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arXiv:quant-ph/0501112 · Physical Review A

Continuous-variable Gaussian analog of cluster states

Jing Zhang, Samuel L. Braunstein

Published Mar 16, 2006• 310 citations
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Abstract

We present a continuous-variable (CV) Gaussian analog of cluster states, a new class of CV multipartite entangled states that can be generated from squeezing and quantum nondemolition coupling ${H}_{I}=\ensuremath{\hbar}\ensuremath{\chi}{X}_{A}{X}_{B}$. The entanglement properties of these states are studied in terms of classical communication and local operations. The graph states as general forms of the cluster states are presented. A chain for a one-dimensional example of cluster states can be readily experimentally produced only with squeezed light and beam splitters.

DOI
10.1103/physreva.73.032318

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We present a continuous-variable (CV) Gaussian analog of cluster states, a new class of CV multipartite entangled states that can be generated from squeezing and quantum nondemolition coupling HI=\ensuremathℏ\ensuremathχXAXB{H}_{I}=\ensuremath{\hbar}\ensuremath{\chi}{X}_{A}{X}_{B}HI​=\ensuremathℏ\ensuremathχXA​XB​. The entanglement properties of these states are studied in terms of classical communication and local operations. The graph states as general forms of the cluster states are presented. A chain for a one-dimensional example of cluster states can be readily experimentally produced only with squeezed light and beam splitters.
Continuous-variable (CV) cluster states are a universal quantum computing platform that has experimentally out-scaled qubit platforms by orders of magnitude. The implementation of CV cluster states uses multimode squeezed Gaussian states and fault tolerance thresholds for CV quantum computing have been predicted for realistic squeezing. An outstanding question has been whether any Gaussian state can be expressed as a cluster state. We show here that Gaussian states can present hidden entanglement, which is not captured by the corresponding cluster state graph and therefore invalidates this graph. Hidden entanglement vanishes in the infinite squeezing limit, which should therefore be used with extreme caution. We derive a general procedure to detect hidden entanglement in an arbitrary Gaussian state and validate it as a CV cluster state. These results are relevant to all experimental implementations of CV cluster states to date.
We provide a unified graphical calculus for all Gaussian pure states, including graph transformation rules for all local and semilocal Gaussian unitary operations, as well as local quadrature measurements. We then use this graphical calculus to analyze continuous-variable (CV) cluster states, the essential resource for one-way quantum computing with CV systems. Current graphical approaches to CV cluster states are only valid in the unphysical limit of infinite squeezing, and the associated graph transformation rules only apply when the initial and final states are of this form. Our formalism applies to all Gaussian pure states and subsumes these rules in a natural way. In addition, the term ``CV graph state'' currently has several inequivalent definitions in use. Using this formalism we provide a single unifying definition that encompasses all of them. We provide many examples of how the formalism may be used in the context of CV cluster states: defining the ``closest'' CV cluster state to a given Gaussian pure state and quantifying the error in the approximation due to finite squeezing; analyzing the optimality of certain methods of generating CV cluster states; drawing connections between this graphical formalism and bosonic Hamiltonians with Gaussian ground states, including those useful for CV one-way quantum computing; and deriving a graphical measure of bipartite entanglement for certain classes of CV cluster states. We mention other possible applications of this formalism and conclude with a brief note on fault tolerance in CV one-way quantum computing.
Continuous-variable Gaussian cluster states are a potential resource for universal quantum computation. They can be efficiently and unconditionally built from sources of squeezed light using beam splitters. Here we report on the generation of three different kinds of continuous-variable four-mode cluster states. In our realization, the resulting cluster-type correlations are such that no corrections other than simple phase-space displacements would be needed when quantum information propagates through these states. At the same time, the inevitable imperfections from the finitely squeezed resource states and from additional thermal noise are minimized, as no antisqueezing components are left in the cluster states.
We investigate continuous-variable (CV) multipartite unlockable bound entangled states. Comparing with the qubit multipartite unlockable bound entangled states, CV multipartite unlockable bound entangled states present some new and different properties. CV multipartite unlockable bound entangled states may serve as a useful quantum resource for new multiparty communication schemes. The experimental protocol for generating CV unlockable bound entangled states is proposed with a setup that is, at present, accessible.
Continuous-variable (CV) cluster states are a universal quantum computing platform that has experimentally outscaled qubit platforms by orders of magnitude.Room-temperature implementation of CV cluster states has been achieved with quantum optics by using multimode squeezed Gaussian states.It has also been proven that fault tolerance thresholds for CV quantum computing can be reached at realistic squeezing levels.In this paper, we show that standard approaches to design and characterize CV cluster states can miss entanglement present in the system.Such hidden entanglement may be used to increase the power of a quantum computer but it can also, if undetected, hinder the successful implementation of a quantum algorithm.By a detailed analysis of the structure of Gaussian states, we derive an algorithm that reveals hidden entanglement in an arbitrary Gaussian state and optimizes its use for one-way quantum computing.
The quantum fusion technique enables the bonding of two cluster states together, and therefore provides a promising method to grow large-scale cluster states. Quantum fusion was first proposed in a discrete-variable regime. In terms of the resource usage, quantum fusion can be divided into two categories, which are type-I and type-II fusions. Type-I fusion consumes one qubit in the fusion procedure, while type-II consumes two qubits. However, for continuous-variable (CV) analogs, concrete schemes for realizing quantum fusions of cluster states with arbitrary topologies have not been shown yet. In this paper, we present the CV analog of quantum fusion schemes. We show that CV type-I fusion can be realized by an ancilla and two quantum nondemolition interactions, and CV type-II fusion can be realized by an entanglement swapping scheme. With finite squeezing considered, we analyze the extra errors added by the fusion procedures in CV cluster-state quantum computation. Our schemes are universal and can be generalized to fuse cluster states with arbitrary topologies.