1. Information-Theoretic Techniques in Many-Body Physics Day 1 Fernando G.S.L. Brandão UCL Based on joint work with A. Harrow and M. Horodecki New Mathematical Directions for Quantum Info

2. Quantum Many-Body Systems Quantum Hamiltonian Interested in computing properties such as minimum energy, correlations functions at zero and finite temperature, dynamical properties, …

3. Constraint Satisfaction Problems vs Local Hamiltonians k-arity CSP: Variables {x 1, …, x n }, alphabet Σ Constraints: Assignment: Unsat :=

4. Constraint Satisfaction Problems vs Local Hamiltonians k-arity CSP: Variables {x 1, …, x n }, alphabet Σ Constraints: Assignment: Unsat := k-local Hamiltonian H: n qudits in Constraints: qUnsat := E 0 : min eigenvalue H1H1 qudit

5. Classical vs Quantum Optimal Assignments Finding optimal assignment of CSPs is usually hard (NP-hard) Finding optimal assignment of quantum CSPs (groundstates) seems even harder (QMA-hard; See Daniel Nagaj’s talk) Main difference: Optimal Assignment can be a highly entangled state (unit vector in )

6. The Plan Today: Product-State Approximations to Groundstates - de Finetti theorem - information theory approach (entropies, chain rule, Pinsker’s inequality, info-complete meas., …) Tomorrow: Groundstates in 1D - area laws and matrix product states - information theory approach (decoupling, state merging, single-shot protocols, …)

7. Approximation Scale We want to approximate the minimum energy (i.e. minimum eigenvalue of H): Small total error: Small extensive error: today

8. Mean-Field… …consists in approximating the groundstate by a product state is a CSP

9. Mean-Field… …consists in approximating the groundstate by a product state is a CSP

10. Mean-Field… …consists in approximating the groundstate by a product state is a CSP Successful heuristic in Quantum Chemistry (Hartree-Fock) Condensed matter (e.g. BCS theory) Intuition: Mean-Field good when Many-particle interactions Low entanglement in state It’s a mapping from quantum Hamiltonians to CSPs

11. Hamiltonian on the Complete Graph Consider a Hamiltonian on the complete graph G of size n H ij The Hamiltonian is permutation symmetric: with

12. Quantum de Finetti Theorems (Christandl, Koenig, Mitchison, Renner ‘05) Let ρ 1,…,n be a permutation-symmetric state. Then (Stormer ’69, Hudson, Moody ’76) Infinite Quantum de Finetti Theorem (Raggio, Werner ’89) Connection of Infinite Quantum de Finetti with Mean-Field (Caves, Fuchs, Sachs ’01) Proof infinite de Finetti using info-complete measurements (Koenig, Renner ’05) Finite Quantum de Finneti Theorem (remember Graeme Mitchison’s talk)

13. (Christandl, Koenig, Mitchison, Renner ‘05) Let ρ 1,…,n be a permutation-symmetric state. Then Product-States Approximation and de Finetti Theorem

14. (Christandl, Koenig, Mitchison, Renner ‘05) Let ρ 1,…,n be a permutation-symmetric state. Then By de Finetti: Product-States Approximation and de Finetti Theorem

15. (Christandl, Koenig, Mitchison, Renner ‘05) Let ρ 1,…,n be a permutation-symmetric state. Then By de Finetti: So Product-states achieve error 2d 2 /n for mean-energy Product-States Approximation and de Finetti Theorem

16. The Role of Permutation Symmetry To apply quantum de Finetti we need a permutation-invariant Hamiltonian. Can we relax this assumption? Can we show product states do a good job for models not on the complete graph?

