2003/07/31 by Pablo Arrighi, Christophe Patricot · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Mathematics #Open quantum system #Physics #Pure mathematics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum algorithm #Quantum dynamics #Quantum information #Quantum mechanics #Quantum network #Quantum operation #Quantum process #Quantum state #Theoretical physics #hep-th #quant-ph
paper · pdf · doi:10.1016/j.aop.2003.11.005
published as Annals Phys. 311 (2004) 26-52 · Revtex, 14 pages, v4: changed citations
openalex publication_date 2003/12/22 · arxiv created 2005/09/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We formalize the correspondence between quantum states and quantum operations isometrically, and harness its consequences. This correspondence was already implicit in the various proofs of the operator sum representation of Completely Positive-preserving linear maps; we go further and show that all of the important theorems concerning quantum operations can be derived directly from those concerning quantum states. As we do so the discussion first provides an elegant and original review of the main features of quantum operations. Next (in the second half of the paper) we find more results stemming from our formulation of the correspondence. Thus we provide a factorizability condition for quantum operations, and give two novel Schmidt-type decompositions of bipartite pure states. By translating the composition law of quantum operations, we define a group structure upon the set of totally entangled states. The question whether the correspondence is merely mathematical or can be given a physical interpretation is addressed throughout the text: we provide formulae which suggest quantum states inherently define a quantum operation between two of their subsystems, and which turn out to have applications in quantum cryptography. Keywords: Kraus, CP-maps, superoperators, extremality, trace-preserving, factorizable, triangular.