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 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 Uµ+ν ⊂ Uµ ⊗ Uν of the unitary group U(d), for highest weights µ, ν and µ + ν. Let ξµ be the state obtained by tracing out Uν. Then ξµ is close to a convex combination of the coherent states Uµ(g)|vµ , where g ∈ U(d) and |vµ is the highest weight vector in Uµ.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.

Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements

“…Another open question is, in Theorem 1, for a state supported on the symmetric subspace (also known as the Bose-symmetric state), whether its reduced states have pure-state approximations of the form R φ ⊗k dμðφÞ with φ pure. We notice that this is, indeed, the case for the de Finetti theorem of [8] and a similar statement holds for [9]. However, our method, as well as that of [10], seems to require that the state φ must be generally mixed.…”

“…A series of works have established analogs of this theorem in the quantum domain [3][4][5][6][7][8][9][10], where a classical probability distribution is replaced by a quantum state and the situation is more complicated and interesting due to entanglement and the existence of many different ways to distinguish states of multipartite systems. These quantum de Finetti theorems are appealing not only due to their own elegance on the characterization of symmetric states, but also because of the successful applications in many-body physics [5,11,12], quantum information [9,13,14], and computational complexity theory [10,15,16].…”

Quantum de Finetti theorem in phase-space representation

“…This was proven in Ref. [8], where a counterexample was exhibited: the n-dimensional generalization of the singlet state 1/ √ n! π sign(π) π(|0 ⊗ |1 ⊗ · · · ⊗ |n − 1 ) is symmetric but any bipartite part, being a mixture of singlet states, cannot be approximated by a mixture of i.i.d.…”

“…Attempts at characterizing the speed of convergence towards an i.i.d. state are more recent, both in the classical case [6] and quantum case [7,8]: the trace distance between the partial trace over (n−k) parties of an n-partite symmetric state and a mixture of k-partite i.i.d. states is bounded from above by 2d 2 k/n, where d is the dimension of the Hilbert space.…”

Equilibration and prethermalization in the Bose-Hubbard and Fermi-Hubbard models

“…The hierarchy (13) is related to the quantum de Finetti theorem [82], the generalized cumulant expansion [83], and the Bogoliubov-BornGreen-Kirkwood-Yvon (BBGKY) hierarchy [84], but we are considering lattice sites instead of particles. As an example for the four-point correlator, let us consider observables µ ,B ν ,Ĉ κ , andD λ at four different lattice sites, which have vanishing on-site expectation values  µ = B ν = Ĉ κ = D λ = 0.…”