17. Product-State Approximation without Symmetry (B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E) and |E| local terms. Let {X i } be a partition of the sites with each X i having m sites. E i : expectation over X i Deg : degree of G S(X i ) : entropy of groundstate in X i X1X1 X2X2 size m

18. Product-State Approximation without Symmetry (B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E) and |E| local terms. Let {X i } be a partition of the sites with each X i having m sites. Then there are states ψ i in X i s.t. E i : expectation over X i Deg : degree of G S(X i ) : entropy of groundstate in X i X1X1 X2X2 size m

19. Approximation in terms of degree Implications to the quantum PCP problem (whether to compute is QMA-hard ): Shows that attempts to quantize Dinur’s proof of the PCP theorem cannot work. Also gives a no-go for “quantum PCP + parallel repetition of qCSP”

20. Approximation in terms of degree Implications to the quantum PCP problem (whether to compute is QMA-hard ): Shows that attempts to quantize Dinur’s proof of the PCP theorem cannot work. Also gives a no-go for “quantum PCP + parallel repetition of qCSP” Bound: Φ G ½) Obs: Restricted to 2-local models (Aharonov, Lior ‘13) k-local, commuting models

21. Approximation in terms of degree 1-D 2-D 3-D ∞-D …shows mean field becomes exact in high dim See (Cirac, Kraus, Lewenstein) for rotationally invariant systems

22. Approximation in terms of average entanglement Product-states do a good job if entanglement of groundstate satisfies a subvolume law: X1X1 X3X3 X2X2 m < O(log(n))

23. Approximation in terms of average entanglement If, Pinsker’s inequality shows product states give error

24. Approximation in terms of average entanglement If, Pinsker’s inequality shows product states give error In constrast, if merely, the theorem shows product states give error

25. When does it fail? Expander graph G(V, E) with expansion Φ G E.g.

26. Intuition: Monogamy of Entanglement Quantum correlations are non-shareable (see Aram Harrow’s and Thomas Vidick’s talks) Cannot be highly entangled with too many neighbors S(X i ) quantifies how much entangled X i can be with the rest

27. Intuition: Monogamy of Entanglement Quantum correlations are non-shareable Cannot be highly entangled with too many neighbors S(X i ) quantifies how much entangled X i can be with the rest (see Aram Harrow’s and Thomas Vidick’s talks) Proof uses information-theoretic techniques to make this intuition precise Inspired by classical information-theoretic ideas for bounding convergence of Sum-Of-Squares hierarchy for CSPs (Tan, Raghavendra ’10; Barak, Raghavendra, Steurer ‘10)

28. Mutual Information 1.Mutual Information 1.Pinsker’s inequality 1.Conditional MI 1.Chain Rule 5.Upper bound 4+5 for some t ≤ k

29. Quantum Mutual Information 1.Mutual Information 1.Pinsker’s inequality 1.Conditional MI 1.Chain Rule 5.Upper bound 4+5 for some t ≤ k

30. But… …conditioning on quantum is problematic For X, Y, Z random variables No similar interpretation is known for I(X:Y|Z) with quantum Z

31. Conditioning Decouples Idea that almost works. Suppose we have a distribution p(z 1,…,z n ) 1. Choose i, j 1, …, j k at random from {1, …, n}. Then there exists t<k such that Define So i j1j1 j2j2 jkjk

32. Conditioning Decouples 2. Conditioning on subsystems j 1, …, j t causes, on average, error <k/n and leaves a distribution q for which, and so By Pinsker: j1j1 jtjt j2j2 Choosing k = εn

33. Informationally Complete Measurements There exists a POVM M( ρ ) = Σ k tr(M k ρ ) |k><k| s.t. for all k and ρ 1…k, σ 1…k in D((C d )  k ) (Lacien, Winter ‘12, Montanaro ‘12)

34. Proof Overview 1.Measure εn qudits with M and condition on outcomes. Incur error ε. 2.Most pairs of other qudits would have mutual information ≤ log(d) / ε deg(G) if measured. 3.Thus their state is within distance d 3 (log(d) / ε deg(G)) 1/2 of product. 4.Witness is a global product state. Total error is ε + d 6 (log(d) / ε deg(G)) 1/2. Choose ε to balance these terms. 5.General case follows by coarse graining sites (can replace log(d) by E i H(X i ))

35. Proof Overview Let … previous argument q : probability distribution obtained conditioning on z j1, …, z jt

36. Proof Overview (σ: state obtained by measuring M on j 1, …, j t and conditioning on the outcome). Choosing k = εn info complete measurement

37. Other Applications 1: New Classical Algorithms for Q. Hamiltonians Following same approach one obtains polynomial time algorithms for approximating the groundstate energy of 1.Planar Hamiltonians, improving on (Bansal, Bravyi, Terhal ‘07) 2.Dense Hamiltonians, improving on (Gharibian, Kempe ‘10) 3.Hamiltonians on graphs with low threshold rank, building on (Barak, Raghavendra, Steurer ‘10) In all cases we prove that a product state does a good job and use efficient algorithms for CSPs.

