Relation between symmetric subspaces and $n$-exchangeable density matrices

Question

Let us consider $n$ elements, each taken from the set $\{1, 2, \ldots, d\}$ and let $S_n$ be the set of all permutations on these $n$ elements.

Define a permutation operator on the set of $n$ qudits as

$$ P_d(\pi) = \sum_{i_1, i_2, \ldots, i_n \in [d]} |i_{\pi(1)}, \ldots, i_{\pi(n)}\rangle \langle i_1, \ldots i_n|. $$

Define the symmetric subspace on $n$ qudits as \begin{equation} V^{n}(\mathbb{C}^{d}) = \text{span}\{|\phi\rangle \in (\mathbb{C}^{d})^{\otimes n} : P_d(\pi)|\phi\rangle = |\phi\rangle~~\text{for all}~~\pi \in S_n \}. \end{equation}

Define an $n$ exchangeable density matrix $\rho \in \text{Density}\bigg(\big(\mathbb{C}^{d}\big)^{\otimes n}\bigg)$ as

$$ P_d(\pi) \rho P_d(\pi) = \rho, ~\text{for all} ~\pi \in S_n. $$

The notations are borrowed from here.

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It is easy to see that any linear combination like

$$ \rho = \sum_i \alpha_i |\phi_i\rangle\langle \phi_i|, $$ where each $|\phi_i\rangle \in V^{n}(\mathbb{C}^{d})$ and the $\alpha_i$s are arbitrary complex numbers is an $n$ exchangeable state.

I could not prove the converse, however. Is it true that any $n$ exchangeable state is a linear combination of density matrices of states in the symmetric subspace? If not, what is an example to the contrary?

Answer 1

The answer is: no, it is not true that any $n$ exchangeable state is a linear combination of density matrices of states in the symmetric subspace (that is supported on the symmetric subspace). Actually, there are even pure state counterexamples when $n=2$. Consider the state

$$ \rho = |\phi\rangle\langle \phi|, $$

where

$$ |\phi\rangle = \frac{1}{\sqrt{2}}(|0 \rangle \otimes |1 \rangle - |1 \rangle \otimes |0 \rangle). $$

In this case the permutation group is simply $S_2$. The action of the trivial permutation obviously maps $|\phi\rangle$ to $|\phi\rangle$; while for the nontrivial permutation $\pi=(12)$ (which is simply the swap operation), we have $P_{\pi} |\phi\rangle=-|\phi\rangle$. This means that $P_{\pi}|\phi\rangle \langle \phi|P_{\pi} = (-1)^2 |\phi\rangle \langle \phi|= |\phi\rangle \langle \phi| $, which implies that (according to your definition) $\rho$ is exchangeable. However, $\rho$ is clearly not supported on the symmetric subspace (it is actually supported on the antisymmetric subspace), thus it cannot be written as a linear combination of density matrices of states in the symmetric subspace.

Also in the $n > 2$ case there are states that are not supported on the symmetric subspace (i.e., their density matrices cannot be written as the requested linear combination in the question). However, when $n >d$ (i.e., when there is no totally antisymmetric subspace) these states must be mixed, and in this case (when $n>d$) every $n$-exchangeable pure state is supported on the symmetric subspace.