bibkey: wu2014randomized authors: Jun-Yi Wu; Matteo Rossi; Hermann Kampermann; Simone Severini; Leong Chuan Kwek; Chiara Macchiavello; Dagmar Bruß year: 2014 title: “Randomized Graph States and their Entanglement Properties” doi: 10.1103/PhysRevA.89.052335 url: https://arxiv.org/abs/1403.3828v3 claim: “For a graph G and a probability p, the randomized graph state is rho_G^p = sum over spanning subgraphs F of G of p^|E_F| (1-p)^(|E_G| - |E_F|) |F><F|, where |F> = prod over the edges of F of CZ |+>^n is the graph state of F. Section V defines the negativity N(rho_AB) = (||rho_AB^Gamma_A|| - 1)/2 with the trace norm ||X|| = Tr sqrt(X^dagger X), reports numerically that it is monotone in p for the complete graphs K_n and the star graphs S_n with n <= 4, and states that it is an open question whether the monotonic behaviour of the negativity in p is a common feature to all randomized graph states.” strata_touched:

• D5/S3/Quantum/Entanglement/RandomizedGraphNegativityRefutation license: citation-only triage: anchor

Randomized Graph States and their Entanglement Properties

Jun-Yi Wu, Matteo Rossi, Hermann Kampermann, Simone Severini, Leong Chuan Kwek, Chiara Macchiavello, Dagmar Bruß, Phys. Rev. A 89, 052335 (2014), arXiv:1403.3828v3 (quant-ph). Quotations are from the arXiv v3 source, with its macros kept as printed.

The graph state (Section II C):

$|G⟩:={\prod }_{\{i_1,i_2\}\in E}(\text{CZ}{)}_{i_1{i}_2}|+{⟩}^{\otimes n}.$

with $\mathrm{CZ}=\mathrm{diag}(1,1,1,-1)$ in the computational basis.

The randomized graph state (Section III, Definition of the randomization operator):

R_{p}(|G\rangle ):=\sum_{F \emph{spans} G}p^{|E_{F}|}\left( 1-p\right)^{\left\vert E_{G}\backslash E_{F}\right\vert }|F\rangle \langle F|,

where $F$ are spanning subgraphs of $G$, $E_F$ and $E_G$ are the sets of

edges of $F$ and $G$; the randomized graph state is ${\rho }_G^p:=R_p(|G⟩)$.

The negativity (Section V):

We finally quantify the amount of bipartite entanglement by considering the negativity, evaluated with respect to all possible bipartitions of the qubits. The \emph{negativity} of a bipartite state ${\rho }_{AB}$ is defined \cite{VidalWerner2002-02} as $N({\rho }_{AB})=\frac{||{\rho }_{AB}^{{\Gamma }_A}||-1}{2},$ where ${\Gamma }_A$ represents the partial transposition with respect to the subsystem $A$, and $||X||=\text{Tr}[\sqrt{X^{†}X}]$ is the trace norm.

The open question (Section V, last paragraph):

We have evaluated the negativity numerically for some RG states composed of a small number of qubits. The results for the negativity of states corresponding to the complete graph $K_n$ and the star graph $S_n$ up to $n=4$ vertices are reported in Fig. \ref{fig::monotomicity_of_negativity}. As can be seen, in the studied cases the negativity exhibits a monotonic behaviour in terms of the randomness parameter $p$. This suggests that the entanglement content might increase monotonically in $p$ with respect to any bipartition. Actually, since for the extreme cases $p=0$ and $p=1$ we have a fully separable state and an entangled state, respectively, one might expect that, as the weight of entangled subgraph states in ${\rho }_G^p$ increases with increasing $p$, a corresponding growth of the entanglement content of the RG state ${\rho }_G^p$. However, even though this conjecture is supported by numerical evidence, it is an open question whether the monotonic behavior of the negativity in terms of the randomness $p$ is a common feature to all RG states.

The encoding indexes the qubits by Fin n and the computational basis by Fin n → Bool, takes the graph state of an edge set to be its CZ phases times $|+{⟩}^{\otimes n}$, sums over the subsets of the edge set of $G$, and uses the frozen partial transposition transposePart of an arbitrary set of qubits and the frozen trace norm $\operatorname{re}\operatorname{Tr} \sqrt{X^\dagger X}$.

Verified locator

• DOI: https://doi.org/10.1103/PhysRevA.89.052335 (Phys. Rev. A 89, 052335, 2014).
• URL: https://arxiv.org/abs/1403.3828v3 (v3, 2014-06-03, the latest version, marked as the published version; source Random_graph_states_and_their_entanglement_properties_published.tex, md5 3c212afcc2a0e40f0e7a9b4a2082292e): the graph state (l. 262–267), the randomization operator (l. 328–336), the negativity (l. 632–640) and the open question (l. 644–657).