38. Other Applications 2: New de Finetti Theorems - Classical de Finetti without symmetry: For p(x 1,…,x n ) with - Q. version using info-complete measurement - Q. version using locality constrained norms (see Aram’s talk) - Version replacing uniform randomness by Santa-Vazirani source (Ramanathan et al ‘13)

39. Thank you!

40. Information-Theoretic Techniques in Many-Body Physics Day 2 Fernando G.S.L. Brandão UCL Based on joint work with A. Harrow and M. Horodecki New Mathematical Directions for Quantum Info

41. The Plan Yesterday: Product-State Approximations to Groundstates - de Finetti theorem - information theory approach (entropies, chain rule, Pinsker’s inequality, info-complete meas.) Today: Groundstates in 1D - matrix product states - area law and exponential decay of correlations - information theory approach (decoupling, state merging, single-shot protocols) (see Nilanjana Datta’s talk)

42. Quantum Many-Body Systems Quantum Hamiltonian Interested in computing properties such as minimum energy, correlations functions, etc…

43. Approximation Scale We want to approximate the minimum energy (i.e. minimum eigenvalue of H): Small total error: Small extensive error: today

44. Matrix Product States (Fannes, Nachtergaele, Werner ‘92) D : bond dimension Only nD 2 parameters. Local expectation values computed in poly(D, n) time Variational class of states for powerful DMRG Generalization of product states (MPS with D=1)

45. Area Law in 1D Let be a n-qubit quantum state Entanglement Entropy: Area Law: For all partitions of the chain (X, Y) X Y (Bekenstein ‘73, ….…, Eisert, Cramer, Plenio ’10)

46. MPS Area Law XY For MPS,

47. MPS Area Law XY If is s.t. then it has a MPS description of bound dim. D (Fannes, Nachtergaele, Werner ‘92, Vidal ’03, Jozsa ‘06) For MPS,

48. MPS Area Law XY If is s.t. then it has a MPS description of bound dim. D (Fannes, Nachtergaele, Werner ‘92, Vidal ’03, Jozsa ‘06) (Approx. version) If is s.t. then it can be approximated by a MPS of bound dim. D up to error ε For MPS, Def:

49. Exponential Decay of Correlations Let be a n-qubit quantum state Correlation Function: Exponential Decay of Correlations: There is ξ > 0 s.t. for all cuts X, Y, Z with |Y| = l X Z Y l

50. MPS EDC ≈ Let Define (w.l.o.g. ) and let λ j be the second largest eingenvalue of and λ := max |λ j | If λ is independent of n we say is a gMPS has (1/log(1/|λ|)) -EDC

51. How good are MPS? Negative results: (Aharonov, Gottesman, Irani, Kempe ‘07) 1D Hamiltonians can be QMA-hard (see Daniel Nagaj’s talk) (Irani ’09; Gottesman, Hastings ‘09) 1D Hamiltonians with volume scaling of entanglement (Irani, Gottesman ‘09) 1D Hamiltonians with translational-invariance still hard … is there hope?

52. 1D gapped models Given Let Then H n is gapped if (remember Toby Cubbit’s talk)

53. 1D gapped models Area LawgMPS EDC (Hastings ’07 Arad, Kitaev, Landau, Vazirani ’12) Groundstate Gapped model (Hastings ’05) (Landau, Vidick, Vazirani ‘12)

54. gMPSEDC Area Law E.g. there are gapless Hamiltonians with a mobility gap/dynamical localization, which imply EDC in the groundstate (Hastings ’10; Hamza, Sims, Stolz ‘11) Do we need the gap?

