How to evaluate the commutator of direct product?

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To evaluate the commutator of direct products of operators, the expression [A⊗B, C⊗D] is defined as (A⊗B)(C⊗D) - (C⊗D)(A⊗B). This simplifies to AC⊗BD - CA⊗DB. The resulting form highlights the interaction between the operators A, B, C, and D in the context of their tensor products. Understanding this commutator is crucial for analyzing the properties of composite systems in quantum mechanics. The discussion emphasizes the mathematical structure underlying operator algebra in this framework.
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assuming that A B C D are all n\times n operators
how to evaluate the commutator of direct product?
[A\otimes B, C\otimes D]
 
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[A\otimes B, C\otimes D] is defined as A\otimes B \cdot C\otimes D - C\otimes D \cdot A\otimes B which is AC\otimes BD - CA\otimes DB.
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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