Multipartite entanglement sudden death and birth in randomized hypergraph states

Abstract

We introduce and analyze the entanglement properties of randomized hypergraph states, as an extended notion of the randomization procedure in the quantum logic gates for the usual graph states, recently proposed in the literature. The probabilities of applying imperfect generalized controlled-$Z$ gates simulate the noisy operations over the qubits. We obtain entanglement measures as negativity, concurrence, and genuine multiparticle negativity, and show that entanglement exhibits a non-monotonic behavior in terms of the randomness parameters, which is a consequence of the non-uniformity of the associated hypergraphs, reinforcing the claim that the entanglement of randomized graph states is monotonic since they are related to $2$-uniform hypergraphs. Moreover, we observed the phenomena of entanglement sudden death and entanglement sudden birth in RH states. This work revels a connection between the non-uniformity of hypergraphs and loss of entanglement.

Figures (5)

• Figure 1: Hypergraphs that are of interest in this paper. The hypergraphs $H_4^{(1)}$, $H_4^{(2)}$, and $H_{4}^{(3)}$ are cases of special interest because their reduced single-qubit matrices are maximally mixed JPA.47.335303.2014. $H_{n}^{(i)}$ represents the $i$th hypergraph on $n$ vertices.
• Figure 2: Process of randomization of the $\ket{H_{3}^{(2)}}$. Dashed lines represent noisy $C_e$ gates, in which with probability $p_{|e|}$ the gate succeeds and with probability $(1-p_{|e|})$ the gate fails. When the gate fails, it has the same effect as an identity operator. Continuum lines are a successful creation of hyperedges. (a) $F_1$ subhypergraph with probability $p_2p_3$; (b) $F_2$ subhypergraph with probability $(1-p_2)p_3$; (c) $F_3$ subhypergraph with probability $p_2(1-p_3)$; and (d) $F_4$ subhypergraph with probability $(1-p_2)(1-p_3)$ [see Eq. \ref{['eq:Rp']}].
• Figure 3: Negativity for RH states listed in Fig. \ref{['fig:fig1']}. The notation "$\{v_1,\ldots\}|\{u_1,\ldots\}$" represents the bipartition used to calculate the negativity. (a)--(d) Negativity of 3-qubit RH states for bipartition $\{3\}|\{1,2\}$. (e)--(f) Negativity of 4-qubit RH states for bipartition $\{2\}|\{1,3,4\}$. (g)--(h) Negativity of 4-qubit RH states for bipartition $\{1,2\}|\{3,4\}$.
• Figure 4: Concurrence of RH states listed in Fig. \ref{['fig:fig1']}. (a)--(d) Concurrence of 3-qubit RH states between qubits $1$ and $3$; (e), (f) concurrence of 4-qubit RH states between qubits $3$ and $4$; and (g), (h) concurrence of 4-qubit RH states between qubits $1$ and $3$. The light gray part indicates the zero value of concurrence.
• Figure 5: GMN of RH states listed in Fig. \ref{['fig:fig1']}. (a)--(d) GMN for 3-qubit RH states. (e)--(h) GMN for 4-qubit RH states. The light gray part indicates the zero value of GMN.

Theorems & Definitions (1)

• Definition 3.1: Randomized hypergraph state