1. Information-Theoretical Analysis of the Topological Entanglement Entropy and Multipartite correlations
Kohtaro Kato (The University of Tokyo)
based on PRA, 93, (2016)
joint work with
Fabian Furrer (NTT Basic Research Laboratories)
Mio Murao (The University of Tokyo)

2. Topologically ordered phases
Topologically ordered phase (TOP)
A new kind of quantum phases in a gapped system
Many interesting properties
Degenerated ground states (g.s.) which are locally indistinguishable
The g.s. degeneracy depends on the spatial topology
Anyonic excitations
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

3. Topologically ordered phases
Topologically ordered phase (TOP)
A new kind of quantum phases in a gapped system
Many interesting properties
Degenerated ground states (g.s.) which are locally indistinguishable
The g.s. degeneracy depends on the spatial topology
Anyonic excitations
Robust against any local perturbations
Topological Entanglement Entropy
(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

4. Area law & Topological entanglement entropy [1/2]
A ground state in a gapped system typically obeys area law
𝐴 𝑐 𝐴
𝑆 𝐴 𝜌 =𝛼 𝜕𝐴 −𝛾+𝒪 𝜕𝐴 −1
𝑆 𝐴 𝜌 ≔−𝑇𝑟 𝜌 𝐴 log 2 𝜌 𝐴
𝛾: The topological entanglement entropy
𝛾=0 for conventional gapped phases, but 𝛾>0 for TOP.

5. Area law & Topological entanglement entropy [1/2]
A ground state in a gapped system typically obeys area law
𝐴 𝑐 𝐴
𝑆 𝐴 𝜌 =𝛼 𝜕𝐴 −𝛾+𝒪 𝜕𝐴 −1
𝑆 𝐴 𝜌 ≔−𝑇𝑟 𝜌 𝐴 log 2 𝜌 𝐴
𝛾: The topological entanglement entropy
This talk:
Reveals information-theoretical meanings of 𝛾
𝛾=0 for conventional gapped phases, but 𝛾>0 for TOP.

6. Area law & Topological entanglement entropy [2/2]
Another form of the topological entanglement entropy
𝑆 topo ≔𝑆 𝐴𝐵 𝜌 +𝑆 𝐵𝐶 𝜌 +𝑆 𝐶𝐴 𝜌 −𝑆 𝐴 𝜌 −𝑆 𝐵 𝜌 −𝑆 𝐶 𝜌 −𝑆 𝐴𝐵𝐶 𝜌 𝐵 A 𝐶 𝐵 A 𝐶 𝐵 A 𝐶
𝑆 topo ≈𝛾
𝑆 topo ≈2𝛾
𝑆 topo is equivalent to the interaction information in info.-theory.
(McGill ‘54, Bell ‘03)

7. Interaction information
𝑆 topo 𝐴 𝐵 𝐶
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.
𝑆 topo can be negative
𝑆 topo =0 for all pure states
No geometrical/operational meaning is known
Point: we are only interested in gapped ground states

8. Useful properties of gapped ground states
For simplicity, we assume the exact type of area law:
𝑆 𝐴 𝜌 =𝛼 𝜕𝐴 −𝛾.
If two regions 𝐴 and 𝐵 are separated, then
𝐼 𝐴:𝐵 𝜌 ≔𝑆 𝐴 𝜌 +𝑆 𝐵 𝜌 −𝑆 𝐴𝐵 𝜌 =0.
If regions 𝐴 and 𝐶 are indirectly connected through 𝐵, then
𝐼 𝐴:𝐶 𝐵 𝜌 ≔𝐼 𝐴:𝐵𝐶 𝜌 −𝐼 𝐴:𝐵 𝜌 =0. 𝐴 𝐵 𝐴 𝐵 𝐶

9. Our approach
Showing the equivalence of 𝑆 topo and other information-theoretical quantities under the assumptions of area law.
Irreducible correlation: 𝐶 3 ( 𝜌 𝐴𝐵𝐶 )
A geometrical measure of multipartite correlations
Optimal rate of a secret sharing protocol: 𝐶 3 ( 𝜌 𝐴𝐵𝐶 )
An operational measure of multipartite correlations

