SLIDE 1
Operator algebras and data hiding in topologically ordered systems
Leander Fiedler Pieter Naaijkens Tobias Osborne UC Davis & RWTH Aachen 9 October 2016 QMath 13
arXiv:1608.02618
Operator algebras and data hiding in topologically ordered systems - - PowerPoint PPT Presentation
Operator algebras and data hiding in topologically ordered systems - - PowerPoint PPT Presentation
Operator algebras and data hiding in topologically ordered systems Leander Fiedler Pieter Naaijkens Tobias Osborne UC Davis & RWTH Aachen arXiv:1608.02618 9 October 2016 QMath 13 Topological order Topological phases
SLIDE 2
SLIDE 3
Topological order
SLIDE 4
Topological phases
SLIDE 5
Quantum phase outside of Landau theory
Topological phases
SLIDE 6
Quantum phase outside of Landau theory ground space degeneracy
Topological phases
SLIDE 7
Quantum phase outside of Landau theory ground space degeneracy long range entanglement
Topological phases
SLIDE 8
Quantum phase outside of Landau theory ground space degeneracy long range entanglement anyonic excitations
Topological phases
SLIDE 9
Quantum phase outside of Landau theory ground space degeneracy long range entanglement anyonic excitations modular tensor category / TQFT
Topological phases
SLIDE 10
Quantum phase outside of Landau theory ground space degeneracy long range entanglement anyonic excitations modular tensor category / TQFT
Topological phases
SLIDE 11
Modular tensor category
Describes all properties of the anyons, e.g. fusion, braiding, charge conjugation, …
SLIDE 12
Modular tensor category
Describes all properties of the anyons, e.g. fusion, braiding, charge conjugation, … Irreducible objects anyons ρi ⇔
SLIDE 13
Quantum dimension D2 =
X
i
d(ρi)2
Modular tensor category
Describes all properties of the anyons, e.g. fusion, braiding, charge conjugation, … Irreducible objects anyons ρi ⇔
SLIDE 14
SLIDE 15
Topological entanglement entropy SΛ = α|∂Λ| − γ + · · ·
Kitaev & Preskill (06), Levin & Wen (06)
Area law for top. ordered states:
SLIDE 16
Topological entanglement entropy SΛ = α|∂Λ| − γ + · · ·
Kitaev & Preskill (06), Levin & Wen (06)
Area law for top. ordered states:
γ = log D
Universal constant:
SLIDE 17
SLIDE 18
Technical framework
SLIDE 19
SLIDE 20
A(Λ) = O
x∈Λ
Md(C)
Quasi-local algebra A =
[
Λ
A(Λ)
k·k
SLIDE 21
A(Λ) = O
x∈Λ
Md(C)
and local Hamiltonians HΛ ∈ A(Λ)
SLIDE 22
A(Λ) = O
x∈Λ
Md(C)
ground state representation π0
SLIDE 23
Example: toric code
is a single excitation state ω0 ρ
SLIDE 24
Example: toric code
is a single excitation state ω0 ρ describes
- bservables in
presence of background charge π0 ρ
SLIDE 25
Quantum dimension
SLIDE 26
SLIDE 27
RA = π0(A(A))00
SLIDE 28
RA = π0(A(A))00 RB
SLIDE 29
RAB = RA ∨ RB RA = π0(A(A))00 RB
SLIDE 30
b RAB = π0(A((A ∪ B)c))0
SLIDE 31
Locality: RAB ⊂ b RAB
SLIDE 32
Locality: RAB ⊂ b RAB but: RAB $ b RAB
SLIDE 33
SLIDE 34
SLIDE 35
SLIDE 36
Weak-operator limit is in b
RAB
SLIDE 37
Weak-operator limit is in b
RAB
Jones-Kosaki-Longo index [ b RAB : RAB]
SLIDE 38
Theorem The number of excitation types is bounded by If all excitations have conjugates, is equal to the total quantum dimension. µπ0 µπ0 = sup
A∪B
[ b RAB : RAB]
PN, J. Math. Phys. ’13 Kawahigashi, Longo & Müger, Commun. Math. Phys. ’01
SLIDE 39
SLIDE 40
Data hiding
SLIDE 41
A data hiding task
Alice Bob Eve
SLIDE 42
A data hiding task
Alice Bob Eve Operations in are invisible to Eve
b RAB
SLIDE 43
A data hiding task
Alice Bob Eve and can be used to create charge pairs
SLIDE 44
A data hiding task
Kato, Furrer & Murao, Phys. Rev. A., ’16
Similar conclusion: TEE as a secret sharing capacity
SLIDE 45
SLIDE 46
Distinguishing states Alice prepares a mixed state :
SLIDE 47
Distinguishing states Alice prepares a mixed state : …and sends it to Bob
SLIDE 48
Distinguishing states Alice prepares a mixed state : …and sends it to Bob Can Bob recover ?
SLIDE 49
Holevo 𝜓 quantity In general not exactly:
SLIDE 50
Holevo 𝜓 quantity In general not exactly:
SLIDE 51
Holevo 𝜓 quantity In general not exactly: Generalisation of Shannon information
SLIDE 52
Holevo 𝜓 quantity In general not exactly: Generalisation of Shannon information
SLIDE 53
Optimal strategy
Want to compare and :
SLIDE 54
Optimal strategy
Want to compare and :
SLIDE 55
Optimal strategy
Want to compare and :
SLIDE 56
Optimal strategy
Want to compare and : Shirokov & Holevo, arXiv:1608.02203
SLIDE 57
A quantum channel
For finite index inclusion
SLIDE 58
A quantum channel
For finite index inclusion quantum channel, describes the restriction of operations
SLIDE 59
Quantum dimension and entropy
Hiai, J. Operator Theory, ’90; J. Math. Soc. Japan, ‘91
SLIDE 60
Quantum dimension and entropy
gives an information-theoretic interpretation to quantum dimension
Hiai, J. Operator Theory, ’90; J. Math. Soc. Japan, ‘91
SLIDE 61
Quantum dimension and entropy
gives an information-theoretic interpretation to quantum dimension Completely different methods from Kato/Furrer/ Murao, PRA 93, 022317 (2016)
Hiai, J. Operator Theory, ’90; J. Math. Soc. Japan, ‘91
SLIDE 62
Some remarks
SLIDE 63
Only classical information can be stored
Some remarks
SLIDE 64
Only classical information can be stored Different methods compared to Kato et al.
Some remarks
SLIDE 65
Only classical information can be stored Different methods compared to Kato et al. No finite dimensional analogue to index
Some remarks
SLIDE 66
Only classical information can be stored Different methods compared to Kato et al. No finite dimensional analogue to index Can use powerful methods from mathematics
Some remarks
SLIDE 67
Only classical information can be stored Different methods compared to Kato et al. No finite dimensional analogue to index Can use powerful methods from mathematics Right framework to study stability?
Some remarks
SLIDE 68