General bounds on holographic complexity

Abstract

We prove a positive volume theorem for asymptotically AdS spacetimes: the maximal volume slice has nonnegative vacuum-subtracted volume, and the vacuum-subtracted volume vanishes if and only if the spacetime is identically pure AdS. Under the Complexity=Volume proposal, this constitutes a positive holographic complexity theorem. The result features a number of parallels with the positive energy theorem, including the assumption of an energy condition that excludes false vacuum decay (the AdS weak energy condition). Our proof is rigorously established in broad generality in four bulk dimensions, and we provide strong evidence in favor of a generalization to arbitrary dimensions. Our techniques also yield a holographic proof of Lloyd’s bound for a class of bulk spacetimes. We further establish a partial rigidity result for wormholes: wormholes with a given throat size are more complex than AdS-Schwarzschild with the same throat size.

Holographic complexity and de Sitter space

“…Although the volume is always positive, note that the expression (6.4) can become negative. This contrasts with the theorem proven in [84] for asymptotically AdS spacetimes in four dimensions. It would be interesting to use our example to find limitations on possible generalizations of this theorem to different dimensions.…”

Complexity of Scalar Collapse in Anti-de Sitter Spacetime

“…If it holds as the amplitude vanishes, the scaling of complexity fluctuations allows us to comment on the validity of Lloyd's bound, |dC/dτ | M, which [44] proved recently for volume complexity with = 1. 3 In particular, the validity of Lloyd's bound depends on the reference scale chosen in the definition of C V .…”

Quantum Computational Complexity -- From Quantum Information to Black Holes and Back

“…In all these examples the complexity of formation is non-negative. This property was proven in general in asymptotically AdS spaces in d = 3 and in some symmetric spaces in other dimensions in [95].…”

“…This however would still be compatible with a putative bound given by 2T S. In fact a counterexample was given already in the initial paper [14,15]: the bound is violated for large charged black holes 41 and this violation is most pronounced close to extremality, but in general such black holes are unstable to the emission of light charged particles. Recently a version of the holographic Lloyd's bound was proven for the case of CV: it was shown [95] that under certain energy conditions, in asymptotically-AdS spacetimes in d ≥ 3, the rate of growth of C V is bounded by 8πM d−1 f (M ), where f (M ) is a function equal to 1 for M ≤ M with M a mass scale near the Hawking-Page transition, and f (M ) = 1+2(M/ M ) 1/(d−2) for M > M . We will comment further on the bounds on the rate of computation in the discussion section.…”