Abstract
We define circuits given by unitary representations of Lorentzian conformal field theory in 3 and 4 dimensions. Our circuits start from a spinning primary state, allowing us to generalize formulas for the circuit complexity obtained from circuits starting from scalar primary states. These results are nicely reproduced in terms of the geometry of coadjoint orbits of the conformal group. In contrast to the complexity geometry obtained from scalar primary states, the geometry is more complicated and the existence of conjugate points, signaling the saturation of complexity, remains open.
Highlights
The connectivity of space and entanglement was described in [7]
Our circuits start from a spinning primary state, allowing us to generalize formulas for the circuit complexity obtained from circuits starting from scalar primary states
Computational complexity can be estimated by distances in the manifold of unitary operators and in this way the description of complexity is naturally couched in the language of differential geometry [16,17,18]
Summary
The matrix elements needed for an evaluation of the Fubini-Study metric can be obtained by taking appropriate derivatives of the above result. All dependence on λR− and λL− drops out and we obtain a metric defined on an 8 dimensional space with complex coordinates αμ. This space can be identified with the coset space SO(2,4)/SO(2)×SO(4) [53, 57], where we recognize the stability group of a scalar primary which is SO(2)×SO(4). The stability group of a spinning primary is SO(2)×SO(2)×SO(2) In this case the metric is defined on an 12 dimensional space with complex coordinates αμ, λR− and λL−.
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