Kavli Affiliate: Gang Su
| First 5 Authors: Shuo Qi, Shuo Qi, , ,
| Summary:
Multipartite entanglement offers a powerful framework for understanding the
complex collective phenomena in quantum many-body systems that are often beyond
the description of conventional bipartite entanglement measures. Here, we
propose a class of multipartite entanglement measures that incorporate the
matrix product state (MPS) representation, enabling the characterization of the
optimality of quantum circuits for state preparation. These measures are
defined as the minimal distances from a target state to the manifolds of MPSs
with specified virtual bond dimensions $chi$, and thus are dubbed as
$chi$-specified matrix product entanglement ($chi$-MPE). We demonstrate
superlinear, linear, and sublinear scaling behaviors of $chi$-MPE with respect
to the negative logarithmic fidelity $F$ in state preparation, which correspond
to excessive, optimal, and insufficient circuit depth $D$ for preparing
$chi$-virtual-dimensional MPSs, respectively. Specifically, a linearly-growing
$chi$-MPE with $F$ suggests $mathcalH_chi simeq mathcalH_D$, where
$mathcalH_chi$ denotes the manifold of the $chi$-virtual-dimensional
MPSs and $mathcalH_D$ denotes that of the states accessible by the $D$
layer circuits. We provide an exact proof that $mathcalH_chi=2 equiv
mathcalH_D=1$. Our results establish tensor networks as a powerful and
general tool for developing parametrized measures of multipartite entanglement.
The matrix product form adopted in $chi$-MPE can be readily extended to other
tensor network ans"atze, whose scaling behaviors are expected to assess the
optimality of quantum circuit in preparing the corresponding tensor network
states.
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