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arXiv:1006.3974 · Physical Review A

Graphical rule of transforming continuous-variable graph states by local homodyne detection

Jing Zhang

Published Sep 17, 2010• 7 citations
arXivHTMLPDF

Abstract

Graphical rule, describing that any single-mode homodyne detection turns a given continuous-variable (CV) graph state into a new one, is presented. Employing two simple graphical rules---local complement operation and vertex deletion (single quadrature-amplitude $\mathrm{x\ifmmode \hat{}\else \^{}\fi{}}$ measurement)---the graphical rule for any single-mode quadrature component measurement can be obtained. The shape of CV weighted graph state may be designed and constructed easily from a given larger graph state by applying this graphical rule.

DOI
10.1103/physreva.82.034303

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Graphical rule, describing that any single-mode homodyne detection turns a given continuous-variable (CV) graph state into a new one, is presented. Employing two simple graphical rules---local complement operation and vertex deletion (single quadrature-amplitude x\ifmmode^\elseˆ\fi\mathrm{x\ifmmode \hat{}\else \^{}\fi{}}x\ifmmode^\elseˆ\fi measurement)---the graphical rule for any single-mode quadrature component measurement can be obtained. The shape of CV weighted graph state may be designed and constructed easily from a given larger graph state by applying this graphical rule.
The local complementation rule is applied for continuous-variable (CV) graph states in the paper, which is an elementary graph transformation rule and successive application of which generates the orbit of any graph states. The corresponding local Gaussian transformations of local complementation for four-mode unweighted graph states were found, which do not mirror the form of the local Clifford unitary of qubit exactly. This work is an important step to characterize the local Gaussian equivalence classes of CV graph states.
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.
The form of a local Clifford [LC, also called local Gaussian (LG)] operation for the continuous-variable (CV) weighted graph states is presented in this paper, which is the counterpart of the LC operation of local complementation for qubit graph states. The property of the CV weighted graph states is shown, which can be expressed by the stabilizer formalism. It is distinctively different from the qubit weighted graph states, which cannot be expressed by the stabilizer formalism. The corresponding graph rule, stated in purely graph theoretical terms, is described, which completely characterizes the evolution of CV weighted graph states under this LC operation. This LC operation may be applied repeatedly on a CV weighted graph state, which can generate the infinite LC equivalent graph states of this graph state. This work is an important step to characterize the LC equivalence class of CV weighted graph states.
We provide a graphical method to describe and analyze non-Gaussian quantum states using a hypergraph framework. These states are pivotal resources for quantum computing, communication, and metrology, but their characterization is hindered by their complex high-order correlations. The framework encapsulates transformation rules for a series of typical Gaussian unitary operation and local quadrature measurement, offering a visually intuitive tool for manipulating such states through experimentally feasible pathways. Notably, we develop methods for the generation of complex hypergraph states with more or higher-order hyperedges from simple structures through Gaussian operations only, facilitated by our graphical rules. We present illustrative examples on the preparation of non-Gaussian states rooted in these graph-based formalisms, revealing their potential to advance continuous-variable general quantum computing capabilities.
We demonstrate experimentally how to remove an arbitrary node from a continuous-variable cluster state and how to shorten any quantum wires of such a state. These two basic operations, performed in an unconditional fashion, are a manifestation of quantum erasure and can be employed to obtain various graph states from an initial cluster state. Starting with a sufficiently large cluster, the resulting graph states can then be used for universal quantum information processing. In the experiment, all variations of this cluster shaping are demonstrated on a four-mode linear cluster state through homodyne measurements and feedforward.
Multimode entangled pure Gaussian states are an essential resource for continuous variable (CV) quantum information processing. Menicucci et al. [2011] have shown how to describe them as graphs with complex adjacency matrices. Among such complex graph states (CGSs), quantum optics experimental setups straightforwardly display correlations described by purely imaginary graph states (iRGSs) whereas theoretical literature favors off-diagonal real graph state (RGS) descriptions, which actually reflect the entanglement structure in the infinite squeezing limit [Gonzalez-Arciniegas et al. 2021]. We study them beyond this ideal case, thus establishing in the finite squeezing regime an equivalence criterion for the generic case of CGSs through a graph propagating set of conditions which either finds a Gaussian Local Unitary (GLU) converting one state to another or gives a proof of impossibility if such a GLU does not exist. In the experimentally relevant case of the conversion between iRGSs and RGSs, the set of conditions results in simple relations between the topologies of the equivalent real and imaginary graphs. Finally, we show that the canonical forms of iRGSs and RGSs imply that the topology of the CV real and imaginary graph states are too rigid to admit any exact qumode local-addition analogue to the qubit complementation or qudit local-addition, thus demonstrating the non-existence of any exact qumode local-addition for these states. We however define an approximate one when the squeezing is strong enough.