Erratum: Unknown quantum states: The quantum de Finetti representation [J. Math. Phys. 44, 4537 (2002)] Free

We defined exchangeability for probability distributions and quantum states in the following ways.

Probability distributions: A probability distribution $p(x1,x2,…,xN)$ is said to be symmetric (or finitely exchangeable) if it is invariant under permutations of its arguments, i.e., if

for any permutation π of the set ${1,…,N}$⁠. The distribution $p(x1,x2,…,xN)$ is called exchangeable (or infinitely exchangeable) if it is symmetric and if for any integer $M>0$⁠, there is a symmetric distribution $pN+M(x1,x2,…,xN+M)$ such that

Quantum states: . . . a joint state $ρ(N)$ of $N$ systems is said to be symmetric (or finitely exchangeable) if it is invariant under any permutation of the systems. . . . The state $ρ(N)$ is said to be exchangeable (or infinitely exchangeable) if it is symmetric and if, for any $M>0$⁠, there is a symmetric state $ρ(N+M)$ of $N+M$ systems such that the marginal density operator for $N$ systems is $ρ(N)$⁠, i.e.,

where the trace is taken over the additional $M$ systems.

In place of the conditions (2.3) and (3.13), we should have stated that the distributions $p(x1,x2,…,xN)$ and the quantum states $ρ(N)$ are to be considered as elements of infinite sequences, $pN(x1,x2,…,xN)$ and $ρ(N)$ for $N=1,2,…,∞$⁠, whose elements satisfy the consistency conditions that for all $N$⁠, $p(x1,x2,…,xN)$ is the marginal of $p(x1,x2,…,xN,xN+1)$ and $ρ(N)$ is the marginal of $ρ(N+1)$⁠, i.e., $ρ(N)=tr1ρ(N+1)$⁠.

The proof of the theorem in Sec. IV was formulated with the correct definition in mind and therefore remains unaffected. Thus, in Eq. (4.12), $ρ(N)$ is to be considered as part of an infinite sequence as described above. This gives rise to an infinite sequence $ρ(N)(α)$⁠, which leads to the existence of a unique probability density $P(p)$ as claimed. Equations (4.12) to (4.17) are valid for an arbitrary integer $N$⁠. The final expression (4.17) holds indeed for the entire sequence. The remainder of the proof makes explicit use of the limit $N→∞$⁠.

If one uses the incorrect wording in our original definitions, one can easily find counterexamples to the uniqueness of the measure in both the classical and quantum de Finetti representation theorems. Although the discussion in the Introduction makes clear that the statements of the theorems should concern infinite sequences, we unfortunately left that out of the formal definitions.

We thank Matthias Christandl for bringing this point to our attention.