# 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 http://arxiv.org/abs/quant-ph/0602130.

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

@unpublished{CKMR06,
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 http://arxiv.org/abs/quant-ph/0602130},
}