Is there a PPA with the latest version of LibreOffice?

There is an official LibreOffice PPA here.

You need to add ppa:libreoffice/ppa in your software sources to add it to the software center, just follow the instructions for the GUI way.

More information:

• Is the ppa of LibreOffice to be considered safe for the system?

To add ppa and upgrade libre office using command line:

sudo add-apt-repository ppa:libreoffice/ppa
sudo apt-get update  
sudo apt-get upgrade  

UPDATE 2

As of July 25th 2013, the following PPA offers the Release version of LibreOffice:

sudo add-apt-repository ppa:libreoffice/ppa
sudo apt-get update  
sudo apt-get upgrade  

That will leave you with LibreOffice 4.1.

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UPDATE 1

A PPA is now available for 4.1.0. To add it, just press Ctrl+Alt+T on your keyboard to open Terminal. When it opens, run the command(s) below:

sudo add-apt-repository ppa:libreoffice/libreoffice-prereleases

For complete instructions on how to install the beta2 of Libreoffice 4.1.0, check out LibreOffice Pre-Releases

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ORIGINAL ANSWER

I have searched high and low, and I can't find a PPA for 4.1.x. Now if you would like to install it, (the documentation doesn't mention removing prior installation) I would recommend that you remove previous versions before installing this one.

To install LibreOffice 4.1.0 Beta 1 on Ubuntu 13.04, 12.10, and 12.04, do this. Just press Ctrl+Alt+T on your keyboard to open Terminal. When it opens, run the command(s) below:

For 32 bit systems:

wget -c dev-builds.libreoffice.org/pre-releases/deb/x86/LibreOfficeDev_4.1.0.0.beta1_Linux_x86_deb.tar.gz

For 64 bit systems:

wget -c dev-builds.libreoffice.org/pre-releases/deb/x86_64/LibreOfficeDev_4.1.0.0.beta1_Linux_x86-64_deb.tar.gz

Extract the archive:

eDev_4.1.0.tar -zxvf LibreOffic0.beta1_Linux_x86*_deb.tar.gz

Install LibreOffice 4.1.0:

cd LibreOfficeDev*/DEBS
sudo dpkg -i *

As of June 13, 2013 it still not available in the pre-release PPA. Once available will update the answer.

^{Source:LinuxG}

The main LibreOffice PPA is for "test builds and backports", including "alpha and beta releases!".
UPDATE As of 2016-07, I see it's "LibreOffice fresh", the latest release of the newest series (but no alpha/beta releases, which are now in separate PPA).

If you want less-bleeding edge, it includes links to per-version stable PPAs, e.g. https://launchpad.net/~libreoffice/+archive/libreoffice-4-1 for LibreOffice 4.1.