10. The irreducible correlation [1/3]
Any quantum state 𝜌∈𝒮( ℋ ⊗𝑛 ) has a Gibbs representation
𝜌= 𝑒 − 𝐻 𝜌
for some “Hamiltonian” 𝐻 𝜌 .
One can classify multipartite correlations in quantum states by locality of 𝐻 𝜌 .
The (𝑘th-order) irreducible correlation
(Linden, et al.,’02, Zhou ’08)
𝐶 𝑘 𝜌 ≔ inf 𝐻∈ 𝐸 𝑘 𝑆 𝜌|| 𝑒 −𝐻 ,
𝐸 𝑘 ≔ 𝐻 𝑘 : 𝑘−𝑙𝑜𝑐𝑎𝑙 Hamiltonian .
(Note: C 1 (𝜌) is the distance from the set of product states)

11. The irreducible correlation [2/3]*
The irreducible correlation has another interpretation through Jaynes’s maximum entropy principle (Jaynes ‘57).
For instance, for any tripartite state 𝜌 𝐴𝐵𝐶 ,
𝐶 𝜌 𝐴𝐵𝐶 =𝑆 𝜌 𝐴𝐵𝐶 −𝑆( 𝜌 𝐴𝐵𝐶 )
The inference
𝜌 𝐴𝐵𝐶 ≔ argmax 𝜎∈ 𝑅 𝜌 2 𝑆( 𝜎 𝐴𝐵𝐶 )
𝑅 𝜌 2 ≔ 𝜎 𝐴𝐵𝐶 𝜎 𝐴𝐵 = 𝜌 𝐴𝐵 , 𝜎 𝐵𝐶 = 𝜌 𝐵𝐶 , 𝜎 𝐶𝐴 = 𝜌 𝐶𝐴 }
The amount of information (correlation) in 𝝆 𝑨𝑩𝑪 that is not contained in all bipartite reduced density matrices (RDM).

12. The irreducible correlation [3/3]
Examples)
|𝐺𝐻𝑍 = |000 + |111
𝐶 (2) =2, 𝐶 (3) =1 .
|W = ( |001 + |010 + |100 )
𝐶 (2) ≈2.8, 𝐶 3 =0 .
(Note: If we apply 𝑆 topo to these states, its 0 for both cases )

13. Equivalence of TEE and IC
Result 1
Under the assumption, it holds that
C (3) 𝜌 𝐴𝐵𝐶 = 𝑆 topo
for every regions where 𝑆 topo is defined.
Proof: Explicitly construct 𝜌 𝐴𝐵𝐶 by using properties of states satisfying 𝐼 𝐴:𝐶 𝐵 =0. 𝐵 A 𝐶
KP type region
LW type region

14. Equivalence of TEE and IC
Result 1
Under the assumption, it holds that
C (3) 𝜌 𝐴𝐵𝐶 = 𝑆 topo
for every regions where 𝑆 topo is defined.
Proof: Explicitly construct 𝜌 𝐴𝐵𝐶 by using properties of states satisfying 𝐼 𝐴:𝐶 𝐵 =0.
The Gibbs state representation =
New characterization of TOP
𝐻 𝐴𝐵𝐶 ≔−log 𝜌 𝐴𝐵𝐶
𝛾=0
→ contains only n.n. interactions.
𝛾>0
→contains a 3−body interaction. 𝐵 A 𝐶
KP type region
LW type region

15. 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) 1
SUM= odd
The secret can be read out only when a sufficient number of parties collaborate together.

