Entangled states from simple quantum graphs

TL;DR

The paper addresses how to generate and control entanglement between two open quantum graphs by using a coherent controlled scattering operation that ties the entanglement to graph topology and tunable parameters. The authors model two QGs with two leads and derive explicit conditions for maximal entanglement, showing that maximal EE occurs when $|r_A|^2 = |t_A|^2 = \tfrac{1}{2}$ and appropriate relations hold for Bob’s amplitudes; introducing a controlled phase $\varphi$ enables a controlled-$Z$-like operation with maximal entanglement under $\varphi = (2n+1)\pi$. They illustrate gate realizations on a star graph $S_4$ and map parameter choices to identity, Pauli, and Hadamard gates, and they analyze a two-graph setup to map the EE landscape as a function of wave numbers and phase. The work suggests experimental feasibility with microwave networks and outlines avenues for extending the framework to multi-graph and multi-lead configurations, highlighting the potential of QG-based entanglement generation as a complementary quantum-information resource.

Abstract

Entanglement is a fundamental resource for many applications in quantum information processing. Here, we investigate how quantum transport in simple quantum graphs, modeled as controlled two-level quantum systems, can be utilized to generate entangled states through coherent control operations between two simple quantum graphs. A controlled operation is defined such that the scattering behavior of one quantum graph dynamically modifies the other. Our analysis reveals the precise conditions under which maximal entanglement or separability arises, including configurations that can be implemented via phase shifts in graph structures. Our findings demonstrate that the maximal entanglement in this system is closely related to recent results on randomized quantum graphs. These results provide new pathways for engineering entanglement using simple quantum graphs and suggest experimental feasibility using microwave networks.

Figures (7)

• Figure 1: The scattering in QGs as an information model. Initially, there is an input state in a given QG $\Gamma_G$, where a particle enters by one of its leads, called $l_0$, and thus the particle is found to be in the state $\ket{0}$. After the scattering in $\Gamma_G$, there is an output state where the particle has two scattering amplitudes: $r_G$ to reflect to the same lead $l_0$ and thus it is in the state $\ket{0}$; and $t_G$ to transmit through the QG and be found in the other lead $l_1$, being in the state $\ket{1}$.
• Figure 2: The coherent control operation between two QGs $\Gamma_A$ and $\Gamma_B$ by the controlled scattering operator, which may change $\Gamma_B$ to $\Gamma_{B'}$.
• Figure 3: The controlled scattering system of two QGs $\Gamma_A$ and $\Gamma_B$, where the controlled operation is a phase $\varphi$ applied to the scattering channel $l_{1,B}$ by a controlled scattering operator $CS_{B,B'}^{A}$ in the scattering channel $l_{1,A}$.
• Figure 4: The EE as a function of the transmission probability $|t_A|^2$ and $|t_B|^2$ in each QG when a controlled phase $\pi$ is applied in Bob's QG, with $t_{B'}=-t_{B}$.
• Figure 5: (a) A star QG with four vertices ($S_4$) and two scattering channels ($\Gamma_{S_4}$) which are connected to the vertices labeled as $v_1$ and $v_4$. Each edge which connects a pair of vertices $\{i,j\}$ in the QG has a given length $\ell_{i,j}$. (b) An equivalent QG where a phase shift operation $\alpha$ is used to mimic the phase obtained due to the length $\ell_{1,2}$, and the same is done to $\ell_{2,4}$, being replaced by $\beta$. The distance between the vertices $v_2$ and $v_3$ is $\ell$ and a possible phase shift operation along this edge is defined by $\varphi$.