The typicality of symmetry-induced entanglement

Abstract

In the presence of a globally conserved charge $N$, a natural question is whether a given separable state can be separated into charge-conserving components. We dub this problem the Symmetric Separability Problem (SSP). On random states, the SSP is answered negatively with probability one for almost all $N$. Using a witness to the failure of symmetric separability, namely the number entanglement (NE) introduced in arXiv:2110.09388, we show that most symmetric and separable states are actually far from being symmetrically separable, with the NE featuring Gaussian concentration around a strictly positive mean value. We discuss some consequences of our results for quantum tasks in the presence of a superselection rule or in the absence of a common reference frame. Progress is made on the question of the size of the separable space constrained by $N$. We also touch upon the question of the complexity of SSP, and multiparty entanglement.

Figures (6)

• Figure 1: Pictorial representation of some classes of states considered in this work. Class $\mathscr{D}$ contains all states on $\mathscr{H}_A \otimes \mathscr{H}_B$. A lower index $\hat{N}$ indicates symmetries, and an upper index 'sep' or 'symsep' indicates that states are separable or symmetrically separable, respectively. We algebraically show that $\mathscr{D}^{\textup{sep}}_{\hat{N}_{\textup{local}}} = \mathscr{D}^{\textup{symsep}}_{\hat{N}_{\textup{local}}} = \mathscr{D}^{\textup{symsep}}_{\hat{N}}$. For a degenerate $\hat{N}$, $\mathscr{D}^{\textup{sep}}_{\hat{N}_{\textup{local}}}$ is of measure zero in $\mathscr{D}^{\text{sep}}_{\hat{N}}$. Thus, separable and $\hat{N}$-symmetric states fail to be symmetrically separable with probability one. On the other hand, we show that $\text{vol}\mathscr{D}^{\textup{sep}}_{\hat{N}_{\textup{local}}} / \text{vol}\mathscr{D}_{\hat{N}_{\textup{local}}}$ is nonzero. As the figure suggests, $\mathscr{D}^{\text{sep}}_{\hat{N}}$ is of positive measure in $\mathscr{D}_{\hat{N}}$ when $\hat{N} = \hat{N}_A\otimes 1_B + 1_A \otimes \hat{N}_B$.
• Figure 2: Sets and maps for the construction of the purifying manifold.$i$ is a $C^{\infty}$-diffeomorphism, $j$ is $C^{\infty}$ injective, while $\pi, \tau$, and $\text{Tr}_{\mathscr{H}_A \otimes \mathscr{H}_B}$ are $C^{\infty}$ surjective, but not injective. $(\cdot)\lvert_{\text{ran } i}$ means restriction to the range of $i$. Rigorously speaking, the map $i$ is not from the sphere but from the complex projective space. This subtlety makes no difference in the argument.
• Figure 3: Distribution of NE values for randomly generated 2-qudit states in $\mathscr{D}^{\text{sep}}_{\hat{N}}$, showing concentration around the mean as dimension increases. Each subsystem $A$, $B$ supports a single qudit with $d \in \{3,...,8\}$, and $\hat{N} = \hat{N}_A \otimes 1_B + 1_A \otimes \hat{N}_B$ with $\hat{N}_A$, $\hat{N}_B$ possessing nondegenerate eigenvalues $\{0,1,...,d-1\}$, i.e. single-qudit level number on each subsystem. The distributions are fitted to an empirical chi distribution of order $k$ that depends on dimension.
• Figure 4: Convex hull of two compact convex sets belonging to mutually orthogonal spaces.$C_1\subset W$ and $C_2\subset W^{\perp}$ are compact convex sets in $W$ and $W^{\perp}$, respectively, two mutually orthogonal subspaces of $\mathbb{R}^n$. (In the figure, $W=\textup{span}\{z\}$ and $W^{\perp}=\textup{span}\{x,y\}$.) $S_1$ and $S_2$ are convex subsets of $C_1$ and $C_2$, respectively. If both $\textup{vol}\, S_1 / \textup{vol}\, C_1$ and $\textup{vol}\, S_2/\textup{vol}\, C_2$ tend to zero, so does $\textup{vol conv} (S_1\cup S_2) / \textup{vol conv} (C_1\cup C_2)$. In general, it is not enough that only one among $\textup{vol}\,S_1, \textup{vol}\,S_2$ be small.
• Figure 5: Distribution of NE values for 2-qudit states. Each subsystem $A$, $B$ supports a single qudit of dimension $d_A = d_B =d \in \{2,3,4\}$, and $\hat{N} = \hat{N}_A \otimes \hat{N}_B$ with $\hat{N}_A$, $\hat{N}_B$ possessing nondegenerate eigenvalues $\{1,2,...,d\}$, e.g. single-qudit level-number on each subsystem. The distributions are fitted to an empirical chi distribution of order $k$ that depends on dimension.

Theorems & Definitions (42)

• Proposition 1
• Proposition 2
• Proposition 3
• Corollary 1
• Lemma 1
• Lemma 2: Lévy's Lemma
• Lemma 3
• Proposition 4
• Proposition 5
• Proposition 6