2006/02/28 by Matthias Christandl, Robert Koenig, Robert König +2 · 6 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Combinatorics #Connection (principal bundle) #Discrete mathematics #Invariant (physics) #Mathematical physics #Mathematics #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #State (computer science) #Symmetric group #quant-ph
paper · pdf · doi:10.1007/s00220-007-0189-3
published as Comm. Math. Phys., 273 (2), 473-498, (2007) · 14 pages, 3 figures, v4: minor additions (including figures), published version
openalex publication_date 2007/03/12 · arxiv created 2008/10/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We prove a new kind of quantum de Finetti theorem for representations of the unitary group U(d). Consider a pure state that lies in the irreducible representation Umu+nu for Young diagrams mu and nu. Umu+nu is contained in the tensor product of Umu and Unu; let xi be the state obtained by tracing out Unu. We show that xi is close to a convex combination of states Uv, where U is in U(d) and v is the highest weight vector in Umu. When Umu+nu is the symmetric representation, this yields the conventional quantum de Finetti theorem for symmetric states, and our method of proof gives near-optimal bounds for the approximation of xi by a convex combination of product states. For the class of symmetric Werner states, we give a second de Finetti-style theorem (our 'half' theorem); the de Finetti-approximation in this case takes a particularly simple form, involving only product states with a fixed spectrum. Our proof uses purely group theoretic methods, and makes a link with the shifted Schur functions. It also provides some useful examples, and gives some insight into the structure of the set of convex combinations of product states.