Entanglement in cyclic sign invariant quantum states

TL;DR

This work introduces Local Cyclic Sign Invariant (LCSI) quantum states, a symmetry-driven class that sits between full unitary/orthogonal-invariant families and diagonal-invariant subspaces. By parameterizing LCSI states with a triple of circulant vectors $(a,b,c)$ and leveraging the discrete Fourier transform, the authors provide simple, sharp criteria for positivity, PPT, and separability, culminating in Circulant Triplewise Completely Positive (TCP) conditions that exactly characterize separability within this class. They apply these tools to cyclic mixtures of Dicke states, proving PPT entanglement exists for all dimensions $d\ge 5$, and giving complete or partial descriptions of the PPT and separable cones up to $d=5$ (with partial results for $d=6,7$). A new PPT-entangled family $X_{A,C}$ is presented, with PPT/separability linked to correlation structures and copositive/circulant-cone duals, expanding the toolkit for identifying entanglement under symmetry constraints. Overall, the paper reveals a rich entanglement structure within highly symmetric bosonic subspaces and suggests efficient geometric and algebraic avenues for entanglement detection in symmetry-reduced quantum state spaces.

Abstract

We introduce and study bipartite quantum states that are invariant under the local action of the cyclic sign group. Due to symmetry, these states are sparse and can be parameterized by a triple of vectors. Their important semi-definite properties, such as positivity and positivity under partial transpose (PPT), can be simply characterized in terms of these vectors and their discrete Fourier transforms. We study in detail the entanglement properties of this family of symmetric states, showing that it contains PPT entangled states. For states that are diagonal in the Dicke basis, deciding separability is equivalent to a circulant version of the complete positivity problem. In local dimension d <= 5, we completely characterize these sets and construct entanglement witnesses; some partial results are also obtained for d = 6, 7. We construct a new family of states for which the properties of PPT and separability can be characterized for all dimensions, generalizing results from the literature. Our results show that this class of symmetric states has a rich entanglement structure, even in the bosonic subspace.

Figures (13)

• Figure 1: The slice $(2,x,y,x)$ through the cones $\overrightarrow{\mathsf{DNN}}_{4}=\overrightarrow{\mathsf{CP}}_{4}$. In blue, $10^4$ randomly generated points inside $\overrightarrow{\mathsf{CP}}_{4}$; in red, the 4 extreme points of $\overrightarrow{\mathsf{DNN}}_{4}$ from \ref{['sec:facets-circDNN']} and the polyhedral slice they generate
• Figure 2: The slice $(1,x,y,y,x)$ through the cones $\overrightarrow{\mathsf{SPN}}_{5} \subsetneq \overrightarrow{\mathsf{COP}}_{5}$. The red dots are the 2 extreme points $(1,-\cos(\pi/5), \cos(2\pi/5),\cos(2\pi/5),-\cos(\pi/5))$, $(1,\cos(2\pi/5), -\cos(\pi/5),-\cos(\pi/5),\cos(2\pi/5))$ of $\overrightarrow{\mathsf{SPN}}_{5}$ belonging to this slice. The red dashed lines correspond to the boundary of the (unbounded) slice. Solid green curves correspond to the extremal rays of $\overrightarrow{\mathsf{COP}}_{5}$ belonging to this slice: $h_\theta$ and $h'_\theta$ for $\theta \in [0,\pi/5]$. Dashed green lines depict the boundary of $\overrightarrow{\mathsf{COP}}_{5}$.
• Figure 3: The slice $(2,x,y,y,x)$ through the cones $\overrightarrow{\mathsf{CP}}_{5} \subsetneq \overrightarrow{\mathsf{DNN}}_{5}$. In blue, $10^4$ randomly generated points inside $\overrightarrow{\mathsf{CP}}_{5}$. In red, the 4 extreme points of $\overrightarrow{\mathsf{DNN}}_{5}$ from \ref{['sec:facets-circDNN']} and the polyhedral slice they generate. Solid green curves correspond to the extremal rays $x_\theta$ and $x'_\theta$ for $\theta \in [0,\pi/5]$, while the three green points are the extremal rays in the directions $(1,0,0,0,0)$, $(1,1,1,1,1)$, $(2,1,0,0,1)$, $(2,0,1,1,0)$. Dashed green lines are non-extremal elements of $\overrightarrow{\mathsf{CP}}_{5}$ corresponding to $x_\theta$ and $x'_\theta$ in the parameter range $\theta \in (\pi/5,\pi/2]$. The two brown lines fill in the missing (non-extremal) part of the boundary of $\overrightarrow{\mathsf{CP}}_{5}$.
• Figure 4: The slice $(2,x,y,0,y,x)$ through the cones $\overrightarrow{\mathsf{CP}}_{6}$ and $\overrightarrow{\mathsf{DNN}}_{6}$. In blue, randomly generated points inside the $\overrightarrow{\mathsf{CP}}_{6}$ cone. In red, the four extreme points of $\overrightarrow{\mathsf{DNN}}_{6}$ on this face, and the polytope they generate. The solid green curve is the continuous family of extremal points, while the four other green points are also extremal. The $\mathsf{PPT}$ entangled states are present in the small "corner" formed by the continuous family extremal rays of $\overrightarrow{\mathsf{CP}}_{6}$ and the point $(2,3/2,1/2,0,1/2,3/2)$ (which is the only extremal $\overrightarrow{\mathsf{DNN}}_{6}$ point that is not in $\overrightarrow{\mathsf{CP}}_{6}).$
• Figure 5: The slice $(2,x,y,0,0,y,x)$ through the cones $\overrightarrow{\mathsf{CP}}_{7}$ (corresponding to separable matrices) and $\overrightarrow{\mathsf{DNN}}_{7}$ (corresponding to $\mathsf{PPT}$ matrices). In blue, randomly generated points inside the face of the $\overrightarrow{\mathsf{CP}}_{7}$ cone. In red, the four extreme points of $\overrightarrow{\mathsf{DNN}}_{7}$ on this face, and the polytope they generate. The solid green curve is the continuous family of extremal points $x_\theta^{(3)}, \theta \in [0, \pi/2]$, while the three other green points (big green circles) are also extremal. All the extremal $\overrightarrow{\mathsf{DNN}}_{d}$ states (red dots) except $\ket{0}$ are not in $\overrightarrow{\mathsf{CP}}_{d}$. For $\overrightarrow{\mathsf{DNN}}_{7}$ and $\overrightarrow{\mathsf{CP}}_{7}$, all the other slices $(2,0,x,y,y,x,0)$ and $(2,y,0,x,x,0,y)$ are identical to the one above, up to permutation of the coordinates.

Theorems & Definitions (100)

• Definition 2.1
• Definition 2.2: Separability Problem
• Definition 2.3
• Remark 2.4
• Proposition 2.5: davis1979circulant
• proof
• Remark 2.6
• Definition 2.7
• Definition 2.8
• Definition 2.9