We study information retrieval from evaporating black holes, assuming that the internal dynamics of a black hole is unitary and rapidly mixing, and assuming that the retriever has unlimited control over the emitted Hawking radiation. If the evaporation of the black hole has already proceeded past the ``half-way'' point, where half of the initial entropy has been radiated away, then additional quantum information deposited in the black hole is revealed in the Hawking radiation very rapidly. Information deposited prior to the half-way point remains concealed until the half-way point, and then emerges quickly. These conclusions hold because typical local quantum circuits are efficient encoders for quantum error-correcting codes that nearly achieve the capacity of the quantum erasure channel. Our estimate of a black hole's information retention time, based on speculative dynamical assumptions, is just barely compatible with the black hole complementarity hypothesis.

http://absimage.aps.org/image/MAR08/MWS_MAR08-2007-003180.pdf

Black holes as mirrors: quantum information in random subsystems

We examine the circuit complexity of coherent states in a free scalar field theory, applying Nielsen’s geometric approach as in [1]. The complexity of the coherent states have the same UV divergences as the vacuum state complexity and so we consider the finite increase of the complexity of these states over the vacuum state. One observation is that generally, the optimal circuits introduce entanglement between the normal modes at intermediate stages even though our reference state and target states are not entangled in this basis. We also compare our results from Nielsen’s approach with those found using the Fubini-Study method of [2]. For general coherent states, we find that the complexities, as well as the optimal circuits, derived from these two approaches, are different.

/pdf/circuit-complexity-for-coherent-states-qt79jfp6e8.pdf

Circuit Complexity for Coherent States

Motivated by holographic complexity proposals as novel probes of black hole spacetimes, we explore circuit complexity for thermofield double (TFD) states in free scalar quantum field theories using the Nielsen approach. For TFD states at t = 0, we show that the complexity of formation is proportional to the thermodynamic entropy, in qualitative agreement with holographic complexity proposals. For TFD states at t > 0, we demonstrate that the complexity evolves in time and saturates after a time of the order of the inverse temperature. The latter feature, which is in contrast with the results of holographic proposals, is due to the Gaussian nature of the TFD state of the free bosonic QFT. A novel technical aspect of our work is framing complexity calculations in the language of covariance matrices and the associated symplectic transformations, which provide a natural language for dealing with Gaussian states. Furthermore, for free QFTs in 1+1 dimension, we compare the dynamics of circuit complexity with the time dependence of the entanglement entropy for simple bipartitions of TFDs. We relate our results for the entanglement entropy to previous studies on non-equilibrium entanglement evolution following quenches. We also present a new analytic derivation of a logarithmic contribution due to the zero momentum mode in the limit of vanishing mass for a subsystem containing a single degree of freedom on each side of the TFD and argue why a similar logarithmic growth should be present for larger subsystems.

/pdf/complexity-and-entanglement-for-thermofield-double-states-2vxq6e3337.pdf

Complexity and entanglement for thermofield double states

We apply the recently developed notion of complexity for field theory to a quantum quench through a critical point in 1+1 dimensions. We begin with a toy model consisting of a quantum harmonic oscillator, and show that complexity exhibits universal scalings in both the slow and fast quench regimes. We then generalize our results to a one-dimensional harmonic chain, and show that preservation of these scaling behaviors in free field theory depends on the choice of norm. Applying our setup to the case of two oscillators, we quantify the complexity of purification associated with a subregion, and demonstrate that complexity is capable of probing features to which the entanglement entropy is insensitive. We find that the complexity of subregions is subadditive, and comment on potential implications for holography.

Complexity as a Novel Probe of Quantum Quenches: Universal Scalings and Purifications

We revisit the complexity = action proposal for charged black holes. We investigate the complexity for a dyonic black hole, and we find the surprising feature that the late-time growth is sensitive to the ratio between electric and magnetic charges. In particular, the late-time growth rate vanishes when the black hole carries only a magnetic charge. If the dyonic black hole is perturbed by a light shock wave, a similar feature appears for the switchback effect, e.g. it is absent for purely magnetic black holes. We then show how the inclusion of a surface term to the action can put the electric and magnetic charges on an equal footing, or more generally change the value of the late-time growt rate. Next, we investigate how the causal structure influences the late-time growth with and without the surface term for charged black holes in a family of Einstein-Maxwell-Dilaton theories. Finally, we connect the previous discussion to the complexity=action proposal for the two-dimensional Jackiw-Teitelboim theory. Since the two-dimensional theory is obtained by a dimensional reduction from Einstein-Maxwell theory in higher dimensions in a near-extremal and near-horizon limit, the choices of parent action and parent background solution determine the behaviour of holographic complexity in two dimensions.

