Bundling of bipartite entanglement

TL;DR

The paper reveals a bundling phenomenon in which entanglement spectra across many bipartitions coincide when the system state is constrained to a subspace defined by embeddings such as parity or minor. It provides a rigorous framework based on an equivalence relation over subsystems, proving that equal spectra hold for all states in the constrained subspace whenever two subsystems are equivalent under this relation. It further develops an operator-based formulation and shows that, in parity-embedded scenarios, equivalence verification can be done in polynomial time, with concrete algorithms and bundle classifications demonstrated for embedding-based quantum optimization. The work offers practical implications for efficiently characterizing and measuring entanglement in quantum optimization devices, and it extends to mixed states and various embedding schemes, broadening the applicability of entanglement-based analyses in quantum many-body physics.

Abstract

We investigate bipartite entanglement and prove that in constrained energy subspaces, the entanglement spectra of multiple bipartitions are the same across the whole subspace. We show that in quantum many-body systems the bipartite entanglement entropy is affected in such a way that it forms "bundles" under unitary time evolution. Leveraging the structure of the subspace, we present methods to verify whether the entanglement spectrum of two bipartitions is identical throughout the entire subspace. For the subspace defined by the parity embedding, we further provide an algorithm that can determine this in polynomial time.

Figures (5)

• Figure 1: Dynamics of von Neumann entanglement entropy $S_{A}(s)$ are presented in the top row for three different annealing processes with linear schedule and final annealing times (a)${t_f=800}$, (b)${t_f=100}$ and (c)${t_f=11}$ of a system with ten physical qubits laid out on a square lattice realizing a parity embedding of a complete graph with five vertices. Here, we used the parity constraint strengths in (a)-(b)${C=4}$ and in (c)${C=1}$. Each process presents the entanglement dynamics of in total $266$ different unordered bipartitions $\left\lbrace A, A^c\right\rbrace$ of sizes ${\left\lvert A\right\rvert \in \left\lbrace 2,3, 4, 5\right\rbrace}$. Figure \ref{['fig:GroupEntanglementDynamics']}(d) illustrates a few of the bipartitions of the physical qubits in a square lattice, which belong to one of the five visible bundles in Fig. \ref{['fig:GroupEntanglementDynamics']}(b). The ten physical qubits are presented with circles labeled with the physical indices from $1$ to $10$. Each triangle represents one possible bipartition whose qubits are filled with the color of its associated bundle's entanglement entropy. Additionally, the bipartitions corresponding to the same final entanglement value are framed by dashed lines in the same color. On the far right, two examples of the lowest bundle (violet) can be seen, in the middle one example of the second lowest (green-blue), and on the left four examples of the bundle in the middle (orange). All the smaller drawn bipartitions are subsets of the larger bipartition $\left\lbrace A, A^c\right\rbrace$ drawn on the far left.
• Figure 2: Bundles and dynamics of the von Neumann entanglement entropy $S_{A}(s)$ for the example with nine physical qubits given in constraintpaper. As sum constraint we chose ${\langle\tilde{\sigma}_{z}^{(3,4)} + \tilde{\sigma}_{z}^{(1,4)} + \tilde{\sigma}_{z}^{(1,6)} \rangle = 3}$ and the initial state is set to ${\ket{\Psi(0)} = \ket{0 1 1 1 1 1 0 1 0}}$. Figure \ref{['fig:GroupEntanglementDynamicsConstrainedPaper']} (a) illustrates the entanglement bundles containing all $255$ bipartitions $\left\lbrace A,A^c\right\rbrace$ of size ${\left\lvert A\right\rvert\in\left\lbrace 1,2,3,4\right\rbrace}$. Each bar corresponds to one entanglement bundle enumerated from $0$ to $15$ where the bundles are sorted by their maximal entanglement entropy value in ascending order. The $y$-axis shows the number of bipartitions of each bundle and the colors correspond to the size of $A$. Figure \ref{['fig:GroupEntanglementDynamicsConstrainedPaper']} (b) shows the dynamics of the von Neumann entanglement entropy of all $255$ unordered bipartitions. The final annealing time of the adiabatic annealing process is ${t_f=500}$. Further details of the annealing process and the driver Hamiltonian are available in constraintpaper.
• Figure 3: Parity encoding applied to the optimization problem used for the numerical simulations in Fig. \ref{['fig:GroupEntanglementDynamics']}. The left graph illustrates the initial optimization problem, the middle graph the corresponding logical graph and the right picture the physical graph embedded by the parity encoding. The logical spins and the physical qubits are presented by black cycles. The logical vertices are labeled by indices ${0,1,2,3,4}$ inside the cycles and the vertices of the physical graph are labeled in red above the cycles with indices ${1, \ldots, 10}$. The indices inside the vertices of the physical graph indicate the corresponding edges of the logical problem graph. The local fields of the Ising Hamiltonian are presented by interactions with a logical ancilla spin with index $0$. The red triangles (blue squares) are three-body (four-body) parity constraints. The logical line of vertex with index $3$ is marked in green.
• Figure 4: Bundles of von Neumann entanglement entropy. On the left, all numerically detected entanglement entropy bundles for all $511$ bipartitions for the first numerical example in Sec. \ref{['sec:numVis']} are represented with bars. The bundles are enumerated from $0$ to $50$ along the $x$-axis and sorted by their maximal entanglement entropy value in ascending order. The $y$-axis shows the number of bipartitions of each bundle. The different colors of the bars indicate the number of bipartitions of varying size $\left\lvert A\right\rvert$. On the right, a zoomed-in view of the first $50$ bundles from the left figure is shown.
• Figure 5: Entanglement spectrum dynamics for all bipartitions of the largest bipartition class given in the example at the beginning of Sec. \ref{['sec:numVis']}, which contains all bipartitions of the highest entropy bundle in Fig. \ref{['fig:GroupEntanglementDynamics']}(b). We present the entanglement spectra for all elements of this class for each of the different annealing process Fig. \ref{['fig:entanglementSpectraLargestClass']}(a) (adiabatic and restricted to $\Pi$), Fig. \ref{['fig:entanglementSpectraLargestClass']}(b) (non-adiabatic and restricted to $\Pi$) and Fig. \ref{['fig:entanglementSpectraLargestClass']}(c) (non-adiabatic and not restricted to the $\Pi$).

Theorems & Definitions (45)

• Definition 3.1
• Definition 3.2
• Theorem 1
• Definition 4.1: Generator set
• Definition 4.2: Bipartite operator set
• Theorem 2
• Lemma 1
• Lemma 1
• Theorem 3
• Lemma 2