INFORMATIONTHEORETICAL ANALYSIS OF THE TOPOLOGICAL ENTANGLEMENT ENTROPY AND

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INFORMATION-THEORETICAL ANALYSIS OF THE TOPOLOGICAL ENTANGLEMENT ENTROPY AND MULTIPARTITE CORRELATIONS Kohtaro Kato (The University

INFORMATION-THEORETICAL ANALYSIS OF THE TOPOLOGICAL ENTANGLEMENT ENTROPY AND MULTIPARTITE CORRELATIONS Kohtaro Kato (The University of Tokyo) based on PRA, 93, 022317 (2016) joint work with Fabian Furrer (NTT Basic Research Laboratories) Mio Murao (The University of Tokyo)

Topologically ordered phases Topologically ordered phase (TOP) Ø A new kind of quantum phases

Topologically ordered phases Topologically ordered phase (TOP) Ø A new kind of quantum phases in a gapped system Many interesting properties n Degenerated ground states (g. s. ) which are locally indistinguishable n The g. s. degeneracy depends on the spatial topology n Anyonic excitations n Robust against any local perturbations Can be utilized for topological quantum computation! Symmetry-breaking phases: Characterized by local order parameters Topologically ordered phases: No local order parameters

Topologically ordered phases Topologically ordered phase (TOP) Ø A new kind of quantum phases

Topologically ordered phases Topologically ordered phase (TOP) Ø A new kind of quantum phases in a gapped system Many interesting properties n Degenerated ground states (g. s. ) which are locally indistinguishable n The g. s. degeneracy depends on the spatial topology n Anyonic excitations Topological Entanglement Entropy n Robust against any local perturbations (Kitaev & Preskill ‘ 06, Levin & Wen ‘ 06) Can be utilized for topological quantum computation! Symmetry-breaking phases: Characterized by local order parameters Topologically ordered phases: No local order parameters

Area law & Topological entanglement entropy [1/2] A ground state in a gapped system

Area law & Topological entanglement entropy [1/2] A ground state in a gapped system typically obeys area law

Area law & Topological entanglement entropy [1/2] A ground state in a gapped system

Area law & Topological entanglement entropy [1/2] A ground state in a gapped system typically obeys area law

Area law & Topological entanglement entropy [2/2] •

Area law & Topological entanglement entropy [2/2] •

Interaction information • The interaction information is one of generalizations of mutual information for

Interaction information • The interaction information is one of generalizations of mutual information for multipartite situations. • Unfortunately, the definition contains several disadvantages as a measure of correlations for general states/distributions. Point: we are only interested in gapped ground states

Useful properties of gapped ground states •

Useful properties of gapped ground states •

Our approach •

Our approach •

The irreducible correlation [1/3] •

The irreducible correlation [1/3] •

The irreducible correlation [2/3]* • The irreducible correlation has another interpretation through Jaynes’s maximum

The irreducible correlation [2/3]* • The irreducible correlation has another interpretation through Jaynes’s maximum entropy principle (Jaynes ‘ 57). The inference

The irreducible correlation [3/3] •

The irreducible correlation [3/3] •

Equivalence of TEE and IC KP type region LW type region

Equivalence of TEE and IC KP type region LW type region

Equivalence of TEE and IC The Gibbs state representation = New characterization of TOP

Equivalence of TEE and IC The Gibbs state representation = New characterization of TOP KP type region LW type region

Relation to Secret Sharing Protocol [1/3] Result 1 also implies that the characteristic correlations

Relation to Secret Sharing Protocol [1/3] Result 1 also implies that the characteristic correlations in TOP are hidden from all 2 -RDMs. Similar to secret sharing protocols! (Shamir ‘ 79, Blakley’ 79) The secret can be read out only when a sufficient number of parties collaborate together.

Relation to Secret Sharing Protocol [2/3] • The setup (Zhou et al. , ‘

Relation to Secret Sharing Protocol [2/3] • The setup (Zhou et al. , ‘ 07)

Relation to Secret Sharing Protocol [3/3] Ex. ) Toric code model • A z-string

Relation to Secret Sharing Protocol [3/3] Ex. ) Toric code model • A z-string (x-string) operator creates a corresponding anyon pair at the ends. • The type of an anyon is measured by interferometry measurements surrounding it. Apply z-string Apply x-string Apply both

Summary • Under an area law + zero-correlation length, we show that The TEE

Summary • Under an area law + zero-correlation length, we show that The TEE = The 3 rd-order irreducible correlation (a geometrical meaning) = The optimal rate of a SS protocol (an operational meaning) Open questions • Approximately holds for finite correlation length cases? (Joint work with F. Brandao, in preparation) • Can we quantify the quantum contribution of the IC? • IC = the optimal rate of SS protocol for general states? Thank you for your attention!

Properties of RDMs of gapped ground states •

Properties of RDMs of gapped ground states •

Quantum Markov States

Quantum Markov States

Merging two QMSs

Merging two QMSs

Proof sketch [1/3]

Proof sketch [1/3]

Proof sketch [2/3]

Proof sketch [2/3]

Proof sketch [3/3] Consider N-copy state Only grows polynomially for N

Proof sketch [3/3] Consider N-copy state Only grows polynomially for N

Relation to Secret Sharing Protocol [3/3] Kitaev-Preskill type Apply x-string Apply z-string or

Relation to Secret Sharing Protocol [3/3] Kitaev-Preskill type Apply x-string Apply z-string or