Efficiency of quantum energy teleportation within spin-$\frac{1}{2}$ particle pairs

Abstract

A protocol for quantum energy teleportation (QET) is known for a so-called minimal spin-$\frac{1}{2}$ particle pair model. We extend this protocol to explicitly admit quantum weak measurements at its first stage. The extended protocol is applied beyond the minimal model to spin-$\frac{1}{2}$ particle pairs whose Hamiltonians are of a general class characterized by orthogonal pairs of entangled eigenstates. The energy transfer efficiency of the extended QET protocol is derived for this setting, and we show that weaker measurement yields greater efficiency. In the minimal particle pair model, for example, the efficiency can be doubled by this means. We also show that the QET protocol’s transfer efficiency never exceeds 100 %, supporting the understanding that quantum energy teleportation is, indeed, an energy transfer protocol, rather than a protocol for remotely catalyzing local extraction of system energy already present.

Appendix

We prove that the QET protocol’s efficiency $\eta (\pi ,s)$ in (15) is a non-increasing function of the measurement strength $s$ for any particle pair $\mathcal{P}$ initially in the ground state $\rho = |E_0\rangle \langle E_0|$ with parameters $\pi$. The derivative of $\eta (\pi ,s)$ in (16) is

$$\begin{aligned} \frac{\partial \eta (\pi ,s)}{\partial s} = - \frac{\tilde{s}^{\prime }}{\tilde{s}^2 P_3}\frac{L-R}{\sqrt{s^2 m^2 P_1+(P_2+\tilde{s} Q)^2}} \end{aligned}$$

(34)

where

$$\begin{aligned} L = \tilde{s} m^2 P_1+P_2(P_2 +\tilde{s} Q), \quad R = P_2\sqrt{s^2 P_1+(P_2+\tilde{s} Q)^2} \end{aligned}$$

(35)

with $P_1,P_2,P_3$ and $Q$ given as in (17), and $\tilde{s}^{\prime }=s/\sqrt{1-s^2} \ge 0$. We have $E_0 \le E_1 \le E_2 \le E_3$, so

$$\begin{aligned} P_2+\tilde{s} Q= & {} \left( \frac{E_{10}}{2} + \frac{E_{21}}{4}(2-\tilde{m}) \right) + \tilde{s} \frac{E_{32}-E_{10}}{4} \nonumber \\= & {} (2-\tilde{s})\frac{E_{10}}{4}+(2-\tilde{m})\frac{E_{21}}{4}+\tilde{s} \frac{E_{32}}{4} \nonumber \\\ge & {} 0. \end{aligned}$$

Therefore, $L \ge 0$ in (35), and the derivative of $\eta (\pi ,s)$ in (34) is non-positive if and only if $L^2 \ge R^2$. So consider

$$\begin{aligned} L^2-R^2= & {} \left( \tilde{s} m^2 P_1+P_2^2 +\tilde{s} P_2 Q \right) ^2 - P_2^2 \left( s^2 m^2 P_1+(P_2+\tilde{s} Q)^2 \right) \nonumber \\= & {} \tilde{s}^2 m^4 P_1^2+2\tilde{s} P_1 P_2 (P_2+\tilde{s} Q) - s^2 P_1 P_2^2 \nonumber \\= & {} m^2 P_1 \left( \tilde{s}^2 m^2 P_1+(2\tilde{s} -s^2) P_2^2 + 2\tilde{s}^2 P_2 Q \right) \nonumber \\= & {} \tilde{s}^2 m^2 P_1 \left( m^2 P_1+P_2(P_2+2Q) \right) . \end{aligned}$$

(36)

In (36), we have

$$\begin{aligned} P_2+2 Q= & {} \left( \frac{E_{10}}{2} + \frac{E_{21}}{4}(2-\tilde{m}) \right) + 2 \frac{E_{32}-E_{10}}{4} \nonumber \\= & {} (2-\tilde{m})\frac{E_{21}}{4}+ 2\frac{E_{32}}{4} \nonumber \\\ge & {} 0. \end{aligned}$$

Therefore, $L^2 - R^2 \ge 0$ in (36), so the derivative in (34) is non-positive. $\square$