Abstract
We introduce “binding complexity”, a new notion of circuit complexity which quantifies the difficulty of distributing entanglement among multiple parties, each consisting of many local degrees of freedom. We define binding complexity of a given state as the minimal number of quantum gates that must act between parties to prepare it. To illustrate the new notion we compute it in a toy model for a scalar field theory, using certain multiparty entangled states which are analogous to configurations that are known in AdS/CFT to correspond to multiboundary wormholes. Pursuing this analogy, we show that our states can be prepared by the Euclidean path integral in (0 + 1)-dimensional quantum mechanics on graphs with wormhole-like structure. We compute the binding complexity of our states by adapting the Euler-Arnold approach to Nielsen’s geometrization of gate counting, and find a scaling with entropy that resembles a result for the interior volume of holographic multiboundary wormholes. We also compute the binding complexity of general coherent states in perturbation theory, and show that for “double-trace deformations” of the Hamiltonian the effects resemble expansion of a wormhole interior in holographic theories.
Highlights
The GHZ states are separable upon tracing out any subset of the parties, whereas the W states are not
To illustrate the new notion we compute it in a toy model for a scalar field theory, using certain multiparty entangled states which are analogous to configurations that are known in AdS/CFT to correspond to multiboundary wormholes
The “stretching” means that we include a thin region just outside the horizons in the volume computation. We address this conjecture by computing the binding complexity for a natural class of multiparty entangled states in our toy model, and showing that it has a linear dependence on entanglement entropy like the interior volume of the multiboundary wormholes of [35, 44, 45]
Summary
We will demonstrate some elementary lower bounds on binding complexity in terms of other measures of the entanglement structure of a state, such as the entanglement entropy, separability, etc. Turning to our original state ψ in (2.2), we repeatedly employ operator Schmidt decomposition to cut all the two-party gates which act across the partition HA ⊗ HB, while leaving all other gates untouched This allows us to rewrite the state in the form (see figure 3). While this upper bound is satisfied by every quantum circuit which constructs ψ from the given gate set G, the bound will be the tightest for the circuit which minimizes nAB This bound shows that the binding complexity of the state with respect to a bipartition is lower bounded by the entanglement entropy. It is clear from this expression that if we trace out A, the number of negative eigenvalues of ρΓBC will be upper bounded by the maximum allowed rank of ρBC minus one (there needs to be at least one positive eigenvalue so the trace can be one), i.e., (JGnAB+2nBC+nCA −1). The bound in equation (2.21) shows that the binding complexity is a much more fine grained probe of the entanglement structure than the entanglement entropy, and in particular is sensitive to multiparty entanglement measures such as separability
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