16. Relation to Secret Sharing Protocol [2/3]
The setup (Zhou et al., ‘07)
We encode a secret 𝑖 to 𝜌 ⊗𝑁 by a unitary 𝑈 𝑖 which does not change the 2-RDMs of 𝜌 ⊗𝑁 .
𝜌 𝐴𝐵𝐶 ⊗𝑁 → 𝑈 𝑖 𝜌 𝐴𝐵𝐶 ⊗𝑁 𝑈 𝑖 † ∈ 𝑅 𝜌 ⊗𝑁 2
→ To read out the secret 𝑖, a global joint measurement is needed.
𝐶 3 ( 𝜌 𝐴𝐵𝐶 ):= the optimal asymptotic rate of secret bits we can encode
Result 2
If 𝜌 𝐴𝐵𝐶 satisfies 𝐶 𝜌 𝐴𝐵𝐶 = 𝑆 topo , it holds that
𝐶 𝜌 𝐴𝐵𝐶 = 𝑆 topo .

17. 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. 𝒁 𝒁 𝒁 𝑿 𝑿
Apply z-string
Apply x-string
Apply both 𝐵 A 𝐶
𝑆 topo =2𝛾= log 2 4

18. Summary Thank you for your attention!
Under an area law + zero-correlation length, we show that
The TEE = The 3rd-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!

19. Properties of RDMs of gapped ground states
An area law + zero correlation length imply the following
If two regions 𝐴 and 𝐵 are separated, then
𝐼 𝜌 𝐴:𝐵 ≔ 𝑆 𝜌 𝐴 + 𝑆 𝜌 𝐵 − 𝑆 𝜌 𝐴𝐵 =0.
If region 𝐴 and 𝐶 are indirectly connected through 𝐵
and 𝐴𝐵𝐶 has no holes, then
𝐼 𝜌 𝐴:𝐶 𝐵 ≔ 𝐼 𝜌 𝐴:𝐵𝐶 − 𝐼 𝜌 (𝐴:𝐵)=0.
Properties 𝒂&𝒃 implies... 𝐵 A 𝐶
𝐵 1 A 𝐶
𝐵 2
𝐼 𝜌 𝐴: 𝐵 2 = 𝐼 𝜌 𝐴:𝐶 = 𝐼 𝜌 𝐵 1 :𝐶 =0
𝜌 AC is a product state
𝐼 𝜌 𝐴: 𝐵 2 𝐵 1 = 𝐼 𝜌 𝐵 1 :𝐶| 𝐵 2 =0
𝜌 𝐴 𝐵 1 𝐵 2 & 𝜌 𝐵 1 𝐵 2 𝐶 are QMSs

20. Quantum Markov States Quantum Markov State
𝜌 𝐴𝐵𝐶 is a quantum Markov state conditioned on 𝐵 iff
𝐼 𝜌 𝐴:𝐶 𝐵 =0.
This condition is equivalent to the following (Hayden et al., ‘04)
There is a CPTP-map Λ 𝐵→𝐵𝐶 s.t.
Λ 𝐵→𝐵𝐶 ( 𝜌 𝐴𝐵 ) = 𝜌 𝐴𝐵𝐶 .
There is a decomposition ℋ 𝐵 = 𝑖 ℋ 𝐵 𝑖 𝐿 ⊗ ℋ 𝐵 𝑖 𝑅 s.t.
𝜌 𝐴𝐵𝐶 = ⊕ 𝑖 𝑝 𝑖 𝜌 𝐴 𝐵 𝑖 𝐿 ⊗ 𝜌 𝐵 𝑖 𝑅 𝐶 . A 𝐵 𝐶 A
𝐵 𝐿
𝐵 𝑅 𝐶

