The Complexity of Being Entangled
- URL: http://arxiv.org/abs/2311.04277v1
- Date: Tue, 7 Nov 2023 19:00:02 GMT
- Title: The Complexity of Being Entangled
- Authors: Stefano Baiguera, Shira Chapman, Giuseppe Policastro and Tal
Schwartzman
- Abstract summary: Nielsen's approach to quantum state complexity relates the minimal number of quantum gates required to prepare a state to the length of geodesics computed with a certain norm on the manifold of unitary transformations.
For a bipartite system, we investigate binding complexity, which corresponds to norms in which gates acting on a single subsystem are free of cost.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Nielsen's approach to quantum state complexity relates the minimal number of
quantum gates required to prepare a state to the length of geodesics computed
with a certain norm on the manifold of unitary transformations. For a bipartite
system, we investigate binding complexity, which corresponds to norms in which
gates acting on a single subsystem are free of cost. We reduce the problem to
the study of geodesics on the manifold of Schmidt coefficients, equipped with
an appropriate metric. Binding complexity is closely related to other
quantities such as distributed computing and quantum communication complexity,
and has a proposed holographic dual in the context of AdS/CFT. For finite
dimensional systems with a Riemannian norm, we find an exact relation between
binding complexity and the minimal R\'enyi entropy. We also find analytic
results for the most commonly used non-Riemannian norm (the so-called $F_1$
norm) and provide lower bounds for the associated notion of state complexity
ubiquitous in quantum computation and holography. We argue that our results are
valid for a large class of penalty factors assigned to generators acting across
the subsystems. We demonstrate that our results can be borrowed to study the
usual complexity (not-binding) for a single spin for the case of the $F_1$ norm
which was previously lacking from the literature. Finally, we derive bounds for
multi-partite binding complexities and the related (continuous) circuit
complexity where the circuit contains at most $2$-local interactions.
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