Systems and Symmetry Breaking Koji Umemoto (YITP) Based on Arpan - PowerPoint PPT Presentation

QIST 2019 June 6th, 2019 Entanglement of Purification in Many Body Systems and Symmetry Breaking Koji Umemoto (YITP) Based on Arpan Bhattacharyya (YITP), Alexander Jahn (Freie U.) Tadashi Takayanagi (YITP) and KU [1902.02369]

𝐹 𝑋 = 𝐹 𝑄 conjecture Takayanagi- KU ’17, Nguyen -Devakul-Halbasch-Zaletel-Swingle ’17 𝐵 𝐶 min Σ 𝐵𝐶 Entanglement wedge cross section Area(Σ 𝐵𝐶 ) 𝐹 𝑋 𝜍 𝐵𝐶 ≔ min 4𝐻 𝑂 Σ 𝐵𝐶 2 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

𝐹 𝑋 = 𝐹 𝑄 conjecture Takayanagi- KU ’17, Nguyen -Devakul-Halbasch-Zaletel-Swingle ’17 𝜔 𝐵𝐵′𝐶𝐶′ 𝑇 𝐵𝐵 ′ 𝐹 𝑄 𝜍 𝐵𝐶 ≔ min ？ Entanglement of purification = min 𝐵 𝐶 Σ 𝐵𝐶 Kudler- Flam and Ryu ’18 𝑈 𝐵 1 ℰ 𝑂 𝜍 𝐵𝐶 ≔ log 𝜍 𝐵𝐶 3 Logarithmic negativity ( 2 𝐹 𝑋 ) Tamaoka ’18 𝑈 𝐵 𝑜 𝑝 − 1] 𝐹 𝑋 𝜍 𝐵𝐶 Tr(𝜍 𝐵𝐶 𝑇 𝑝 𝜍 𝐵𝐶 ≔ lim 1 − 𝑜 𝑝 𝑜 𝑝 :𝑝𝑒𝑒→1 Odd entanglement entropy ( 𝐹 𝑋 + 𝑇 𝐵𝐶 ) Dutta and Faulkner ’19 √𝜍 𝐵𝐶 𝑇 𝑆 𝜍 𝐵𝐶 ≔ 𝑇 𝐵𝐵 ∗ Reflected entropy ( 2𝐹 𝑋 ) 3 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Entanglement of Purification (EoP) Definition: (𝜍 𝐵𝐵 ′ ≔ 𝐹 𝑄 𝜍 𝐵𝐶 ≔ min 𝑇(𝜍 𝐵𝐵 ′ ) Tr 𝐶𝐶 ′ [ 𝜔 𝜔 𝐵𝐵 ′ 𝐶𝐶 ′ ]) 𝜔 𝐵𝐵′𝐶𝐶′ 4 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Entanglement of Purification (EoP) Definition: (𝜍 𝐵𝐵 ′ ≔ 𝐹 𝑄 𝜍 𝐵𝐶 ≔ min 𝑇(𝜍 𝐵𝐵 ′ ) Tr 𝐶𝐶 ′ [ 𝜔 𝜔 𝐵𝐵 ′ 𝐶𝐶 ′ ]) 𝜔 𝐵𝐵′𝐶𝐶′ • In practice, hard to compute • Thus we still don’t know much about EoP in physical many body systems e.g. CFTs Related papers: Terhal-Horodecki-Leung- DiVincenzo ’02 , Chen- Winter ‘12 Nguyen-Devakul-Halbasch-Zaletel-Swingle ’17, … 5 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Our goal 𝐹 𝑄 𝜍 𝐵𝐶 ≔ min 𝑇(𝜍 𝐵𝐵 ′ ) 𝜔 𝐵𝐵′𝐶𝐶′ To compute EoP in many body systems (on a lattice) by numerically performing the minimization 6 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Our Targets • 2d (massless) free scalar field theory on a lattice Method: Minimal Gaussian purification ansatz Ground state reduced matrix 𝜍 𝐵𝐶 is Gaussian Purifications Ψ 𝐵𝐵 ′ 𝐶𝐶 ′ is assumed to be Gaussian with 𝐵 ′ 𝐶 ′ ≔ |𝐵𝐶| • 2d transverse-field (critical) Ising model Method: Full minimization without ansatz A theorem [Ibinson-Linden- Winter ’06] guarantees that it is sufficient to search dimℋ 𝐵 ′ ≤ rank𝜍 𝐵𝐶 , dimℋ 𝐶 ′ ≤ rank𝜍 𝐵𝐶 for minimizing 𝑇 𝐵𝐵 ′ E.g. 2 qubits 𝐵 ′ ∪ 𝐶′ 𝐵 ∪ 𝐶 𝐵 𝐶 dimℋ 𝐵 ′ = dimℋ 𝐶 ′ = 4 rank𝜍 𝐵𝐶 = 4 7 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Our Targets EoP behaves similarly in both models. Common results  EoP can increase with the physical distance.  