Trace Inequalities and Quantum Entropy: An
... For now, we mention that strong subadditivity of the entropy is an example of a trace inequality for density matrices acting on a tensor product of Hilbert spaces – H1 ⊗ H2 ⊗ H3 in this case – that involves partial traces. The different Hilbert spaces correspond to different parts of the quantum mec ...
... For now, we mention that strong subadditivity of the entropy is an example of a trace inequality for density matrices acting on a tensor product of Hilbert spaces – H1 ⊗ H2 ⊗ H3 in this case – that involves partial traces. The different Hilbert spaces correspond to different parts of the quantum mec ...
Lectures on Modules over Principal Ideal Domains
... Therefore πi (N ) is non zero ideal in R. Thus it is free of rank 1. Also, ker πi ∩ N is a submodule of ker πi . By induction hypothesis, rank of ker πi ∩ N is ≤ n − 1. Let α be a generator for πi (N ) and v ∈ N be any element such that πi (v) = α. It is an easy exercise to show that N = ker πi ∩ N ...
... Therefore πi (N ) is non zero ideal in R. Thus it is free of rank 1. Also, ker πi ∩ N is a submodule of ker πi . By induction hypothesis, rank of ker πi ∩ N is ≤ n − 1. Let α be a generator for πi (N ) and v ∈ N be any element such that πi (v) = α. It is an easy exercise to show that N = ker πi ∩ N ...