21. Merging two QMSs 𝛾= 𝐶 3 ( 𝜌 𝐴𝐵𝐶 )
𝐴 B 𝐵 𝐶
𝜌 𝐴 𝐵 1 𝐵 2
𝜌 𝐴 𝐵 1 𝐵 2 𝐶 ≔ Λ 𝐵 2 → 𝐵 2 𝐶 ( 𝜌 𝐴 𝐵 1 𝐵 2 )
𝜌 𝐵 1 𝐵 2 𝐶 = Λ 𝐵 2 → 𝐵 2 𝐶 ( 𝜌 𝐵 1 𝐵 2 )
𝜌 𝐴 𝐵 1 𝐵 2 = ⊕ 𝑖 𝑝 1 𝑖 𝜌 𝐴 𝐵 1𝑖 𝐿 ⊗ 𝜌 𝐵 1𝑖 𝑅 𝐵 2
𝜌 𝐵 1 𝐵 2 𝐶 = ⊕ 𝑗 𝑝 2 𝑗 𝜌 𝐵 1 𝐵 2𝑖 𝐿 ⊗ 𝜌 𝐵 2𝑖 𝑅 𝐶
𝜌 𝐴 𝐵 1 𝐵 2 𝐶 = ⊕ 𝑖,𝑗 𝑝 1 𝑖 𝑝 2 𝑗 𝑖 𝜌 𝐴 𝐵 1𝑖 𝐿 ⊗ 𝜌 𝐵 1𝑖 𝑅 𝐵 2𝑗 𝐿 ⊗ 𝜌 𝐵 2𝑗 𝑅 𝐶
𝜌 𝐴𝐶 = tr B 2 Λ 𝐵 2 → 𝐵 2 𝐶 ( 𝜌 𝐴 𝐵 2 )=𝜌 𝐴 ⊗ tr B 2 Λ 𝐵 2 → 𝐵 2 𝐶 𝜌 𝐵 2 = 𝜌 𝐴 ⊗ 𝜌 𝐶
𝜎∈ 𝑅 𝜌 2 → 𝑆 𝜎 𝐴𝐵𝐶 ≤ 𝑆 𝜌 𝐴𝐵 + 𝑆 𝜌 𝐵𝐶 − 𝑆 𝜌 𝐵 = 𝑆 𝜌 (𝐴𝐵𝐶)
𝛾= 𝐶 3 ( 𝜌 𝐴𝐵𝐶 )

22. Proof sketch [1/3] 𝜌 𝐴𝐵𝐶 𝜌 𝐴𝐵𝐶 = ⊕ 𝑖 𝑝 𝑖 𝜌 𝐴 𝐵 𝑖 𝐿 ⊗ 𝜌 𝐵 𝑖 𝑅 𝐶 A
𝜌 𝐴𝐵𝐶 = ⊕ 𝑖 𝑝 𝑖 𝜌 𝐴 𝐵 𝑖 𝐿 ⊗ 𝜌 𝐵 𝑖 𝑅 𝐶
𝜌 𝐴𝐵 = 𝜌 𝐴𝐵 , 𝜌 𝐵𝐶 = 𝜌 𝐵𝐶 , 𝜌 𝐶𝐴 = 𝜌 𝐶𝐴
𝜌 𝐴 𝐵 𝐿 = ⊕ 𝑖 , 𝐾 𝑖 𝑝 𝑖 𝜆 𝐾 𝑖 Π 𝐾 𝑖 , Π K i = Σ 𝑚 𝐾 𝑖 | 𝐾 𝑖 , 𝑚 𝐾 𝑖 𝐾 𝑖 , 𝑚 𝐾 𝑖 |
𝒰= 𝑈 𝐴 𝐵 𝐿 : A set of encoding unitaries s.t. 𝑈 𝐴 𝐵 𝐿 = ⊕ 𝑖, 𝐾 𝑖 𝑈 𝐴 𝐵 𝑖 𝐿 𝐾 𝑖 , where
𝑈 𝐴 𝐵 𝑖 𝐿 𝐾 𝑖 is a element of a exact 1-design on supp 𝜌 𝐴𝐵 𝑖 𝐿 .
𝒰 1 𝒰 𝑈 𝐴 𝐵 𝐿 | 𝐾 𝑖 , 𝑚 𝐾 𝑖 𝐿 𝑗 , 𝑛 𝐿𝑗 | 𝑈 𝐴 𝐵 𝐿 † = 𝛿 𝐾 𝑖 𝐿 𝑗 𝑑 𝐾 𝑖 Π 𝐾 𝑖
𝑈 𝐴 𝐵 𝐿 𝜌 𝐴𝐵 𝑈 𝐴 𝐵 𝐿 † = 𝜌 𝐴𝐵

