The Complexity of Being Entangled
arXiv:2311.04277 · doi:10.22331/q-2024-09-12-1472
Abstract
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 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 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 -local interactions.
arXiv comment64 pages, 7 figures; v2: journal version on Quantum, minor changes
References in corpus (96)
- Complexity Equals Action
- Complexity and Shock Wave Geometries
- Time Evolution of Entanglement Entropy from Black Hole Interiors
- Geometric measure of entanglement and applications to bipartite and multipartite quantum states
- Entanglement monotones
- Complexity, action, and black holes
- Non-locality and Communication Complexity
- Quantum Computation as Geometry
- Qudits and high-dimensional quantum computing
- A Theory of Trotter Error
- Circuit complexity in quantum field theory
- Towards Complexity for Quantum Field Theory States
- Entanglement spectrum in one-dimensional systems
- Gravitational action with null boundaries
- The Gravity Dual of Renyi Entropy
- Quantum Brachistochrone
- The Second Law of Quantum Complexity
- Comments on Holographic Complexity
- Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT
- On the Time Dependence of Holographic Complexity
- Holographic Complexity
- Noether charge, black hole volume, and complexity
- Complexity of Formation in Holography
- Holographic Calculations of Renyi Entropy
- Circuit Complexity in Fermionic Field Theory
- Efficient Distributed Quantum Computing
- Complexity and entanglement for thermofield double states
- Quantum Computation as Gravity
- Complexity Growth for AdS Black Holes
- Entanglement capabilities of non-local Hamiltonians
- Entwinement and the emergence of spacetime
- Quantum Complexity and Negative Curvature
- Holographic Complexity in Vaidya Spacetimes II
- Quantum Computational Complexity -- From Quantum Information to Black Holes and Back
- Does Complexity Equal Anything?
- Circuit Complexity for Coherent States
- Complexity as a novel probe of quantum quenches: universal scalings and purifications
- Path-Integral Complexity for Perturbed CFTs
- Subsystem Complexity and Holography
- Quantum brachistochrone curves as geodesics: obtaining accurate control protocols for time-optimal quantum gates
- Holographic Complexity in Vaidya Spacetimes I
- Quantum Complexity of Time Evolution with Chaotic Hamiltonians
- Complexity of Mixed States in QFT and Holography
- Entanglement rates and area laws
- Holographic complexity is nonlocal
- Holography and wormholes in 2+1 dimensions
- Complexity and the bulk volume, a new York time story
- The Complexity Geometry of a Single Qubit
- The First Law of Complexity
- Topological Complexity in AdS3/CFT2
- Holographic Subregion Complexity from Kinematic Space
- Holographic Complexity And Cosmological Singularities
- Complexity for Conformal Field Theories in General Dimensions
- Complexity Equals Anything II
- Geometry of Complexity in Conformal Field Theory
- Holographic Complexity for Defects Distinguishes Action from Volume
- Half the entanglement in critical systems is distillable from a single specimen
- Upper bounds on entangling rates of bipartite Hamiltonians
- Arithmetic on a Distributed-Memory Quantum Multicomputer
- Subregion Action and Complexity
- Casting Shadows on Holographic Reconstruction
- Complexity Growth in Integrable and Chaotic Models
- Entangling power and quantum circuit complexity
- Aspects of The First Law of Complexity
- Conformal field theory complexity from Euler-Arnold equations
- Holographic Complexity in dS
- Circuit Complexity for Fermionic Thermofield Double states
- Complexity in the presence of a boundary
- Circuit Complexity and 2D Bosonisation
- Holographic Complexity of Quantum Black Holes
- Does Boundary Distinguish Complexities?
- A relation between Krylov and Nielsen complexity
- Binding Complexity and Multiparty Entanglement
- Complexity of mixed Gaussian states from Fisher information geometry
- Exact Gravity Duals for Simple Quantum Circuits
- Complexity=Anything: Singularity Probes
- Holographic Purification Complexity
- Bounds on quantum evolution complexity via lattice cryptography
- The Cosmological Switchback Effect
- Charged Complexity and the Thermofield Double State
- Integrability and complexity in quantum spin chains
- Quantum Complexity as Hydrodynamics
- Is Action Complexity better for de Sitter space in Jackiw-Teitelboim gravity?
- On volume subregion complexity in Vaidya spacetime
- On the Role of Counterterms in Holographic Complexity
- On subregion action complexity in AdS and in the BTZ black hole
- Volume complexity for Janus geometries
- Quantum speed limit helps interpret geometric measure of entanglement
- Volume complexity of dS bubbles
- Regularizations of Action-Complexity for a Pure BTZ Black Hole Microstate
- Action complexity in the presence of defects and boundaries
- Volume complexity for the non-supersymmetric Janus AdS geometry
- Universality in long-distance geometry and quantum complexity
- Complexity and Multi-boundary Wormholes
- Polynomial Equivalence of Complexity Geometries
- Straddling-gates problem in multipartite quantum systems