The question of "Is this application/feature/fix ready for release?" is one that can't be answered accurately, because - as the OP said - there's no way to be sure that there are no bugs in the software.
More than that, depending on the situation, the business could have reasons for releasing despite the concerns of QA or testers that the software is not ready for production (I've been there).
Our role in this situation is simply this: Give the business the information they need to make an informed decision whether or not to release. That can be in the form "Yes, I think it's good to go - there are no critical bugs and I've tested everything I can think of", it can be a matrix of test case results, or it can be something else entirely.
Ultimately, we give the business a risk assessment from our perspective, ideally looking at these areas:
- What has been tested - features and functions that have been tested and in how much detail. This includes the outcome of tests and bugs that have been reported and fixed.
- What has not been tested - any feature or function that wasn't tested and why it wasn't tested, as well as the risk of releasing with that part of the software not tested.
- Known bugs - any reported bugs that were not fixed, why they weren't fixed, and if you know it, when the team intends to fix them. If there are workarounds for the known bugs, they get included here as well.
- Any caveats - any quirks of the software that you've come across which aren't actual bugs but could cause support calls - this information often winds up in readme files and goes to the support staff so they know how to advise customers who call about these items (some examples include the software requiring a specific version of a supporting framework, or that it's not compatible with some piece of hardware).
- Risk assessment - the summary of how comfortable we are with releasing given all the information above.
The format you use to communicate the information isn't relevant: what matters is that it's something everyone involved understands.