In this paper, we argue that holographic complexity should be a basis-dependent quantity. Computational complexity of a state is defined as a minimum number of gates required to obtain that state from the reference state. Due to this minimality, it satisfies the triangle inequality and can be regarded as a (discrete version of) distance in the Hilbert space. However, we show a no-go theorem that any basis-independent distance cannot reproduce the behavior of the holographic complexity. Therefore, if holographic complexity is dual to a distance in the Hilbert space, it should be basis dependent; i.e., it is not invariant under a change of the basis of the Hilbert space.

• Received 21 May 2018

DOI:https://doi.org/10.1103/PhysRevD.98.046002

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Published by the American Physical Society