55. gMPSEDC Area Law (B., Horodecki ’12)

56. Do we need the gap? gMPSEDC Area Law (B., Horodecki ’12)

57. Area Law vs. Decay of Correlations Exponential Decay of Correlations suggests Area Law

58. Area Law vs. Decay of Correlations Exponential Decay of Correlations suggests Area Law: X Z Y l = O(ξ) ξ-EDC implies

59. Area Law vs. Decay of Correlations Exponential Decay of Correlations suggests Area Law: X Z Y ξ-EDC implies which implies (by Uhlmann’s theorem) X is only entangled with Y! l = O(ξ)

60. Area Law vs. Decay of Correlations Exponential Decay of Correlations suggests Area Law: X Z Y ξ-EDC implies which implies (by Uhlmann’s theorem) X is only entangled with Y! Alas, the argument is wrong… Uhlmann’s thm require 1-norm: l = O(ξ)

61. Area Law vs. Decay of Correlations Exponential Decay of Correlations suggests Area Law: X Z Y ξ-EDC implies which implies (by Uhlmann’s theorem) X is only entangled with Y! Alas, the argument is wrong… Uhlmann’s thm require 1-norm: l = O(ξ)

62. Data Hiding States Well distinguishable globally, but poorly distinguishable locally Ex. 1 Antisymmetric Werner state ω AB = (I – F)/(d 2 -d) Ex. 2 Random state with |X|=|Z| and |Y|=l (DiVincenzo, Leung, Terhal ’02) X Z Y

63. What data hiding implies? 1.Intuitive explanation is flawed

64. What data hiding implies? 1.Intuitive explanation is flawed 2. No-Go for area law from exponential decaying correlations?

65. What data hiding implies? 1.Intuitive explanation is flawed 2. No-Go for area law from exponential decaying correlations?

66. What data hiding implies? 1.Intuitive explanation is flawed 2. No-Go for area law from exponential decaying correlations? 3. We fixed a partition; EDC gives us more…

67. What data hiding implies? 1.Intuitive explanation is flawed 2. No-Go for area law from exponential decaying correlations? 3. We fixed a partition; EDC gives us more… 4.It’s an interesting quantum information problem: How strong is data hiding in quantum states?

68. Exponential Decaying Correlations Imply Area Law Thm 1 (B., Horodecki ‘12) If has ξ-EDC then for every X and m, X XcXc

69. Exponential Decaying Correlations Imply Area Law Thm 1 (B., Horodecki ‘12) If has ξ-EDC then for every X and m, Obs1: Implies Obs2: Only valid in 1D… Obs3: Reproduces bound of Hastings for GS 1D gapped Ham., using EDC in such states X XcXc

70. Exponential Decaying Correlations Imply Area Law Thm 1 (B., Horodecki ‘12) If has ξ-EDC then for every X and m, Obs1: Implies Obs2: Only valid in 1D… Obs3: Reproduces bound of Hastings for GS 1D gapped Ham., using EDC in such states X XcXc

71. Exponential Decaying Correlations Imply Area Law Thm 1 (B., Horodecki ‘12) If has ξ-EDC then for every X and m, Obs4: Implies stronger form of EDC: For l > exp(O(ξlogξ)) and split ABC with |B|=l X XcXc

72. EDC gMPS (Cor. Thm 1) If has ξ-EDC then for every ε>0 there is gMPS with poly(n, 1/ε) bound dim. s.t. X XcXc

73. Random States Have EDC? : Drawn from Haar measure XZ Y l w.h.p, if size(X) ≈ size(Z): and Small correlations in a fixed partition do not imply area law.

74. Random States Have EDC? : Drawn from Haar measure XZ Y l w.h.p, if size(X) ≈ size(Z): and Small correlations in a fixed partition do not imply area law. But we can move the partition freely...

75. Random States Have Big Correl. : Drawn from Haar measure X Z Y l Let size(XY) < size(Z). W.h.p., X is decoupled from Y.

