock

Quantum information processing is appealing due its ability to solve certain problems quantitatively faster than classical information processing. Most quantum algorithms have been studied in discretely parameterized systems, but many quantum systems are continuously parameterized. The field of quantum optics in particular has sophisticated techniques for manipulating continuously parameterized quantum states of light, but the lack of a code-state formalism has hindered the study of quantum algorithms in these systems. To address this situation, a code-state formalism for the solution of oracle decision problems in continuously-parameterized quantum systems is developed. Quantum information processing is appealing due its ability to solve certain problems quantitatively faster than classical information processing. Most quantum algorithms have been studied in discretely parameterized systems, but many quantum systems are continuously parameterized. The field of quantum optics in particular has sophisticated techniques for manipulating continuously parameterized quantum states of light, but the lack of a code-state formalism has hindered the study of quantum algorithms in these systems. To address this situation, a code-state formalism for the solution of oracle decision problems in continuously-parameterized quantum systems is developed. In the infinite-dimensional case, we study continuous-variable quantum algorithms for the solution of the Deutsch–Jozsa oracle decision problem implemented within a single harmonic-oscillator. Orthogonal states are used as the computational bases, and we show that, contrary to a previous claim in the literature, this implementation of quantum information processing has limitations due to a position-momentum trade-off of the Fourier transform. We further demonstrate that orthogonal encoding bases are not unique, and using the coherent states of the harmonic oscillator as the computational bases, our formalism enables quantifying the relative performances of different choices of the encoding bases. We extend our formalism to include quantum algorithms in the continuously parameterized yet finite-dimensional Hilbert space of a coherent spin system. We show that the highest-squeezed spin state possible can be approximated by a superposition of two states thus transcending the usual model of using a single basis state as algorithm input. As a particular example, we show that the close Hadamard oracle-decision problem, which is related to the Hadamard codewords of digital communications theory, can be solved quantitatively more efficiently using this computational model than by any known classical algorithm.

R. Loudon

Jan Peřina

Aron Pasieka

Quantum state estimation is an important task in the realm of experimental quantum information science. While the common approaches to this task are generally adequate, the maximum-likelihood method is frequently used with parameterizations that do not strictly satisfy the statistical requirements of maximum-likelihood estimation. We show that the source of this issue is in the structure of the parameter space and we introduce a density matrix parameterization that provides direct control over this structure. The parameterization leads to a natural quantum state estimation algorithm that does satisfy the statistical requirements of maximum-likelihood estimation. Finally, we examine the algorithm through the analysis of several data sets and experimental results.

Frank A. Moscatelli