I looked up the specs and that display supports 4K resolutions. 2011 model Macs do not support 4K displays. I understand that you're trying to run it at 2560x1440, which Apple specifically says that model MBP supports for external displays. However, here's the issue you're running into...
To get that 4K resolution, you must have either an HDMI 1.4 or DisplayPort 1.2 source. HDMI 1.4 is definitely out. You cannot get anything higher than 1080p over a MiniDP -> HDMI adapter. So that leaves just DisplayPort 1.2 as your only hope.
I cannot verify whether or not the MiniDP port on 2011 MacBook Pros actually supports DP 1.2. Apple doesn't say on their spec pages and there is argument among tech sites as to whether or not it's does. But let's say for argument's sake that it is supported...
4K displays require the use of a specific feature of the DP 1.2 spec, called Multi-Stream Transport (MST). Even if the MiniDP port on your Mac is DisplayPort 1.2, both the GPU and the driver must also support MST in order to drive a 4K display. Even though you're not trying to run the full 4K resolution on it (which you definitely cannot do), the screen may still be attempting to negotiate an MST link over the DP connection even though it doesn't need to and is failing to do so. That would explain why you don't see anything.
The first thing I would do is check your display's menus to see if there are any settings relating to disabling MST support (maybe called SST mode?). If so that might just be your golden ticket.
If that doesn't work, try running it over DisplayPort in a Windows BootCamp partition with the latest AMD graphics driver. If it can work at all then Windows will be able to do it because the drivers are supplied directly by ATI/AMD and BootCamp Windows does not have to worry about graphics switching with the Intel HD graphics controller.
If you can't turn off MST support and it works in Windows fine, then then it might still be possible to use that screen over DP in OS X. Apple advertised that the new Retina MacBook Pros could support 4K displays but caused kind of a stink because only some 4K displays actually worked (and even then, only at 30Hz refresh instead of the usual 60). This was due to the lack of driver support in OS X for MST streams.
They fixed that as of the Mavericks 10.9.3 update. HOWEVER, that fix is only reported to apply to 2013 Mac Pros and Retina-display MacBook Pros. I'd bet you a sandwich that they didn't backport that fix to earlier models.
So here's the bottom line:
- IF the MiniDP port on the 2011 MacBook Pro is DisplayPort 1.2,
-- and --
- IF Apple backported the MST fix to non-retina MBP models
-- OR --
- You can disable MST mode on your display,
-- Then --
You might be able to use that display the way you want.
There are a lot of ifs there. If none of those things is true, then you may well be screwed.