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Bulletin of the American Physical Society

A General Linear-Optical Quantum State Generator

Dmitry B. Uskov, Nickolas M. VanMeter, Pavel Lougovski, Jonathan P. Dowling

Published Jun 8, 2007• 0 citations
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We introduce schemes for linear-optical quantum state generation. A quantum state generator is a device that prepares a desired quantum state using product inputs from photon sources, linear-optical networks, and postselection using photon counters. We show that this device can be concisely described in terms of polynomial equations and unitary constraints. We illustrate the power of this language by applying the Gr\"obner-basis technique along with the notion of vacuum extensions to solve the problem of how to construct a quantum state generator analytically for any desired state, and use methods of convex optimization to identify bounds to success probabilities. In particular, we disprove a conjecture concerning the preparation of the maximally path-entangled $\ensuremath{\mid}n,0⟩+\ensuremath{\mid}0,n⟩$ (NOON) state by providing a counterexample using these methods, and we derive a new upper bound on the resources required for NOON-state generation.
We present a scheme to generate the polarization-entangled two-photon state 1/2(∣H〉∣V〉+∣V〉∣H〉),1/\sqrt{2}(|H〉|V〉+|V〉|H〉),1/2​(∣H〉∣V〉+∣V〉∣H〉), which is of much interest in the field of quantum information processing. Furthermore, we demonstrate the capability of this concept in respect of a generalization to entangle N-photon states for interferometry and lithography. This scheme requires single-photon sources, linear optical elements, and a multifold coincidence detection.
A major challenge in photonic quantum technologies is developing strategies to prepare suitable discrete-variable quantum states using simple input states, linear optics, and auxiliary photon measurements to identify successful outcomes. Fundamentally, this challenge arises from the lack of strong non-linearities on the single-photon level, meaning that photonic state preparation based on linear optics cannot benefit from the deterministic gate-based approach available to other physical platforms. Instead, the preparation of quantum states can be probabilistically implemented using single photons, linear-optical networks, and photon detection. However, determining whether an input state can be transformed into a target state using a specific measurement pattern - a problem that can be mapped to deciding the feasibility of a system of polynomial equations - is a complex problem in general. To solve it, we apply the Nullstellensatz Linear Algebra algorithm from algebraic geometry to quantum state generation; this can provide definitive no-go results by proving infeasibility when the state preparation task in question has no solution. We demonstrate this capability to validate and establish lower bounds on the physical resource requirements for the realization of several ubiquitous optical states and gates.