Peeter’s lecture notes from class. May not be entirely coherent.
This is apparently covered as a side effect in the text  in one of the advanced material sections. FIXME: what section?
Example, one spin one half particle and one spin one particle. We can describe either quantum mechanically, described by a pair of Hilbert spaces
Recall that a Hilbert space (finite or infinite dimensional) is the set of states that describe the system. There were some additional details (completeness, normalizable, integrable, …) not really covered in the physics curriculum, but available in mathematical descriptions.
We form the composite (Hilbert) space
for any ket in
for any ket in
The composite Hilbert space has dimension
Any ket in can be written
Direct product of kets:
If in cannot be written as , then is said to be “entangled”.
FIXME: insert a concrete example of this, with some low dimension.
With operators and on the respective Hilbert spaces. We’d now like to build
If one defines
Q:Can every operator that can be defined on the composite space have a representation of this form?
Special cases. The identity operators. Suppose that
Can do other operations. Example:
Let’s verify this one. Suppose that our state has the representation
so that the action on this ket from the composite operations are
Our commutator is
Can generalize to
Can also start with and seek factor spaces. If is not prime there are, in general, many ways to find factor spaces
A ket , if unentangled in the first factor space, then it will be in general entangled in a second space. Thus ket entanglement is not a property of the ket itself, but instead is intrinsically related to the space in which it is represented.
 BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.