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arXiv:quant-ph/0612154 · Physical Review A

General linear-optical quantum state generation scheme: Applications to maximally path-entangled states

Nickolas M. VanMeter, Pavel Lougovski, Dmitry B. Uskov, Konrad Kieling, Jens Eisert, Jonathan P. Dowling

Published Dec 12, 2007• 41 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.76.063808

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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∣n,0⟩+\ensuremath∣0,n⟩\ensuremath{\mid}n,0⟩+\ensuremath{\mid}0,n⟩\ensuremath∣n,0⟩+\ensuremath∣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.
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.
We propose a novel linear optical scheme to generate a three-photon W state, which is based on postselection of final outcome events. The success probability of our proposal has been improved to be 21/32, higher than that of the previous methods using the linear optical system. We also discuss the effects of realistic photon resources on its experimental realization and compare our scheme with the nonlinear optical approach to the three-photon W state generation. The idea of our proposal suggests a new way to improve the generation efficiency of some other entangled states and may have a promising application.
Multipartite entanglement is an essential quantum resource for various distributed quantum applications. One promising method for preparing multipartite entanglement is to interfere independent photons at linear optical interference setups. While heralding the successful interference and thereby the state generation is often costly, postselecting entangled states provides an achievable alternative in this framework. We introduce a family of interference schemes for postselecting symmetric qudit Dicke states, useful resources in quantum communication and variational quantum computing. We present schemes with and without ancillary photons and show that using ancillary photons can exceed the upper bound on the success probability of schemes without ancillary photons. Our results accommodate a wide range of linear optical schemes, providing multiple viable approaches for postselecting Dicke states.
Many applications of linear optics in quantum science involve searching for some optimal configuration of a multiport interferometer. For example, to find heralded state preparations, one can optimize the interferometer to maximize the fidelity and success probability of the state preparation. In this paper, we derive a closed analytical formula for the Riemannian gradient of any differentiable function defined over the interferometer unitaries. We use this gradient in Riemannian optimization algorithms, such as gradient descent or BFGS, to find the optimal setup of the interferometer. As an interesting use case, we search for linear optical state preparations, improving some of the success probabilities reported in the literature for NOON or photon catalysis preparations. Finally, we show that our optimizer is orders of magnitude faster than other methods in the literature, opening the door for solving linear optical problems involving more modes and photons.
Entanglement is the basic building block of linear optical quantum computation, and as such understanding how to generate it in detail is of great importance for optical architectures. We prove that Bell states cannot be generated using only three photons in the dual-rail encoding, and give strong numerical evidence for the optimality of the existing four photon schemes. In a setup with a single photon in each input mode, we find a fundamental limit on the possible entanglement between a single mode Alice and arbitrary Bob. We investigate and compare other setups aimed at characterizing entanglement in settings more general than dual-rail encoding. The results draw attention to the trade-off between the entanglement a state has and the probability of postselecting that state, which can give surprising constant bounds on entanglement even with increasing numbers of photons.