If you are already logged in, be sure to quit out to the XMB and back into The Last of Us to receive the patch
Naughty Dog issued a new patch for The Last of Us that is required to play The Last of Us: Left Behind chapter. It adds "Left Behind" to the Main Menu, plus 10 new Trophies (1 Silver, 9 Bronze) for the Left Behind DLC.
In addition, Naughty Dog doubled the rate player movement updates are sent peer-to-peer (it reduced latency in game engine to 66ms). Note that increasing the network update rate will increase network traffic on your Internet connection.
Left Behind is a single-player downloadable story campaign that tells the story of Ellie and Riley - the launch video is available for your viewing pleasure.