76. Random States Have Big Correl. : Drawn from Haar measure X Z Y l Let size(XY) < size(Z). W.h.p., X is decoupled from Y. Extensive entropy, but also large correlations:

77. Random States Have Big Correl. : Drawn from Haar measure X Z Y l Let size(XY) < size(Z). W.h.p., X is decoupled from Y. Extensive entropy, but also large correlations: Maximally entangled state between XZ 1. (Uhlmann’s theorem)

78. Random States Have Big Correl. : Drawn from Haar measure X Z Y l Let size(XY) < size(Z). W.h.p., X is decoupled from Y. Extensive entropy, but also large correlations: Maximally entangled state between XZ 1. Cor(X:Z) ≥ Cor(X:Z 1 ) = Ω(1) >> 2 -Ω(n) : long-range correlations! (Uhlmann’s theorem)

79. Random States Have Big Correl. : Drawn from Haar measure X Z Y l Let size(XY) < size(Z). W.h.p., X is decoupled from Y. Extensive entropy, but also large correlations: Maximally entangled state between XZ 1. Cor(X:Z) ≥ Cor(X:Z 1 ) = Ω(1) >> 2 -Ω(n) : long-range correlations! (Uhlmann’s theorem) This reasoning hints at the idea of the general proof: We’ll show large entropy leads to large correlations by choosing a random measurement that decouples A and B

80. Entanglement Distillation by Decoupling We apply the state merging protocol to show large entropy implies large correlations State merging protocol: Given Alice can distill -S(A|B) = S(B) – S(AB) EPR pairs with Bob by making a random measurement with N≈ 2 I(A:E) elements, with I(A:E) := S(A) + S(E) – S(AE), and communicating the outcome to Bob. (Horodecki, Oppenheim, Winter ‘05) A BE

81. We apply the state merging protocol to show large entropy implies large correlations State merging protocol: Given Alice can distill -S(A|B) = S(B) – S(AB) EPR pairs with Bob by making a random measurement with N≈ 2 I(A:E) elements, with I(A:E) := S(A) + S(E) – S(AE), and communicating the outcome to Bob. (Horodecki, Oppenheim, Winter ‘05) A BE Disclaimer: only works for Let’s cheat for a while and pretend it works for a single copy, and later deal with this issue Entanglement Distillation by Decoupling

82. State merging protocol works by applying a random measurement {P k } to A in order to decouple it from E: log( # of P k ’s ) # EPR pairs: A BE Optimal Decoupling

83. l X Y Z A E B Distillation Bound

84. l X Y Z S(Z) – S(XZ) > 0 (EPR pair distillation by random measurement) Prob. of getting one of the 2 I(X:Y) outcomes in random measurement A E B

85. Proof Strategy We apply previous result to prove EDC -> Area Law in 3 steps: 1. Get area law from EDC under assumption there is a region with “subvolume” law 2. Get region with “subvolume” law from assumption there is a region of “small mutual information” 3. Show there is always a region of “small mutual info”

86. 1. Area Law from Subvolume Law l X Y Z

87. l X Y Z

88. l X Y Z Suppose S(Y) < l/(4ξ) (“subvolume law” assumption)

89. 1. Area Law from Subvolume Law l X Y Z Suppose S(Y) < l/(4ξ) (“subvolume law” assumption) Since I(X:Y) < 2S(Y) < l/(2ξ), ξ-EDC implies Cor(X:Z) < 2 -l/ξ < 2 -I(X:Y)

90. 1. Area Law from Subvolume Law l X Y Z Suppose S(Y) < l/(4ξ) (“subvolume law” assumption) Since I(X:Y) < 2S(Y) < l/(2ξ), ξ-EDC implies Cor(X:Z) < 2 -l/ξ < 2 -I(X:Y) Thus: S(Z) < S(Y)

91. 2. Subvolume Law from Small Mutual Info YLYL YCYC YRYR ll/2

92. YLYL YCYC YRYR R := all except Y L Y C Y R : R R 2. Subvolume Law from Small Mutual Info ll/2 Suppose I(Y C : Y L Y R ) < l/(4ξ) (small mutual information assump.)

