Tensor invariants for multipartite entanglement classification

Abstract

Organising the space of entanglement structures of a multipartite quantum system is a much more challenging task than its bipartite version: while the local unitary (LU) orbit of a bipartite pure state can be conveniently characterized by its entanglement spectrum, invariants of multipartite entanglement structures are comparatively difficult to define and work with. The root cause of this difference is that the bipartite problem can be reduced to the analysis of matrix invariants, while its multipartite version is governed by a much richer space of tensor invariants. The present work explores the latter through the lens of so-called trace-invariants, which are in one-to-one correspondence with combinatorial objects known as colored graphs. We first explain why trace-invariant evaluations can serve as labels of LU-orbits of multipartite pure states, how this strategy extends to random states, and how the effect of local operations (LO) can be analyzed through such data. We then focus on entanglement classification within an (infinite-dimensional) subspace of reference states, whose basic building blocks are GHZ states of various dimensions. We show that relatively simple subclasses of trace-invariants are sufficient to separate the LU-orbits of reference states, and enable a complete (resp. an incomplete) characterization of their relations in the LO (resp. LOCC) resource theory of entanglement. Finally, we investigate how a (still infinite) subclass of reference states of local dimension N can be efficiently distinguished at leading and subleading orders in an asymptotic large-N expansion (among themselves, or from Haar-random states). This analysis relies crucially on combinatorial quantities associated to colored graphs, some of which have already played instrumental roles in the recent literature on random tensors. Results of broader relevance are reported along the way.

Figures (49)

• Figure 1: Graphical representation of $\psi$ and $\overline{\psi}$.
• Figure 2: Left: a graph in $\mathcal{G}_2^{\mathop{\mathrm{c}}\nolimits}$. Right: a graph in $\mathcal{G}_4^{\mathop{\mathrm{c}}\nolimits}$.
• Figure 3: A graph in $\mathcal{G}_3^{\mathop{\mathrm{c}}\nolimits}$ and corresponding trace-invariant $\mathop{\mathrm{Tr}}\nolimits_G$ evaluated on a state $\left\vert \psi \right\rangle$.
• Figure 4: Graphical representation of equation \ref{['eq:proof_sep_ev-eq']}. Multi-indices associated with the color set $B_1$ are represented in red, while those associated with $B_2$ are represented in green. Both $U$ and $V$ are unitary operators from $(\mathcal{H}^\mathsf{F})^{\otimes (k-1)}$ to $\mathcal{H}_{B_2} \otimes \left( \mathcal{H}_{B_1}\otimes (\mathcal{H}^{\mathsf{F}})^{\otimes (k-2)}\right)$, and multi-indices associated with the tensor factor $\mathcal{H}_{B_1}\otimes (\mathcal{H}^{\mathsf{F}})^{\otimes (k-2)}$ are represented in blue. Note that $U$ depends on the permutations $\sigma_2,\ldots, \sigma_D$, while $V$ does not. Any entanglement of $\left\vert \psi \right\rangle$ relative to the bipartition $\mathcal{H}^\mathsf{F}\simeq \mathcal{H}_{B_1}\otimes \mathcal{H}_{B_2}$ necessarily generates entanglement of the state appearing on the left-hand side relative to the bipartition $\mathcal{H}^\mathsf{F} \otimes (\mathcal{H}^\mathsf{F})^{\otimes(k-1)}$ (which is represented by the dotted line).
• Figure 5: On the left, a visual representation of a Bell pair shared between parts $1$ and $2$. On the middle left, the so-called triangle state (see Ref. Iizuka:2025caq) for a tripartite system $1,2$ and $3$. On the middle right and right, a generalizatiof the triangle state to a quantum system made up of $4$ and $6$ subsystems.

Theorems & Definitions (147)

• Lemma 2.1
• proof
• Definition 2.2
• Corollary 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Proposition 2.6
• proof
• Proposition 2.7