Generalized volume-complexity for two-sided hyperscaling violating black branes

Abstract

In this paper, we investigate generalized volume-complexity $$ \mathcal{C} $$ C gen for a two-sided uncharged HV black brane in d + 2 dimensions. This quantity which was recently introduced in [48], is an extension of volume in the Complexity=Volume (CV) proposal, by adding higher curvature corrections with a coupling constant λ to the volume functional. We numerically calculate the growth rate of $$ \mathcal{C} $$ C gen for different values of the hyperscaling violation exponent θ and dynamical exponent z. It is observed that $$ \mathcal{C} $$ C gen always grows linearly at late times provided that we choose λ properly. Moreover, it approaches its late time value from below. For the case λ = 0, we find an analytic expression for the late time growth rate for arbitrary values of θ and z. However, for λ ≠ 0, the late time growth rate can only be calculated analytically for some specific values of θ and z. We also examine the dependence of the growth rate on d, θ, z and λ. Furthermore, we calculate the complexity of formation obtained from volume-complexity and show that it is not UV divergent. We also examine its dependence on the thermal entropy and temperature of the black brane. At the end, we also numerically calculate the growth rate of $$ \mathcal{C} $$ C gen for the case where the higher curvature corrections are a linear combination of the Ricci scalar, square of the Ricci tensor and square of the Riemann tensor. We show that for appropriate values of the coupling constants, the late time growth rate is again linear.

Geometric Aspects of Mixed Quantum States Inside the Bloch Sphere

“…with ds 2 Sj öqvist in Equation (10). In Equation ( 35), r ′ def = dr/dθ and L(r ′ , r, θ) is the Lagrangianlike function defined as L r ′ , r,…”

“…For this reason, there is a certain degree of arbitrariness in selecting a metric when characterizing the physical aspects of quantum states in mixed states. In particular, this freedom can cause metric-dependent explanations of geometric quantities with a clear physical significance, including the complexity [2][3][4][5][6][7][8][9][10] and volume [11][12][13][14][15][16][17][18][19][20][21][22] of quantum states. Two examples of metrics for mixed quantum states are the Bures [23][24][25][26] and the Sjöqvist [27] metrics.…”

Complexity=anything: singularity probes

“…This explicit example was carefully examined in appendix A of both [1,2]. See also [16,17] for the studies of this functional (3.1) in other spacetime backgrounds. We can proceed to evaluate the profile of the extremal surfaces as described above in section 2.1.…”

“…More on circuit complexity. Of course, the key challenge for the complexity=anything proposal is to translate these interesting bulk observables into observables in the boundary theory, 16 and thus establish a dictionary between the geometry of the black hole interior…”

“…For further investigations of how the complexity=action proposal detects and probes black hole singularities, see also[39][40][41][42][43] 16. For example, see[54,55] for probing the singularity inside AdS black holes using geodesics.…”