Quantum Optical Realization of Arbitrary Linear Transformations Allowing for Loss and Gain

N Tischler,C Rockstuhl,K Słowik

doi:10.1103/physrevx.8.021017

Abstract

Unitary transformations are routinely modeled and implemented in the field of quantum optics. In contrast, nonunitary transformations that can involve loss and gain require a different approach. In this theory work, we present a universal method to deal with nonunitary networks. An input to the method is an arbitrary linear transformation matrix of optical modes that does not need to adhere to bosonic commutation relations. The method constructs a transformation that includes the network of interest and accounts for full quantum optical effects related to loss and gain. Furthermore, through a decomposition in terms of simple building blocks it provides a step-by-step implementation recipe, in a manner similar to the decomposition by Reck et al. [Reck et al., Phys. Rev. Lett. 73, 58 (1994)] but applicable to nonunitary transformations. Applications of the method include the implementation of positive-operator-valued measures and the design of probabilistic optical quantum information protocols.

Highlights

Transformations between sets of orthogonal input and output modes are ubiquitous in optics and quantum information technology
On, we drop the expectation values and take T to be a transformation between the annihilation operators of interest, with the understanding that it is an incomplete transformation: Ancilla modes need to be introduced in the mathematical description to faithfully reproduce or predict the full quantum optical transformation. This is straightforward for the simple examples of Y junctions and polarizers, a systematic method to deal with larger-scale problems would be desirable. We investigate whether such a strategy is possible for all linear transformation matrices, how many ancilla modes are needed for any given case, and how a full enlarged quantum optical network can be mathematically represented and physically realized
Each operation here corresponds to a singular value, and each singular value that is different from 1 results in the interaction of a nominal mode with a vacuum ancilla, either through a beam splitter or a parametric amplifier

Summary

Introduction

Transformations between sets of orthogonal input and output modes are ubiquitous in optics and quantum information technology. Linear transformations between the amplitudes of the input and output modes are used to perform a variety of tasks, e.g., to operate single-qubit gates or to model the action of physical elements such as beam splitters [1].

Methods

Results

Discussion

Conclusion