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X-chains reveal substructures of graph states

Jun-Yi Wu, Hermann Kampermann, Dagmar Bruß

Published Apr 13, 2015• 0 citations
OpenAlex

Abstract

A special configuration of graph state stabilizers, which contains only Pauli $\sigma_X$ operators, is studied. The vertex sets $\xi$ associated with such configurations are defined as the X-chains of graph states. The X-chains of a general graph state can be determined efficiently. With the help of X-chains, one obtains the explicit representation of graph states in the X-basis via the so-called X-chain factorization diagram. We show that graph states with different X-chains can have different probability distributions of X-measurement outcomes, which allows to distinguish certain graph states with X-measurements. We provide an approach to find the Schmidt decomposition of graph states in the X-basis. The existence of X-chains in a subsystem facilitates error correction in the entanglement localization of graph states. In all these applications, the difficulty of the task decreases with increasing number of X-chains. Furthermore, we show that the overlap of two graph states can be efficiently determined via X-chains, while its computational complexity with other known methods increases exponentially.

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Group structures and representations of graph states

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A special configuration of graph state stabilizers, which contains only Pauli ${\ensuremath{\sigma}}_{X}$ operators, is studied. The vertex sets $\ensuremath{\xi}$ associated with such configurations are defined as what we call $X$ chains of graph states. The $X$ chains of a general graph state can be determined efficiently. They form a group structure such that one can obtain the explicit representation of graph states in the $X$ basis via the so-called $X$-chain factorization diagram. We show that graph states with different $X$-chain groups can have different probability distributions of $X$-measurement outcomes, which allows one to distinguish certain graph states with $X$ measurements. We provide an approach to find the Schmidt decomposition of graph states in the $X$ basis. The existence of $X$ chains in a subsystem facilitates error correction in the entanglement localization of graph states. In all of these applications, the difficulty of the task decreases with increasing number of $X$ chains. Furthermore, we show that the overlap of two graph states can be efficiently determined via $X$ chains, while its computational complexity with other known methods increases exponentially.

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Determining X -chains in graph states

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Abstract The representation of graph states in the X -basis as well as the calculation of graph state overlaps can efficiently be performed by using the concept of X -chains (Wu et al 2015 Phys. Rev. A 92 012322 ). We present a necessary and sufficient criterion for X -chains and show that they can efficiently be determined by the Bareiss algorithm. An analytical approach for searching X -chain groups of a graph state is proposed. Furthermore we generalize the concept of X -chains to so-called Euler chains, whose induced subgraphs are Eulerian. This approach helps to determine if a given vertex set is an X -chain and we show how Euler chains can be used in the construction of multipartite Bell inequalities for graph states.

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Stabilizer states form a ubiquitous family of quantum states that can be graphically represented through the graph state formalism. A fundamental property of graph states is that applying a local complementation - a well-known and extensively studied graph transformation - results in a graph that represents the same entanglement as the original. In other words, the corresponding graph states are LU-equivalent. This property served as the cornerstone for capturing non-trivial quantum properties in a simple graphical manner, in the study of quantum entanglement but also for developing protocols and models based on graph states and stabilizer states, such as measurement-based quantum computing, secret sharing, error correction, entanglement distribution... However, local complementation fails short to fully characterise entanglement: there exist pairs of graph states that are LU-equivalent but cannot be transformed one into the other using local complementations. Only few is known about the equivalence of graph states beyond local complementation. We introduce a generalisation of local complementation which graphically characterises the LU-equivalence of graph states. We use this characterisation to show the existence of a strict infinite hierarchy of equivalences of graph states. Our approach is based on minimal local sets, which are subsets of vertices that are known to cover any graph, and to be invariant under local complementation and even LU-equivalence. We use these structures to provide a type to each vertex of a graph, leading to a natural standard form in which the LU-equivalence can be exhibited and captured by means of generalised local complementation.

