I have had this issue too.

Please follow these steps and let me know the results, I have tested in my system and it worked as expected:

1. Close QTP
2. Close all Internet Explorers.
3. Open RegEdit
4. Navigate to HKEY_CURRENT_USERSoftwareMicrosoftInternet ExplorerMain
5. Add new Dword key: HKEY_CURRENT_USERSoftwareMicrosoftInternet ExplorerMainTabProcGrowth
Value name=TabProcGrowth, Type=DWORD, Data=0
6. Start QTP and test

When you open two Browsers, normally there will be 3 IE process. (One parent/frame process and two child processes)

This IE's behavior can be changed by IE's registry setting / policy. For example: TabProcGrowth.

The top-level/main windows for both browsers are in the frame process.
Closing frame process will automatically close child processes.
You can use Process Explorer or even the Windows Task Manager to check and verify. (frame process and child processes)

If both browsers do not have the common frame/parent process or they run in single process model, closing one of them surely won't affect the other.

In the default installation of Internet Explorer 8 this key does not exist and is considered by Internet Explorer 8 as set to 1. The key instructs Internet Explorer 8 to open for each tab a separate process if not set or set to 1 (default behaviour), and to run all tabs in the same process if set to 0.

So, by adding this registry we have changed the IE's behavior and if you want to carry its login information from parent browser to child browser, you will need to delete the registry or changed it back to its default value (1).