Even if 𝜍 𝐵𝐶 = 𝜍 𝐶𝐵 , the optimal purification can break its symmetry i.e. 𝜔 ∗ 𝐶𝐶 ′ 𝐵𝐵 ′ . 𝐵𝐵 ′ 𝐶𝐶 ′ ≠ 𝜔 ∗ 8 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Setup 𝑂: 16 𝑂 : total sites number [1+1d, vacuum, periodic boundary condition] 9 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Setup 𝑥 : size of subsystem 𝐵 𝐶 𝑂: 16 𝑥: 2 𝑂 : total sites number [1+1d, vacuum, periodic boundary condition] 10 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Setup 𝑒 : distance between 𝐵 and 𝐶 𝑒 𝑥 : size of subsystem 𝐵 𝐶 𝑂: 16 𝑥: 2 𝑒: 3 𝑂 : total sites number [1+1d, vacuum, periodic boundary condition] 11 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Results E.g. 𝑂 = 60 free scalar Plateau-like behavior EoP Half of mutual information (Mutual information 𝐽 𝐵: 𝐶 = 𝑇 𝐵 + 𝑇 𝐶 − 𝑇 𝐵𝐶 ) 12 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Results E.g. 𝑂 = 60 free scalar Plateau-like behavior EoP Log negativity 𝑈 𝐵 (Log negativity 𝐹 𝑂 = log 𝜍 𝐵𝐶 1 ) 13 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Results E.g. 𝑂 = 60 free scalar (𝑥 = 4) 14 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Results E.g. 𝑂 = 10 Ising (periodic b.c.) (𝑥 = 1) 15 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Non-monotonicity EoP increases with the distance 𝑒 (For small 𝑂 , at 𝑒 = 1 ) Free scalar 𝑥 = 2 Ising model 𝑥 = 2 𝐵𝐶 ) (dimℋ 𝐵 ′ 𝐶 ′ ≔ dimℋ 16 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Non-monotonicity EoP increases with the distance 𝑒 (For small 𝑂 , at 𝑒 = 1 ) Mutual information Ising model 𝑥 = 2 𝐵𝐶 ) (dimℋ 𝐵 ′ 𝐶 ′ ≔ dimℋ 17 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Non-monotonicity It’s so weird … Perhaps the minimization does not work well? We can show this behavior analytically in some cases • E.g) in 𝑂 = 4 Ising model We can show 𝐹 𝑄 (𝑒 = 1) = 𝑇 𝐵 = 𝑇 𝐶 by a thm. and 𝐹 𝑄 (𝑒 = 0) < 𝑇 𝐵 by numerics Theorem Christandl-Winter ’05 If 𝜍 𝐵𝐶 has support only on the (anti-) symmetric subspace of ℋ 𝐵 ⊗ ℋ 𝐶 , 𝐵𝐶 = ℋ then 𝐹 𝑄 𝐵: 𝐶 = 𝑇 𝐵 = 𝑇 𝐶 . 18 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Z2 symmetry breaking The optimal purifications do not necessarily have the exchange symmetry (𝐵𝐵′ ↔ 𝐶𝐶′) But In some cases, 𝐶𝐶 ′ 𝐵𝐵 ′ 𝜔 opt. 𝐵𝐵 ′ 𝐶𝐶 ′ ≠ 𝜔 opt. Optimal purification 𝜍 𝐵𝐶 = 𝜍 𝐶𝐵 19 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Z2 symmetry breaking The optimal purifications do not necessarily have the exchange symmetry (𝐵𝐵′ ↔ 𝐶𝐶′) E.g. Ising model, opt. ) ≠ 𝑇(𝜍 𝐶 ′ opt. ) at 𝑒 = 1 𝑇(𝜍 𝐵 ′ 𝑂 = 10, 𝑥 = 1 𝑇 max ≔ max 𝑇 𝐵 ′ , 𝑇 𝐶 ′ , 𝑇 min ≔ min 𝑇 𝐵 ′ , 𝑇 𝐶 ′ 20 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