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A special configuration of graph state stabilizers, which contains only Pauli σX\sigma_XσX​ operators, is studied. The vertex sets ξ\xiξ associated with such configurations are defined as the X-chains of graph states. The X-chains of a general graph state can be determined efficiently. With the help of X-chains, one obtains the explicit representation of graph states in the X-basis via the so-called X-chain factorization diagram. We show that graph states with different X-chains can have different probability distributions of X-measurement outcomes, which allows to distinguish certain graph states with X-measurements. We provide an approach to find the Schmidt decomposition of graph states in the X-basis. The existence of X-chains in a subsystem facilitates error correction in the entanglement localization of graph states. In all these applications, the difficulty of the task decreases with increasing number of X-chains. Furthermore, we show that the overlap of two graph states can be efficiently determined via X-chains, while its computational complexity with other known methods increases exponentially.
A special configuration of graph state stabilizers, which contains only Pauli ${\ensuremath{\sigma}}_{X}$ operators, is studied. The vertex sets $\ensuremath{\xi}$ associated with such configurations are defined as what we call XXX chains of graph states. The XXX chains of a general graph state can be determined efficiently. They form a group structure such that one can obtain the explicit representation of graph states in the XXX basis via the so-called XXX-chain factorization diagram. We show that graph states with different XXX-chain groups can have different probability distributions of XXX-measurement outcomes, which allows one to distinguish certain graph states with XXX measurements. We provide an approach to find the Schmidt decomposition of graph states in the XXX basis. The existence of XXX chains in a subsystem facilitates error correction in the entanglement localization of graph states. In all of these applications, the difficulty of the task decreases with increasing number of XXX chains. Furthermore, we show that the overlap of two graph states can be efficiently determined via XXX chains, while its computational complexity with other known methods increases exponentially.
Abstract The representation of graph states in the X -basis as well as the calculation of graph state overlaps can efficiently be performed by using the concept of X -chains (Wu et al 2015 Phys. Rev. A 92 012322 ). We present a necessary and sufficient criterion for X -chains and show that they can efficiently be determined by the Bareiss algorithm. An analytical approach for searching X -chain groups of a graph state is proposed. Furthermore we generalize the concept of X -chains to so-called Euler chains, whose induced subgraphs are Eulerian. This approach helps to determine if a given vertex set is an X -chain and we show how Euler chains can be used in the construction of multipartite Bell inequalities for graph states.
Graph states are multiparticle entangled states that correspond to mathematical graphs, where the vertices of the graph take the role of quantum spin systems and edges represent Ising interactions. They are many-body spin states of distributed quantum systems that play a significant role in quantum error correction, multiparty quantum communication, and quantum computation within the framework of the one-way quantum computer. We characterize and quantify the genuine multiparticle entanglement of such graph states in terms of the Schmidt measure, to which we provide upper and lower bounds in graph theoretical terms. Several examples and classes of graphs will be discussed, where these bounds coincide. These examples include trees, cluster states of different dimensions, graphs that occur in quantum error correction, such as the concatenated [7,1,3]-CSS code, and a graph associated with the quantum Fourier transform in the one-way computer. We also present general transformation rules for graphs when local Pauli measurements are applied, and give criteria for the equivalence of two graphs up to local unitary transformations, employing the stabilizer formalism. For graphs of up to seven vertices we provide complete characterization modulo local unitary transformations and graph isomorphisms.
Recently, there are tremendous developments on the number of controllable qubits in several quantum computing systems. For these implementations, it is crucial to determine the entanglement structure of the prepared multipartite quantum state as a basis for further information processing tasks. In reality, evaluation of a multipartite state is in general a very challenging task owing to the exponential increase of the Hilbert space with respect to the number of system components. In this work, we propose a systematic method using very few local measurements to detect multipartite entanglement structures based on the graph state --- one of the most important classes of quantum states for quantum information processing. Thanks to the close connection between the Schmidt coefficient and quantum entropy in graph states, we develop a family of efficient witness operators to detect the entanglement between subsystems under any partitions and hence the entanglement intactness. We show that the number of local measurements equals to the chromatic number of the underlying graph, which is a constant number, independent of the number of qubits. In reality, the optimization problem involved in the witnesses can be challenging with large system size. For several widely-used graph states, such as 1-D and 2-D cluster states and the Greenberger-Horne-Zeilinger state, by taking advantage of the area law of entanglement entropy, we derive analytical solutions for the witnesses, which only employ two local measurements. Our method offers a standard tool for entanglement structure detection to benchmark multipartite quantum systems.
Abstract The objective of Quantum State Verification (QSV) is to evaluate whether an unknown quantum state is sufficiently similar to a predetermined target state at a certain confidence level. The required number of identical and independent copies, N, of an unknown state for such a task is inversely proportional to the spectral gap of a verification operator. Stabilizer states form a foundational class of quantum states, extensively utilized in quantum computing. Defined through a group of stabilizer operators constructed from Pauli matrices in qubit systems, their structural properties not only simplify the mathematical representation and simulation of quantum states but also make them particularly suitable for verification protocols that rely on local measurements. Restricting the verification operator to be those that can be achieved by local measurements, previous researches had determined an upper bound for the spectral gap to be 2/3 in the cases of qubit stabilizer states. In this research, we show a similar upper bound of d/(d+1) in the cases of graph states of d-level systems by a technique from [1] but adapted to d-level graph states. In particular, we show that d-level graph states are maximally entangled across any bipartition of the graph, which allows us to reduce the problem to the case of a bipartite maximally entangled d-level systems for which the maximal spectral gap is known.
Stabilizer states form a ubiquitous family of quantum states that can be graphically represented through the graph state formalism. A fundamental property of graph states is that applying a local complementation - a well-known and extensively studied graph transformation - results in a graph that represents the same entanglement as the original. In other words, the corresponding graph states are LU-equivalent. This property served as the cornerstone for capturing non-trivial quantum properties in a simple graphical manner, in the study of quantum entanglement but also for developing protocols and models based on graph states and stabilizer states, such as measurement-based quantum computing, secret sharing, error correction, entanglement distribution... However, local complementation fails short to fully characterise entanglement: there exist pairs of graph states that are LU-equivalent but cannot be transformed one into the other using local complementations. Only few is known about the equivalence of graph states beyond local complementation. We introduce a generalisation of local complementation which graphically characterises the LU-equivalence of graph states. We use this characterisation to show the existence of a strict infinite hierarchy of equivalences of graph states. Our approach is based on minimal local sets, which are subsets of vertices that are known to cover any graph, and to be invariant under local complementation and even LU-equivalence. We use these structures to provide a type to each vertex of a graph, leading to a natural standard form in which the LU-equivalence can be exhibited and captured by means of generalised local complementation.