23. Proof sketch [2/3] 𝑈 𝐴 𝐵 𝐿 ⊗ 𝐼 𝐵 𝑅 𝐶 𝜌 𝐴𝐵𝐶 𝑈 𝐴 𝐵 𝐿 ⊗ 𝐼 𝐵 𝑅 𝐶 † ∈ 𝑅 𝜌 2
𝑈 𝐴 𝐵 𝐿 ⊗ 𝐼 𝐵 𝑅 𝐶 𝜌 𝐴𝐵𝐶 𝑈 𝐴 𝐵 𝐿 ⊗ 𝐼 𝐵 𝑅 𝐶 † ∈ 𝑅 𝜌 2
𝐴𝐵→ Obvious 𝐵𝐶
𝑇 𝑟 𝐴 𝑈 𝐴 𝐵 𝐿 ⊗ 𝐼 𝐵 𝑅 𝐶 𝜌 𝐴𝐵𝐶 𝑈 𝐴 𝐵 𝐿 ⊗ 𝐼 𝐵 𝑅 𝐶 † = Λ 𝐵 𝐿 ⊗𝑖 𝑑 𝐵 𝑅 𝐶 𝜌 𝐵𝐶
= Λ 𝐵 𝐿 ⊗𝑖 𝑑 𝐵 𝑅 𝐶 𝜌 𝐵𝐶 = ⊕ 𝑖 𝑝 𝑖 Λ 𝐵 𝐿 𝜌 𝐵 𝑖 𝐿 ⊗ 𝜌 𝐵 𝑖 𝑅 𝐶 = ⊕ 𝑖 𝑝 𝑖 𝜌⊗ 𝜌 𝐵 𝑖 𝑅 𝐶 𝐴𝐶
𝑇 𝑟 𝐵 𝑈 𝐴 𝐵 𝐿 ⊗ 𝐼 𝐵 𝑅 𝐶 𝜌 𝐴𝐵𝐶 𝑈 𝐴 𝐵 𝐿 ⊗ 𝐼 𝐵 𝑅 𝐶 † = Λ 𝐴 ⊗𝑖 𝑑 𝐶 𝜌 𝐴𝐶
= Λ 𝐴 𝜌 𝐴 ⊗ 𝜌 𝐶 = 𝜌 𝐴 ⊗ 𝜌 𝐶

24. =𝑆 𝜌 𝐴𝐵𝐶 −𝑆 𝜌 𝐴𝐵𝐶 = 𝐶 3 𝜌 = 𝑆 topo .
Proof sketch [3/3]
𝜌 𝐴𝐵𝐶 = 1 |𝒰| 𝒰 𝑈 𝐴 𝐵 𝐿 ⊗ 𝐼 𝐵 𝑅 𝐶 𝜌 𝐴𝐵𝐶 𝑈 𝐴 𝐵 𝐿 ⊗ 𝐼 𝐵 𝑅 𝐶 †
# of different eigenvalues of 𝜌 𝐴 𝐵 𝐿
𝑆( 𝜌 𝐴𝐵𝐶 )≥𝑆 𝜌 𝐴𝐵𝐶 −log 𝐷 𝐴 𝐵 𝐿
Consider N-copy state
Only grows polynomially for N
𝐷 𝐴 𝐵 𝐿 𝑁 ~ 𝑁+1 𝑑 𝐴 𝐵 𝐿
1 𝑁 log 𝐷 𝐴 𝐵 𝐿 𝑁 →0, 1 𝑁 𝑆 𝜌 𝑁 →𝑆( 𝜌 𝐴𝐵𝐶 )
𝐶 𝜌 = lim 𝑁→∞ 1 𝑁 max 𝜌 𝑁 𝑆 𝜌 𝑁 −𝑆 𝜌 ⊗N
=𝑆 𝜌 𝐴𝐵𝐶 −𝑆 𝜌 𝐴𝐵𝐶 = 𝐶 3 𝜌 = 𝑆 topo .

25. Relation to Secret Sharing Protocol [3/3]
Kitaev-Preskill type
Apply x-string
Apply z-string 𝐵 A 𝐶 or
𝛾= log 2 2