[QInfo broadcasting (1/3)] Support for qml.math density matrix functions

[eddddddy]

Currently the quantum information functions in qml.math and the QNode transforms in qml.qinfo do not support broadcasting. Support for this will be added in a sequence of 3 PRs, of which this is the first. The individual aims of the PRs are roughly:

1. (current) Split qml.math.reduced_dm into two functions, qml.math.reduced_dm_from_sv and qml.math.reduced_dm_from_dm, each of which supports broadcasting. Also begin a deprecation cycle for qml.math.reduced_dm.
2. Support broadcasting for each of the remaining quantum info functions in qml.math. Also change their signatures to accept only a density matrix, and begin a deprecation cycle for passing in state vectors.
3. Support broadcasting for the qml.qinfo QNode transforms.

[github-actions]

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[codecov]

Codecov Report

Merging #4173 (1af71bf) into master (233eec7) will increase coverage by 0.00%.
The diff coverage is 100.00%.

@@           Coverage Diff           @@
##           master    #4173   +/-   ##
=======================================
  Coverage   99.77%   99.77%           
=======================================
  Files         340      340           
  Lines       30566    30595   +29     
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+ Hits        30496    30525   +29     
  Misses         70       70           

Impacted Files	Coverage Δ
pennylane/math/__init__.py	100.00% <ø> (ø)
pennylane/_qubit_device.py	99.29% <100.00%> (ø)
pennylane/devices/default_mixed.py	100.00% <100.00%> (ø)
pennylane/math/quantum.py	100.00% <100.00%> (ø)
pennylane/measurements/state.py	100.00% <100.00%> (ø)
pennylane/qinfo/transforms.py	100.00% <100.00%> (ø)

[timmysilv]

looks great! just a couple of comments

[mudit2812]

Looks good. I think the ongoing discussions should be finished before merging to clear up the roadmap for the broadcasting and potential DensityMatrixMP changes, but otherwise this PR looks merge-ready to me beyond non-blocking suggestions.

[albi3ro]

Thanks !

[timmysilv]

thanks!