/pdf/holographic-complexity-equals-which-action-1rg2gfm93x.pdf

Holographic complexity equals which action

In this paper, we investigate the action growth in various backgrounds in Einstein-Maxwell-dilaton theory. We calculate the full time evolution of action growth in the anti–de Sitter dilaton black hole and find it approaches the late time bound from above. We investigate the black hole which is asymptotically Lifshitz and obtain its late time and full time behavior. We find the violation of Lloyd bound in the late time limit and show the full time behavior approaching the late time bound from above and exhibiting some new features for z sufficiently large.

Effect of the dilaton on holographic complexity growth

We study the effect of the Gauss-Bonnet term on the complexity growth rate of dual field theory using the ``complexity-volume'' (CV) and CV2.0 conjectures. We investigate the late time value and full time evolution of the complexity growth rate of the Gauss-Bonnet black holes with horizons with zero curvature ($k=0$), positive curvature ($k=1$) and negative curvature ($k=\ensuremath{-}1$), respectively. For the $k=0$ and $k=1$ cases, we find that the Gauss-Bonnet term suppresses the growth rate as expected, while in the $k=\ensuremath{-}1$ case the effect of the Gauss-Bonnet term may be opposite to what is expected. The reason for it is briefly discussed, and the comparison of our results to the result obtained by using the ``complexity-action'' (CA) conjecture is also presented. We also briefly investigate two proposals applying some generalized volume functionals dual to the complexity in higher curvature gravity theories, and find their behaviors are different for $k=0$ at late times.

Time Dependence of Holographic Complexity in Gauss-Bonnet Gravity

Based on general and minimal properties of the discrete circuit complexity, we define the complexity in continuous systems in a geometrical way We first show that the Finsler metric naturally emerges in the geometry of the complexity in continuous systems Due to fundamental symmetries of quantum field theories, the Finsler metric is more constrained and consequently, the complexity of SU(n) operators is uniquely determined as a length of a geodesic in the Finsler geometry Our Finsler metric is bi-invariant contrary to the right-invariance of discrete qubit systems We clarify why the bi-invariance is relevant in quantum field theoretic systems After comparing our results with discrete qubit systems we show most results in k-local right-invariant metric can also appear in our framework Based on the bi-invariance of our formalism, we propose a new interpretation for the Schrodinger’s equation in isolated systems – the quantum state evolves by the process of minimizing “computational cost”

/pdf/principles-and-symmetries-of-complexity-in-quantum-field-25szzk5s68.pdf

Principles and symmetries of complexity in quantum field theory

Recently it has been shown that the complexity of SU(n) operator is determined by the geodesic length in a bi-invariant Finsler geometry, which is constrained by some symmetries of quantum field theory. It is based on three axioms and one assumption regarding the complexity in continuous systems. By relaxing one axiom and an assumption, we find that the complexity formula is naturally generalized to the Schatten p-norm type. We also clarify the relation between our complexity and other works. First, we show that our results in a bi-invariant geometry are consistent with the ones in a right-invariant geometry such as k-local geometry. Here, a careful analysis of the sectional curvature is crucial. Second, we show that our complexity can concretely realize the conjectured pattern of the time-evolution of the complexity: the linear growth up to saturation time. The saturation time can be estimated by the relation between the topology and curvature of SU(n) groups.

/pdf/more-on-complexity-of-operators-in-quantum-field-theory-1av3n4oel8.pdf

More on complexity of operators in quantum field theory

Based on three general axioms for complexity, inspired by circuit complexity, and two physical assumptions, we show that the complexity of the SU($n$) group operator is given by the length of a geodesic in bi-invariant Finsler manifolds, of which Finsler metric is uniquely determined. We also provide a new interpretation to Schr\"{o}dinger's equation for isolated systems - the quantum state evolves by the process of minimizing "computational cost." Building on the complexity of the operator, we propose the complexity between states. For pure states, our proposal shows an interesting connection to the complexity of cMERA (continuous multi-scale entanglement renormalization ansatz) tensor networks and implies the results obtained by both the Fubini-Study metric approach and the path-integral approach. Furthermore, our proposal also reproduces the holographic complexity for the thermofield double states by the CV (complexity = volume) and CA (complexity = action) conjectures.

Yu-Sen An

Papers

Effect of the dilaton on holographic complexity growth

Time Dependence of Holographic Complexity in Gauss-Bonnet Gravity

Principles and symmetries of complexity in quantum field theory

More on complexity of operators in quantum field theory

Axiomatic complexity in quantum field theory and its applications