Convex Optimization Approaches to Optimal Teleportation Fidelity in Linear Three-Party Networks

TL;DR

This paper addresses the problem of maximizing the average teleportation fidelity between two distant parties in a three-party network with LOCC via a central mediator. By using separable operations as a tractable superset, the authors formulate convex optimization bounds (SDPs) for the optimal fully entangled fraction and hence the optimal teleportation fidelity, and then introduce a restricted SEP class that significantly reduces problem size while preserving LOCC performance for several two-qubit cases. They show that a maximally entangled-basis measurement by Bob is not universally optimal and provide explicit scenarios where alternative strategies surpass it, including cases with non-maximally entangled measurements and local filtering. The study further extends to linear quantum networks, revealing that different LOCC strategies can achieve the same optimal fidelity while consuming different amounts of entanglement, highlighting nuanced trade-offs in entanglement distribution and paving the way for generalized network-design principles in quantum teleportation.

Abstract

We study the maximum achievable quantum teleportation fidelity between two distant parties, Alice and Charlie, where each of them share a bipartite quantum state only with a common intermediary, Bob, and all parties are allowed to perform {\it Local Operations and Classical Communication} (LOCC). As the structure of LOCC is complicated, we relax the set of free operations to separable (SEP) operations and formulate a convex optimization problem that provides upper bounds on the LOCC achievable fidelity value. We observe that the complexity of such optimization problem reduces significantly if we restrict ourselves to a subclass of SEP operations, where the Kraus operators of either Alice or Charlie are proportional to unitary operators, leading to a simplified convex optimization that matches the general LOCC limit for certain two-qubit states. Through explicit examples, we show that protocols initiated by Bob by performing measurements in a maximally entangled basis are not necessarily optimal, and alternative strategies can outperform them. Finally, we extend our analysis to linear networks and demonstrate that different LOCC strategies can achieve the same optimal fidelity while consuming different amounts of entanglement content.

Figures (9)

• Figure 1: (Color online) Alice-Bob share a two-qubit state $\rho_{AB_1}$, whereas Bob-Charlie share a two-qubit state $\sigma_{B_2 C}$. The task is to obtain the maximum possible teleportation fidelity, or equivalently the optimal fully entangled fraction between Alice and Charlie, where the maximization is over all three-party LOCC. The optimal fully entangled fraction must be a LOCC monotone and should give a higher value than the classical upper bound $\frac{1}{2}$iff some amount of distillable entanglement is distributed between Alice and Charlie.
• Figure 2: Here we have taken the state between Bob-Charlie as a fixed pure non-maximally entangled state with $\beta=0.75$. Here blue curve represents the upper bound $F^*_2=\dfrac{1}{2}\left( 1+C(\rho_{AB_1})~C(\sigma_{B_2C})\right)$, $F^*_P$ is the numerical upper bound from optimization problem represented by an orange curve, $F^*_{\mathscr{P}_1}$ is the fidleity achieved through Bell Measurement by Bob (Protocol 1) denoted by red curve and $F^*_{\{\eta_i\},{\mathscr{P}_2}}$ is the fidelity achieved through some Bell like measurement(Protocol 2) with the green curve. From the figure, it is clear that the orange curve, the red curve, and the blue curve coincide with each other.
• Figure 3: Here we have taken $p_i=q_i$$\forall~i$ with $p_1=p$ and $p_2=p_3=p_4=\frac{1-p}{3}$. Here blue curve represents the upper bound $F^*_1=\min\{F^*_{\rho_{AB_1}}, F^*_{\rho_{B_2 C}}\}$, $F^*_P$ is the bound from optimization problem represented by an orange curve, $F^*_{\mathscr{P}_1}$ is the fidleity achieved through Bell Measurement by Bob (Protocol 1) denoted by the red curve and $F^*_{\{\eta_i\},{\mathscr{P}_2}}$ is the fidelity achieved through some Bell like measurement(Protocol 2) with the green curve. From the figure, it is clear that the orange curve and the red curve coincide with each other.
• Figure 4: Here we have taken the state between Bob-Charlie as a fixed pure non-maximally entangled state with $\beta_0=0.75$ and ADC acts on a maximally entangled state shared between Alice-Bob, i.e., $\alpha_0=0.75$. Here blue curve represents the upper bound $F^*_1=\min\{F^*_{\rho_{AB_1}}, F^*_{\rho_{B_2 C}}\}$, $F^*_P$ is the numerical upper bound from optimization problem represented by an orange curve, $F^*_{\mathscr{P}_1}$ is the fidleity achieved through Bell Measurement by Bob (Protocol 1) denoted by red curve and $F^*_{\{\eta_i\},{\mathscr{P}_2}}$ is the fidelity achieved through some Bell like measurement(Protocol 2) with the green curve. From the figure, it is clear that the orange curve, the red curve, the blue curve, and the green curve all coincide with each other for a certain range of state parameter p of the state between Alice-Bob.
• Figure 5: Here we have taken the state between Bob-Charlie as a fixed two-qubit Werner state with $\lambda=\frac{2}{5}$. Here blue curve represents the upper bound $F^*_1=\min\{F^*_{\rho_{AB_1}}, F^*_{\rho_{B_2 C}}\}$, $F^*_P$ is the numerical upper bound from optimization problem represented by an orange curve, $F^*_{\mathscr{P}_1}$ is the fidleity achieved through Bell Measurement by Bob (Protocol 1) denoted by red curve and $F^*_{\{\eta_i\},{\mathscr{P}_2}}$ is the fidelity achieved through some Bell like measurement(Protocol 2) with the green curve. From the figure, it is clear that initially the red and the orange curve coincide, but as the value of the state parameter p increases, the orange curve is strictly above the red and the green curve. For a certain parameter regime, the red curve also falls below the green curve.

Theorems & Definitions (17)

• Definition 1
• Proposition 1
• proof
• Proposition 2
• proof
• Definition 2
• proof
• proof
• proof
• proof