Enforce the positivity of prob inside ProbMP.process_density_matrix

[JerryChen97]

Context:
This missing line might lead to specific nan error in the downstreaming gradient process especially ag/jax related. Check out https://github.com/PennyLaneAI/pennylane/actions/runs/12360643306/job/34496071685?pr=6684 and https://github.com/PennyLaneAI/pennylane/actions/runs/12360643306/job/34496077651?pr=6684 for concrete examples

P.S. How did the legacy device deal with this issue? It explicitly calls its own analytic prob method and did similar things

pennylane/pennylane/devices/default_mixed.py

Line 386 in 6383dfc

return qnp.where(probs < 0, -probs, probs)

Description of the Change:
use clip with min value eps to enforce the positivity.

Benefits:
More reliability

Possible Drawbacks:
None realized

Related GitHub Issues:

[github-actions]

Hello. You may have forgotten to update the changelog!
Please edit doc/releases/changelog-dev.md with:

• A one-to-two sentence description of the change. You may include a small working example for new features.
• A link back to this PR.
• Your name (or GitHub username) in the contributors section.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 99.62%. Comparing base (579e70d) to head (94bb291).
Report is 2 commits behind head on master.

Additional details and impacted files

@@           Coverage Diff            @@
##           master    #6727    +/-   ##
========================================
  Coverage   99.62%   99.62%            
========================================
  Files         476      477     +1     
  Lines       44733    44890   +157     
========================================
+ Hits        44564    44721   +157     
  Misses        169      169            

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

[JerryChen97]

We will leave this open for discussion furthermore after new year when everyone is back

[JerryChen97]

We will leave this open for discussion furthermore after new year when everyone is back

Concern here: there used to be issues with the use of eps in differentiability debugging before