One-and-a-Half Quantum de Finetti Theorems

Abstract

When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by ${2\frac{kd^2}{n}}$ , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation ${U_{\mu +\nu} \subset U_\mu \otimes U_\nu}$ of the unitary group U(d), for highest weights μ, ν and μ + ν. Let ξ_μ be the state obtained by tracing out U _ν. Then ξ_μ is close to a convex combination of the coherent states ${U_\mu(g)|{v_\mu\rangle}}$ , where ${g\in U(d)}$ and ${|v_\mu\rangle}$ is the highest weight vector in U _μ.

For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our “half” theorem). It arises from a combinatorial formula for the distance of certain special symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1]. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states in this set do not form a polytope in general.