93. YLYL YCYC YRYR R := all except Y L Y C Y R : R R 2. Subvolume Law from Small Mutual Info ll/2 Suppose I(Y C : Y L Y R ) < l/(4ξ) (small mutual information assump.) ξ-EDC implies Cor(Y C : R) < exp(-l/(2ξ)) < exp(-I(Y C :Y L Y R ))

94. YLYL YCYC YRYR R := all except Y L Y C Y R : R R 2. Subvolume Law from Small Mutual Info Suppose I(Y C : Y L Y R ) < l/(4ξ) (small mutual information assump.) ξ-EDC implies Cor(Y C : R) < exp(-l/(2ξ)) < exp(-I(Y C :Y L Y R )) From distillation bound H(Y L Y C Y R ) = H(R) < H(Y L Y R ) ll/2

95. YLYL YCYC YRYR R := all except Y L Y C Y R : R R 2. Subvolume Law from Small Mutual Info Suppose I(Y C : Y L Y R ) < l/(4ξ) (small mutual information assump.) ξ-EDC implies Cor(Y C : R) < exp(-l/(2ξ)) < exp(-I(Y C :Y L Y R )) From distillation bound H(Y L Y C Y R ) = H(R) < H(Y L Y R ) Finally H(Y C ) ≤ H(Y C ) + H(Y L Y R ) – H(Y L Y C Y R ) = I(Y C :Y L Y R ) ≤ l/(4ξ) ll/2

96. Getting Area Law Z To prove area law for Z it suffices to find a not-so-far and not-so-large region Y L Y C Y R with small mutual information We show it with not-so-far = not-so-large = exp(O(ξ))

97. Getting Area Law YLYL YCYC YRYR Since I(Y C : Y L Y R ) < l/(4ξ) with l = exp(O(ξ)) by part 2, H(Y C ) < l/(4ξ) by part 1, H(Z’) < l/(4ξ) by subadditivity and Araki-Lieb: H(X) < exp(O(ξ)) Z’ Z

98. < 2 O(1/ε) 3. Getting Small Mutual Info. X Lemma (Saturation Mutual Info.) Given a site s, for all ε > 0 there is a region Y 2l := Y L,l/2 Y C,l Y R,l/2 of size 2l with 1 < l < 2 O(1/ε) at a distance < 2 O(1/ε) from s s.t. I(Y C,l :Y L,l/2 Y R,l/2 ) < εl Proof: Easy adaptation of result used by Hastings in his area law proof for gapped Hamiltonians (based on successive applications of subadditivity) s YLYL YCYC YRYR

99. Making it Work So far we have cheated, since merging only works for many copies of the state. To make the argument rigorous, we use single-shot information theory (see Nilanjana Datta’s talk) Single-Shot State Merging (Dupuis, Berta, Wullschleger, Renner ‘10) + New bound on correlations by random measurements Saturation max- Mutual Info. Proof much more involved; based on - Quantum substate theorem, - Quantum equipartition property, - Min- and Max-Entropies Calculus - EDC Assumption State Merging Saturation Mutual Info.

100. Overview Condensed Matter (CM) community always knew EDC implies area law

101. Overview Condensed Matter (CM) community always knew EDC implies area law Quantum information (QI) community gave a counterexample (hiding states)

102. Overview Condensed Matter (CM) community always knew EDC implies area law Quantum information (QI) community gave a counterexample (hiding states) QI community sorted out the trouble they gave themselves (this talk)

103. Overview Condensed Matter (CM) community always knew EDC implies area law Quantum information (QI) community gave a counterexample (hiding states) QI community sorted out the trouble they gave themselves (this talk) CM community didn’t notice either of these minor perturbations ”EDC implies Area Law” stays true!

104. Conclusions and Open problems 1.Can we improve the dependency of entropy with correlation length? 2.Can we prove area law for 2D systems? HARD! 3.Can we decide if EDC alone is enough for 2D area law? 4.See arxiv:1206.2947 for more open questions EDC implies Area Law and MPS parametrization in 1D. Proof uses state merging protocol and single-shot information theory: Tools from QIT useful to address problem in quantum many-body physics.