Quantum Information and Quantum Many-body Systems

Transcription

1 Quantum Information and Quantum Many-body Systems Lecture 1 Norbert Schuch California Institute of Technology Institute for Quantum Information

2 Quantum Information and Quantum Many-Body Systems Aim: Understand the physics of quantum systems composed of many particles (systems) In many cases, quantum correlations between particles are not relevant (mean field theory) Strong correlations involved entanglement becomes important Entanglement Theory: - central part of quantum information theory - how can we measure entanglement? - what can we do with entanglement, and what is impossible? Can we use quantum information techniques (in particular entanglement theory) to obtain a better understanding of quantum many-body systems?

3 Entanglement two (and more) qubits: entanglement How much entanglement is in some state e.g.? How much perfect entanglement does it contain? reduced state of Alice : more entanglement more uncertainty in measure of uncertainty (entanglement): von Neumann entropy provides quantitative measure of entanglement entropy = entanglement

4 Quantum many-body systems We consider systems composed of many (N) d-level systems ( spins ) with a locality notion ( lattice geometry) Behavior of system described by local Hamiltonian with a local operator: acts on small region only (e.g. nearest neighbors) Our focus: ground state : with smallest eigenvalue of Questions: - What is? - Which properties does have? (such as correlation functions ) - How do these things depend on?

5 How hard is it to describe the ground state? spins, can we describe the ground state? Problem for large N: But there is hope: has only exponentially large Hilbert space! parameters lives in small region of Hilbert space Can we find an efficient description of ground states from which we can efficiently compute quantities of interest?

6 Physical Hamiltonians Physical guideline for suitable ansatz states? Focus (for the moment) on one-dimensional (1D) systems with local Hamiltonian uniform family of Hamiltonians: acts beween spin and - E.g., translational invariant [either periodic boundary conditions (PBC) or open boundary conditions (OBC)] spectral gap of : second smallest eigenvalue smallest eigenvalue gapped Hamiltonians: uniform in

7 The area law What can we say about ground states of gapped Hamiltonians? Area law for ground states of gapped Hamiltonians: entropy is bounded by a constant Suprising: for random states, we expect Even for gapless systems: Quantum Information: entropy entanglement entanglement located around the boundary construct ansatz from entanglement between adjacent sites

8 An ansatz for states with an area law each site composed of two auxiliary particles ( virtual particles ) forming max. entangled bonds (D: bond dimension ) apply linear map ( projector ) satisfies area law by construction state characterized by parameters family of states: enlarged by increasing

9 Formulation in terms of Matrix Products matrices iterate this for the whole state : Matrix Product State (or for open boundaries)

10 Formulation in terms of Tensor Networks Tensor Network notation: Matrix Product States can be written as with... Tensor Network States

11 Examples any product state, e.g.: the GHZ state the W state (with and on left/right bnd.)

12 AKLT and RVB: rotationally invariant models The AKLT state [Affleck, Kennedy, Lieb & Tasaki, '87] : projector onto subspace (symmetric subspace) rotationally invariant model (different bond can be absorbed in ) Resonating valence bond (RVB) state:

13 When can we write state as MPS? Every state can be written as an MPS: state with entropic area law* efficient MPS approximation exists! *: for Renyi entropies size : linear scaling (poly if ) constant accuracy: