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arXiv:1805.04226 · Physical review. D/Physical review. D.

Thoughts on holographic complexity and its basis dependence

Koji Hashimoto, Norihiro Iizuka, Sotaro Sugishita

Published Aug 6, 2018• 33 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.98.046002

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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.
Abstract As a probe of circuit complexity in holographic field theories, we study sub-system analogues based on the entanglement wedge of the bulk quantities appearing in the “complexity = volume” and “complexity = action” conjectures. We calculate these quantities for one exterior region of an eternal static neutral or charged black hole in general dimensions, dual to a thermal state on one boundary with or without chemical potential respectively, as well as for a shock wave geometry. We then define several analogues of circuit complexity for mixed states, and use tensor networks to gain intuition about them. In the action approach, we find two possible cases depending on an ambiguity in the definition of the action associated with a counterterm. In one case, there is a promising qualitative match between the holographic action and what we call the purification complexity, the minimum number of gates required to prepare an arbitrary purification of the given mixed state. In the other case, the match is to what we call the basis complexity, the minimum number of gates required to prepare the given mixed state starting from a minimal complexity state with the same eigenvalue spectrum. One way to fix this ambiguity is to choose an action definition such that UV divergent part is positive, in which case the best match to the action result is the basis complexity. In contrast, the holographic volume does not appear to match any of our definitions of mixed-state complexity.
Nielsen's approach to quantum state complexity relates the minimal number of quantum gates required to prepare a state to the length of geodesics computed with a certain norm on the manifold of unitary transformations. For a bipartite system, we investigate binding complexity, which corresponds to norms in which gates acting on a single subsystem are free of cost. We reduce the problem to the study of geodesics on the manifold of Schmidt coefficients, equipped with an appropriate metric. Binding complexity is closely related to other quantities such as distributed computing and quantum communication complexity, and has a proposed holographic dual in the context of AdS/CFT. For finite dimensional systems with a Riemannian norm, we find an exact relation between binding complexity and the minimal Rényi entropy. We also find analytic results for the most commonly used non-Riemannian norm (the so-called F1 norm) and provide lower bounds for the associated notion of state complexity ubiquitous in quantum computation and holography. We argue that our results are valid for a large class of penalty factors assigned to generators acting across the subsystems. We demonstrate that our results can be borrowed to study the usual complexity (not-binding) for a single spin for the case of the F1 norm which was previously lacking from the literature. Finally, we derive bounds for multi-partite binding complexities and the related (continuous) circuit complexity where the circuit contains at most 2 -local interactions.
We investigate notions of complexity of states in continuous many-body quantum systems. We focus on Gaussian states which include ground states of free quantum field theories and their approximations encountered in the context of the continuous version of the multiscale entanglement renormalization ansatz. Our proposal for quantifying state complexity is based on the Fubini-Study metric. It leads to counting the number of applications of each gate (infinitesimal generator) in the transformation, subject to a state-dependent metric. We minimize the defined complexity with respect to momentum-preserving quadratic generators which form su(1,1) algebras. On the manifold of Gaussian states generated by these operations, the Fubini-Study metric factorizes into hyperbolic planes with minimal complexity circuits reducing to known geodesics. Despite working with quantum field theories far outside the regime where Einstein gravity duals exist, we find striking similarities between our results and those of holographic complexity proposals.
Abstract Unlike the standard entanglement entropy considered in the holographic con- text, entwinement measures entanglement between degrees of freedom that are not associated to a spatial subregion. Entwinement is defined for two-dimensional CFTs with a discrete ℤN gauge symmetry. Since the Hilbert space of these CFTs does not factorize into tensor products, even the entanglement entropy associated to a spatial subregion cannot be defined as the von Neumann entropy of a reduced density matrix. While earlier works considered embedding the density matrix into a larger, factorizing Hilbert space, we apply a gauge invariant approach by using a density matrix uniquely defined through its relation to the local algebra of observables. We furthermore obtain a fully gauge invariant definition of entwinement valid for general CFTs with ℤN gauge symmetry in terms of all observables acting on the degrees of freedom considered. Holographically, entwinement is dual to the length of non-minimal geodesics present for conical defects or black holes. In this context, we propose a definition of entwinement for thermal states dual to the BTZ black hole. Our results show that “entwinement is enough” to describe the full bulk geometry for the conical defect and provide strong hints that the same holds true for the BTZ black hole. Thus, it provides an alternative to holographic complexity for the theories considered.
This thesis investigates quantum complexity in the AdS/CFT duality. We study a recently proposed measure of quantum complexity, the spread complexity. This is a measure of the number of orthonormal basis vectors needed to describe a given quantum state (its “spread” in the Hilbert space), naturally selecting a preferred basis adapted to the dynamics of the state which minimises this quantity. This thesis demonstrates that the spread complexity provides a quantum information- theoretic quantity which directly connects to bulk dynamics in holography. Quantum complexity has long been studied in the context of holography. One result of these investigations is a conjecture relating the rate of change of the complexity of certain states in the CFT to the radial momentum of a dual massive particle falling along a time-like geodesic in the bulk spacetime. We demonstrate that there is an exact match between the rate of change of the spread complexity of an excited 2D CFT state and the dual particle’s radial momentum, measured in appropriate coordinates. As a toy model of the localised bulk particle state, we show that a similar relationship exists between the spread complexity and momentum of coherent states of the quantum harmonic oscillator. The spread complexity of these CFT states can be expressed analytically by relating them to a discrete series representation of SL(2,R). Additionally, we find analytical results for the spread complexity when the excited CFT state is evolved by the modular Hamiltonian of an interval region in the CFT, and by the half-sided modular Hamiltonian associated with a further sub-interval region. Next, we study the spread complexity of CFT states which describe a particle on the BTZ black hole background. We also study the spread complexity of local states in the semi-classical bulk theory. This provides a direct bulk interpretation of the spread complexity to complement our CFT calculations. Finally, we explore the extensivity of the spread complexity of states and of the Krylov complexity of density matrix operators in non-interacting bipartite systems.
We introduce a new quantity, called pseudo-entropy, as a generalization of entanglement entropy via postselection. We expect this quantity to provide a new class of order parameters in quantum many-body systems. In the anti--de Sitter space (AdS)/conformal field theory (CFT) correspondence, this quantity is dual to areas of minimal area surfaces in time-dependent Euclidean spaces which are asymptotically AdS. We call this geometric computation of pseudo-entropy via the AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT the holographic pseudo-entropy. We study its basic properties and classifications in qubit systems. In specific examples, we provide a quantum information theoretic meaning of this new quantity as an averaged number of Bell pairs when the post-selection is performed. We also present properties of the pseudo-entropy for random states. We then calculate the pseudo-entropy in the presence of local operator excitations for both the two dimensional free massless scalar CFT and two dimensional holographic CFTs. We find a general property in CFTs that the pseudo-entropy is highly reduced when the local operators get closer to the boundary of the subsystem. We also compute the holographic pseudo-entropy for a Janus solution, dual to an exactly marginal perturbation of a two dimensional CFT and find its agreement with a perturbative calculation in the dual CFT. We show the linearity property holds for holographic states, where the holographic pseudo-entropy coincides with a weak value of the area operator. Finally, we propose a mixed state generalization of pseudo-entropy and give its gravity dual.