Information Security and Cryptography Research Group

One-and-a-half quantum de Finetti theorems

Matthias Christandl, Robert Koenig, Graeme Mitchison, and Renato Renner

Feb 2006, Available at

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 U_{mu+nu} for Young diagrams mu and nu. U_{mu+nu} is contained in the tensor product of U_mu and U_nu; let xi be the state obtained by tracing out U_nu. 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 U_mu. When U_{mu+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.

BibTeX Citation

    author       = {Matthias Christandl and Robert Koenig and Graeme Mitchison and Renato Renner},
    title        = {One-and-a-half quantum de {F}inetti theorems},
    year         = {2006},
    month        = {2},
    note         = {Available at},

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