"We have a mandate from the management to lower our bug count, regardless of severities."
Some of this is not possible. At the same time, it is possible to pull together a plan that will reduce the bug count found by QA.
Software is going to have bugs, period. That is the nature of the beast. It is typical to spend 40% or more of any project in QA and fixing bugs before and after release. Considering the totality of any project, some projects spend up to 90% of the budget on finding and fixing bugs. Recently, a bug was found in a commonly used library that had been there for 8 years.
If you have to reduce the bug count found in QA, then most of what QA does has to be moved into the development team. That means that the developer has to work with QA to define the testing that has to happen before submitting to QA. The developer has to do the QA first.
One type of development that reduces the bug count in QA is called "Test Driven Development", or "TDD". It adds a lot of development costs as the developers have to build test beds first and then make sure that the code passes those tests before going to QA. But it does reduce the number of bugs caught by QA.
Another technique could be to have a group review every bug report from QA asking how this bug could have been caught by the team before submitting to QA. This could be a way to help educate the team on techniques to catch bugs, identify patterns of failures and ask for ways to prevent those patterns, and build a plan that could be presented to management on how to change the process so as to reduce the bug count.
So, you can present to management a plan by which the developers go through training on how to do a new process, a change in how stories pass through the development team with perhaps a second developer building the test plan and test bed, and supervisory changes where the supervisor is monitoring how the tests are being built and run during development. There might be a need for follow up training as many people need repeat training to get proficient in a new way of working. This plan will greatly impact the speed at which stories are completed but will reduce the bug count.
In essence, bug count can be reduced, but at a cost. Does the management really want to pay the cost?