The terms "alpha" or "beta" mean less to most players than "get to play it early," which will be the case in September for anyone interested in Battlefield 3 who purchased Medal of Honor: Limited Edition.
The studio is currently rolling into the alpha phase, which is a private and invite-only period meant to test very specific aspects of the online experience. Having purchased Medal of Honor won't help you get into this one, but that doesn't mean you don't have a shot. DICE recommends signing up for the Battlefield 3 newsletter and ensuring your Veteran status is current.
Then, cross your fingers.
What you should really be looking forward to is September's beta. A newsletter recently went out reminding players of the beta, but if you didn't receive that newsletter, there's still time to opt in via your online Electronic Arts profile. There doesn't appear to be a cut-off period, so long as you've got a copy of Medal of Honor in your possession with a Battlefield 3 sticker.
Naturally, that offer doesn't extend to used copies of the game.