Synthetics will outperform non-synthetics at low temperatures while the machinery is warming up to normal operating temperature. So there would be an efficiency improvement
during that time vs. regular mineral oils that don't have the same stability. Though, I'm with Daox and would say the improvement (comparing the same weight oil) would be hard to detect.
You'll also note that none of the major auto lubricant makers claims efficiency improvements in their marketing by going to synthetics (of the same weight).
My opinion is that synth is worth pursuing (for a fractional efficiency gain) if you live in a cold climate, or if you normally only do very short trips.
You're more likely to get an improvement in efficiency by switching to the lowest viscosity lubricant that you can safely use in your application. See:
Effect of gear oil viscosity on transmission efficiency (Metro owners take note)