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## Tensor Product of Quantum State Spaces

Maple

This is about the recent implementation of tensor products of quantum state spaces in the Physics package, in connection with an exchange with the Physics of Information Lab of the University of Waterloo. As usual this development is available to everybody from the Maplesoft R&D Physics webpage. This is the last update for Maple 2017. The updates for Maple 2018, starting with this same material, will begin being distributed through the MapleCloud next week.

Tensor Product of Quantum State Spaces

Basic ideas and design

Suppose  and  are quantum operators and  are, respectively, their eigenkets. The following works since the introduction of the Physics package in Maple

 >
 >
 (1)
 >
 (2)
 >
 (3)

where on the left-hand sides the product operator `*` is used as a sort of inert form (it has all the correct mathematical properties but does not perform the contraction) of the dot product operator `.`, used on the right-hand sides.

Suppose now that  and  act on different, disjointed, Hilbert spaces.

1) To represent that, a new keyword in Setup , is introduced, to indicate which spaces are disjointed, say as in .  We want this notation to pop up at some point as  where the indexation indicates all the operators acting on that Hilbert space. The disjointedspaces keyword has for synonyms disjointedhilbertspaces and hilbertspaces. The display  is not yet implemented.

NOTE: noncommutative quantum operators acting on disjointed spaces commute between themselves, so after setting  - for instance - , automatically,  become quantum operators satisfying (see comment (ii) on page 156 of ref. [1])

2) Product of Kets and Bras (KK, BB, KB and BK) where K and B belong to disjointed spaces, are understood as tensor products satisfying, for instance with disjointed spaces  and  (see footnote on page 154 of ref. [1]),

while of course

 Details

3) All the operators of one disjointed space act transparently over operators, Bras and Kets of the other disjointed spaces, for example

and the same for the Dagger of this equation, that is

And this happens automatically. Hence, when we write the left-hand sides and press enter, they are automatically rewritten (returned) as the right-hand sides.

Note that for the product of an operator times a Bra or a Ket we are not using the notation that expresses the product with the symbol 5.

Regarding the display of Bras and Kets and their tensor products, two enhancements are happening:

 • A new Setup option hideketlabel makes all the labels in Kets and Bras to be hidden when displaying Kets, Bras and Bracket(s), with the indices presented one level up, as if they were a sequence of labels, so that:

is displayed as

 This is the notation used more frequently when working in quantum information. This hideketlabel option is already implemented entering
 • Tensor products formed with operators, or Bras and Kets, that belong to disjointed spaces (set as such using Setup ), are displayed with the symbol 5 in between, as in  instead of , and  instead of .
 Tensor product notation and the hideketlabel option
 The implementation of tensor products using `*` and `.`
 Basic exercising with the new functionality
 Related functionality already in place before these changes

Reference

[1] Cohen-Tannoudji, Diue, Laloe, Quantum Mechanics, Chapter 2, section F.

[2] Griffiths Robert B., Hilbert Space Quantum Mechanics, Quantum Computation and Quantum Information Theory Course, Physics Department, Carnegie Mellon University, 2014.