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Let's say we magically found out that there is an inhabited planet in the Alpha Centauri system and that the inhabitants there have the same level of technology that we do. Similarly, they know of our existance.

Now we both want to communicate with each other. Would it be possible? Obviously the signals would take over 4 years to get either way, but at the end, do we even have the technology to send and receive messages at this distance? Or even just detect that there was some kind of signal?

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Definitely. In 1974, to honor the updating of the Arecibo radio telescope in Puerto Rico, a message was broadcast towards Messier 13, which is about 25,000 light years away. To be sure, just because we broadcast it in that direction, we can't assume any hypothetical aliens there could detect it. But, because the beam is focused directionally and also generated with most of the power at a chosen frequency, it can be as bright at that frequency as some astronomical sources and should have been detectable "just about anywhere in the galaxy" (if we had aimed it in the right direction) with a detector the same size as our transmitter.

Even unintentional messages might be detectable, but it would be hard. Recently a study was done to see whether the cumulative effect of all cell phone signals would be detectable from Barnard's Star, which is about six light years away. The conclusion was that it would be very hard for them to detect. But these signals were not focused or otherwise designed for interstellar communication, and so the effective power is thousands of times less.

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    $\begingroup$ With today's technology, maybe, but not today's equipment. With the loss of Arecibo, the most powerful remaining transmitter is the Goldstone Solar System Radar, less than 2% as powerful as Arecibo. $\endgroup$
    – Mark
    May 14 at 9:46
  • $\begingroup$ Well, Alpha Centauri is much closer than Messier 13. I think Arecibo-class receivers within a few ly could receive Goldstone. I see the Chinese FAST telescope lacks a transmitter, or it would be ideal. $\endgroup$
    – Mark Foskey
    May 15 at 0:38
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Yes, absolutely.

On the listening side, Breakthrough Listen scanned (among other things) 20 nearby stars (from 7 to 143pc distance) and "achieved a minimum detectable flux which would have allowed detections of emissions that were 10−3 to 0.88 times as powerful as the signaling capability of the Arecibo radar transmitter, for the nearest and furthest stars respectively". Alpha Centauri at only 1.3pc away would be rather easier to signal.

(Note that Breakthrough Listen did not scan Alpha Centauri in particular.)

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  • $\begingroup$ So basically Arecibo could have detected Arecibo from Alpha Centauri, but not much further than that. $\endgroup$
    – PearsonArtPhoto ♦
    Oct 3 at 2:06
  • $\begingroup$ 143pc is two orders of magnitude further than 1.3pc...? Slightly more than what I would call "not much further". $\endgroup$
    – TLW
    Oct 3 at 10:52
  • $\begingroup$ I'm assuming that the 7 pc was the 0.88 times as powerful one, and the 143 pc was the 10-3 one. Seems about the right ratios. $\endgroup$
    – PearsonArtPhoto ♦
    Oct 3 at 21:42
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    $\begingroup$ @PearsonArtPhoto I think you have that backwards? The further star would have to signal with a higher power. So the 143pc needs the 0.88×Arecibo, and the 7pc star only needs the 0.001×Arecibo. Unless I'm wrong? $\endgroup$
    – Infinite_Maelstrom
    Oct 3 at 23:52
  • $\begingroup$ @PearsonArtPhoto - I think you have that backwards. It's an inverse-square, to a first approximation. And hence signal strength required is approximately proportional to distance squared. You'd expect given a 'true' inverse-square that the 7pc distance star would need (7/143)^2 times the power as the 143pc distance one - which, if the 143pc one needed 0.88xArecibo, would work out to requiring ~0.002xArecibo for the 7pc one. Close enough. $\endgroup$
    – TLW
    Oct 4 at 12:27

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