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) 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

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

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.

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.

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)

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

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

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

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)

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 3 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).

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

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

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

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.

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 3 𝜌 𝐴𝐵𝐶 = 𝑆 topo , it holds that 𝐶 3 𝜌 𝐴𝐵𝐶 = 𝑆 topo .

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

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!

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

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

Merging two QMSs 𝛾= 𝐶 3 ( 𝜌 𝐴𝐵𝐶 ) 𝐴 B 1 𝐵 2 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 )

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 1 𝑑 𝐾 𝑖 Π 𝐾 𝑖 𝑈 𝐴 𝐵 𝐿 𝜌 𝐴𝐵 𝑈 𝐴 𝐵 𝐿 † = 𝜌 𝐴𝐵

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

=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 3 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 .

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