A qualitative interpretation Try to understand the qualitative aspects of results  Interplay between quantum entanglement and classical correlations Classical correlations: typically in separable states 𝑗 𝑗 ⊗ 𝜍 𝐶 𝜍 𝐵𝐶 = 𝑞 𝑗 𝜍 𝐵 𝑗 Suppose that total correlation, half of MI, is just a sum of them (Of course not precise) 𝐽 𝐵: 𝐶 ~𝐹 𝐵: 𝐶 + 𝐷(𝐵: 𝐶) 2 𝐽 Remainings ( 𝐷 ≔ 2 − 𝐹 𝑡𝑟 ) Squashed entanglement 𝐹 𝑡𝑟 will be a good candidate ( ∵ 𝐹 𝑡𝑟 ≤ 𝐽/2 ) 21 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

A qualitative interpretation Q. What is the coefficients for EoP? 𝐹 𝑄 𝐵: 𝐶 ~ 𝑏 𝐹 𝐵: 𝐶 + 𝑐 𝐷(𝐵: 𝐶) 1) EoP coincides with 𝑇 𝐵 for pure states When 𝐷 𝐵: 𝐶 = 0, 𝐹 𝑄 𝐵: 𝐶 = 𝑇 𝐵 = 𝐹 𝐵: 𝐶 ∴ 𝑏 = 1 2) EoP is at least as large as 𝐽(𝐵: 𝐶) for separable states When 𝐹 𝐵: 𝐶 = 0, 𝐹 𝑄 𝐵: 𝐶 ≥ 𝐽 𝐵: 𝐶 = 2𝐷(𝐵: 𝐶) Terhal-Horodecki-Leung- DiVincenzo ’02 ∴ 𝑐 ≥ 2 𝐹 𝑄 𝐵: 𝐶 ≳ 1 𝐹 𝐵: 𝐶 + 2 𝐷(𝐵: 𝐶) Claim: 22 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

A qualitative interpretation 𝐹 𝑄 𝐵: 𝐶 ≳ 1 𝐹 𝐵: 𝐶 + 2 𝐷(𝐵: 𝐶) 𝐽 𝐵: 𝐶 ~ 1 𝐹 𝐵: 𝐶 + 1 𝐷(𝐵: 𝐶) 2 Classical correlation Entanglement is strong is strong 23 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

A qualitative interpretation A toy model explaining why 𝑎 2 symmetry is broken only at 𝑒 = 1 : Focus on the nearest neighbor entanglement  EoP converts classical correlation into entanglement in the purified system 24 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Summary We computed entanglement of purification in 2d free scalar field • and 2d Ising model by numerically performing the minimization. We found that EoP can increase with the physical distance. • It is quite different from other measures such as mutual information. The optimal purifications are not necessarily symmetric under • exchange 𝐵𝐵 ′ ↔ 𝐶𝐶 ′ even if the original state satisfies 𝜍 𝐵𝐶 = 𝜍 𝐶𝐵 Both can be interpreted as an interplay between entanglement and • classical correlation 25 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Appendices