Many-body quantum resources of graph states

Abstract

Characterizing the non-classical correlations of a complex many-body system is an important part of quantum technologies. A versatile tool for such a task is one that scales well with the size of the system and which can be both easily computed and measured. In this work we focus on graph states, that are promising platforms for quantum computation, simulation and metrology. We consider four topologies, namely the star graph states with edges, Turán graphs, $r$-ary tree graphs, and square grid cluster states, and provide a method to characterise their quantum content: the many-body Bell correlations, non-separability and entanglement depth for an arbitrary number of qubits. We also relate the strength of these many-body correlations to the usefulness of graph states for quantum sensing. Finally, we characterize many-body entanglement depth in graph states with up to $8$ qubits in $146$ classes non-equivalent under local transformations and graph isomorphisms. The technique presented is simple and does not make any assumptions about the multi-qubit state, so it could find applications wherever precise knowledge of many-body quantum correlations is required.

Figures (6)

• Figure 1: Examples of distinct graph state topologies. (a) star graph with $L=12$ nodes, (b) star graph with $L=12$ nodes and $p=4$ edges, (c) Turán graph with $L = 12$ nodes partitioned into equal $K = 2$ groups, (d) $r$-ary tree with $r=3$ roots and $h=2$ depth.The shaded areas show separate star graphs forming GHZ states, which is a useful observation for computation of the Bell correlator. The dashed red circle shows that the tree graph can be cut into closed layers formed by a fractal-like hierarchy of star graphs (see Section \ref{['sec.trees']}).
• Figure 2: The normalized exponent $\gamma/L$ for a star graph with edges, Fig. \ref{['fig:fig1']}(b), is plotted as a function of the number of edges $p$ normalized to $L$, $x=p/L$. The colors correspond to: $L = 4$ (black), $L = 6$ (orange), $L=10^1$ (blue), $L=10^2$ (green) and $L=10^3$ (purple). The dashed horizontal line denotes the Bell limit. All the values of $\gamma/L<1$ imply the presence of many-body entanglement, and the presence of many-body Bell correlations is indicated by values of $1/2 < \gamma/L < 1$.
• Figure 3: The histogram of $\gamma$ for LU non-equivalent graph states with $L=5,6,7,8$ qubits is shown in panels (a)-(d), respectively.
• Figure 4: The exponent $\gamma$ for representatives of graph states in non-equivalent classes under local unitary transformations and graph isomorphisms for up to $L=8$ qubits. Classes No. 1-64.
• Figure 5: The exponent $\gamma$ for representatives of graph states in non-equivalent classes under local unitary transformations and graph isomorphisms for up to $L=8$ qubits. Classes No. 65-128.