2017/04/30 by Pieter Naaijkens
Computer Science · Mathematics · Physics and Astronomy · #Algebraic number #Conformal field theory #Conformal map #Connection (principal bundle) #Context (archaeology) #Dimension (graph theory) #Field (mathematics) #Geometry #Mathematical analysis #Mathematical physics #Mathematics #Operator (biology) #Operator algebra #Physics #Pure mathematics #Quantum #Quantum Computing Algorithms and Architecture #Quantum and electron transport phenomena #Quantum computer #Quantum field theory #Quantum information #Quantum many-body systems #Quantum mechanics #Theoretical physics #Topological quantum field theory #Toric code #Von Neumann architecture #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1090/conm/717/14453
published as Contemporary Mathematics 717, pp. 257-279 (2018) · v2: added more background material, some corrections and clarifications. 23 pages, submitted to QMath 13 (Atlanta, GA) proceedings
openalex publication_date 2018/01/01 · arxiv created 2018/04/03 · arxiv updated 2018/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider quantum information tasks in an operator algebraic setting, where we consider normal states on von Neumann algebras. In particular, we consider subfactors <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper N subset-of German upper M"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">N</mml:mi> </mml:mrow> <mml:mo> ⊂ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">M</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak N ⊂ \mathfrak M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , that is, unital inclusions of von Neumann algebras with trivial center. One can ask the following question: given a normal state <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi> ω </mml:mi> <mml:annotation encoding="application/x-tex">ω</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , how much can one learn by only doing measurements from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ? We argue how the Jones index <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket German upper M colon German upper N right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">M</mml:mi> </mml:mrow> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">N</mml:mi> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[\mathfrak M:\mathfrak N]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be used to give a quantitative answer to this, showing how the rich theory of subfactors can be used in a quantum information context. As an example we discuss how the Jones index can be used in the context of wiretap channels. Subfactors also occur naturally in physics. Here we discuss two examples: rational conformal field theories and Kitaev’s toric code on the plane, a prototypical example of a topologically ordered model. There we can directly relate aspects of the general setting to physical properties such as the quantum dimension of the excitations. In the example of the toric code we also show how we can calculate the index via an approximation with finite dimensional systems. This explicit construction sheds more light on the connection between topological order and the Jones index.