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arXiv:1008.4855 · New Journal of Physics

Bipartite entanglement in continuous variable cluster states

Hugo V. Cable, Dan E. Browne

Published Nov 26, 2010• 11 citations
arXivHTMLPDF

Abstract

A study of the entanglement properties of Gaussian cluster states, proposed as a universal resource for continuous variable (CV) quantum computing is presented in this paper. The central aim is to compare mathematically idealized cluster states defined using quadrature eigenstates, which have infinite squeezing and cannot exist in nature, with Gaussian approximations that are experimentally accessible. Adopting widely used definitions, we first review the key concepts, by analysing a process of teleportation along a CV quantum wire in the language of matrix product states. Next we consider the bipartite entanglement properties of the wire, providing analytic results. We proceed to grid cluster states, which are universal for the qubit case. To extend our analysis of the bipartite entanglement, we adopt the entropic-entanglement width, a specialized entanglement measure introduced recently by Van den Nest et al (2006 Phys. Rev. Lett. 97 150504), adapting their definition to the CV context. Finally, we consider the effects of photonic loss, extending our arguments to mixed states. Cumulatively our results point to key differences in the properties of idealized and Gaussian cluster states. Even modest loss rates are found to strongly limit the amount of entanglement. We discuss the implications for the potential of CV analogues for measurement-based quantum computation.

DOI
10.1088/1367-2630/12/11/113046

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A study of the entanglement properties of Gaussian cluster states, proposed as a universal resource for continuous variable (CV) quantum computing is presented in this paper. The central aim is to compare mathematically idealized cluster states defined using quadrature eigenstates, which have infinite squeezing and cannot exist in nature, with Gaussian approximations that are experimentally accessible. Adopting widely used definitions, we first review the key concepts, by analysing a process of teleportation along a CV quantum wire in the language of matrix product states. Next we consider the bipartite entanglement properties of the wire, providing analytic results. We proceed to grid cluster states, which are universal for the qubit case. To extend our analysis of the bipartite entanglement, we adopt the entropic-entanglement width, a specialized entanglement measure introduced recently by Van den Nest et al (2006 Phys. Rev. Lett. 97 150504), adapting their definition to the CV context. Finally, we consider the effects of photonic loss, extending our arguments to mixed states. Cumulatively our results point to key differences in the properties of idealized and Gaussian cluster states. Even modest loss rates are found to strongly limit the amount of entanglement. We discuss the implications for the potential of CV analogues for measurement-based quantum computation.
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 (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.
Continuous-variable cluster states (CVCSs) can be supplemented with Gottesman-Kitaev-Preskill (GKP) states to form a hybrid cluster state with the power to execute universal, fault-tolerant quantum computing in a measurement-based fashion. As the resource states that comprise a hybrid cluster state are of a very different nature, a natural question arises: Why do GKP states interface so well with CVCSs? To answer this question, we apply the recently introduced subsystem decomposition of a bosonic mode, which divides a mode into logical and gauge-mode subsystems, to three types of cluster state: CVCSs, GKP cluster states, and hybrid continuous-variable (CV)-GKP cluster states. We find that each of these contains a ``hidden'' qubit cluster state across their logical subsystems, which lies at the heart of their utility for measurement-based quantum computing. To complement the analytical approach, we introduce a simple graphical description of these CV-mode cluster states that depicts precisely how the hidden qubit cluster states are entangled with the gauge modes, and we outline how these results would extend to the case of finitely squeezed states. This work provides important insight that is both conceptually satisfying and helps to address important practical issues like when a simpler resource (such as a Gaussian state) can stand in for a more complex one (like a GKP state), leading to more efficient use of the resources available for CV quantum computing.
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.
Quantum properties are soon subject to decoherence once the quantum system interacts with the classical environment. In this paper we experimentally test how propagation losses, in a Gaussian channel, affect the bipartite Gaussian entangled state generated by a subthreshold type-II optical parametric oscillator. Experimental results are discussed in terms of different quantum markers, as teleportation fidelity, quantum discord, and mutual information, and continuous-variable (CV) entanglement criteria. To analyze state properties we have retrieved the composite system covariance matrix by a single homodyne detector. We experimentally found that, even in the presence of a strong decoherence, the generated state never disentangles and keeps breaking the quantum limit for the discord. This result proves that the class of CV entangled states discussed in this paper would allow, in principle, to realize quantum teleportation over an infinitely long Gaussian channel.
We review the theory of continuous-variable entanglement with special emphasis on foundational aspects, conceptual structures and mathematical methods. Much attention is devoted to the discussion of separability criteria and entanglement properties of Gaussian states, for their great practical relevance in applications to quantum optics and quantum information, as well as for the very clean framework that they allow for the study of the structure of nonlocal correlations. We give a self-contained introduction to phase-space and symplectic methods in the study of Gaussian states of infinite-dimensional bosonic systems. We review the most important results on the separability and distillability of Gaussian states and discuss the main properties of bipartite entanglement. These include the extremal entanglement, minimal and maximal, of two-mode mixed Gaussian states, the ordering of two-mode Gaussian states according to different measures of entanglement, the unitary (reversible) localization and the scaling of bipartite entanglement in multimode Gaussian states. We then discuss recent advances in the understanding of entanglement sharing in multimode Gaussian states, including the proof of the monogamy inequality of distributed entanglement for all Gaussian states. Multipartite entanglement of Gaussian states is reviewed by discussing its qualification by different classes of separability, and the main consequences of the monogamy inequality, such as the quantification of genuine tripartite entanglement in three-mode Gaussian states, the promiscuous nature of entanglement sharing in symmetric Gaussian states and the possible coexistence of unlimited bipartite and multipartite entanglement. We finally review recent advances and discuss possible perspectives on the qualification and quantification of entanglement in non-Gaussian states, a field of research that is to a large extent yet to be explored.