Conditional Independence in Quantum Many-Body Systems

In this thesis, I will discuss how information-theoretic arguments can be used to produce sharp bounds in the studies of quantum many-body systems. The main advantage of this approach, as opposed to the conventional field-theoretic argument, is that it depends very little on the precise form of the Hamiltonian. The main idea behind this thesis lies on a number of results concerning the structure of quantum states that are conditionally independent. Depending on the application, some of these statements are generalized to quantum states that are approximately conditionally independent. These structures can be readily used in the studies of gapped quantum many-body systems, especially for the ones in two spatial dimensions. A number of rigorous results are derived, including (i) a universal upper bound for a maximal number of topologically protected states that is expressed in terms of the topological entanglement entropy, (ii) a first-order perturbation bound for the topological entanglement entropy that decays superpolynomially with the size of the subsystem, and (iii) a correlation bound between an arbitrary local operator and a topological operator constructed from a set of local reduced density matrices. I also introduce exactly solvable models supported on a three-dimensional lattice that can be used as a reliable quantum memory.

[1]  M. Hastings,et al.  An area law for one-dimensional quantum systems , 2007, 0705.2024.

[2]  M. B. Hastings,et al.  Lieb-Schultz-Mattis in higher dimensions , 2004 .

[3]  X. Wen.  THEORY OF THE EDGE STATES IN FRACTIONAL QUANTUM HALL EFFECTS , 1992 .

[4]  D. Pérez-García,et al.  PEPS as ground states: Degeneracy and topology , 2010, 1001.3807.

[5]  Daniel A. Lidar,et al.  Entanglement, fidelity, and topological entropy in a quantum phase transition to topological order , 2007, 0705.0026.

[6]  K. Wilson.  The renormalization group and critical phenomena , 1983 .

[7]  M. Hastings.  Quasi-adiabatic Continuation for Disordered Systems: Applications to Correlations, Lieb-Schultz-Mattis, and Hall Conductance , 2010, 1001.5280.

[8]  Wen,et al.  Vacuum degeneracy of chiral spin states in compactified space. , 1989, Physical review. B, Condensed matter.

[9]  Xiao-Gang Wen,et al.  Detecting topological order in a ground state wave function. , 2005, Physical review letters.

[10]  T. Grover.  Quantum Entanglement and Detection of Topological Order in Numerics , 2011, 1112.2215.

[11]  John Preskill,et al.  Logical-operator tradeoff for local quantum codes , 2010, 1011.3529.

[12]  R. Moessner,et al.  Three-dimensional resonating-valence-bond liquids and their excitations , 2003 .

[13]  H. Nagaoka,et al.  Strong Converse and Stein's Lemma in the Quantum Hypothesis Testing , 1999, quant-ph/9906090.

[14]  X. Wen,et al.  Classification of Gapped Symmetric Phases in 1D Spin Systems , 2011 .

[15]  Frank Verstraete,et al.  Matrix product state representations , 2006, Quantum Inf. Comput..

[16]  E. Knill,et al.  Reversing quantum dynamics with near-optimal quantum and classical fidelity , 2000, quant-ph/0004088.

[17]  Zohar Nussinov,et al.  A symmetry principle for topological quantum order , 2007, cond-mat/0702377.

[18]  Sergey Bravyi,et al.  Topological quantum order: Stability under local perturbations , 2010, 1001.0344.

[19]  Matthew B Hastings,et al.  Topological order at nonzero temperature. , 2011, Physical review letters.

[20]  A. Vishwanath,et al.  Entanglement Entropy of Gapped Phases and Topological Order in Three dimensions , 2011, 1108.4038.

[21]  U. Vazirani,et al.  Improved one-dimensional area law for frustration-free systems , 2011, 1111.2970.

[22]  S. Michalakis,et al.  Stability of the Area Law for the Entropy of Entanglement , 2012, 1206.6900.

[23]  J. Cardy,et al.  Entanglement entropy and quantum field theory , 2004, hep-th/0405152.

[24]  N. Regnault,et al.  Real-space entanglement spectrum of quantum Hall states , 2011, 1111.2810.

[25]  I. Devetak,et al.  Exact cost of redistributing multipartite quantum states. , 2006, Physical review letters.

[26]  John Preskill,et al.  Topological entanglement entropy. , 2005, Physical Review Letters.

[27]  Fernando G. S. L. Brandão,et al.  Exponential Decay of Correlations Implies Area Law , 2012, Communications in Mathematical Physics.

[28]  H. Bombin,et al.  Exact topological quantum order in D=3 and beyond : Branyons and brane-net condensates , 2006, cond-mat/0607736.

[29]  Gregory W. Moore,et al.  Nonabelions in the fractional quantum Hall effect , 1991 .

[30]  M. B. Hastings,et al.  A Short Proof of Stability of Topological Order under Local Perturbations , 2010, 1001.4363.

[31]  Raymond Laflamme,et al.  A Theory of Quantum Error-Correcting Codes , 1996 .

[32]  S. Bravyi,et al.  Energy landscape of 3D spin Hamiltonians with topological order. , 2011, Physical review letters.

[33]  Yi Zhang,et al.  Quasiparticle statistics and braiding from ground state entanglement , 2011, 1111.2342.

[34]  K. Audenaert,et al.  Discriminating States: the quantum Chernoff bound. , 2006, Physical review letters.

[35]  G. Vidal.  Class of quantum many-body states that can be efficiently simulated. , 2006, Physical review letters.

[36]  A. Winter,et al.  Robustness of Quantum Markov Chains , 2006, quant-ph/0611057.

[37]  M. B. Hastings,et al.  Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance , 2005 .

[38]  Isaac H. Kim.  Operator extension of strong subadditivity of entropy , 2012, 1210.5190.

[39]  Matthew B Hastings,et al.  Area laws in quantum systems: mutual information and correlations. , 2007, Physical review letters.

[40]  Bruno Nachtergaele,et al.  Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems , 2007, 0712.3820.

[41]  N. Regnault,et al.  Entanglement gap and a new principle of adiabatic continuity. , 2009, Physical review letters.

[42]  Matthew B. Hastings,et al.  Topological entanglement entropy of a Bose-Hubbard spin liquid , 2011, 1102.1721.

[43]  Decay of correlations in Fermi systems at nonzero temperature. , 2004, Physical review letters.

[44]  Beni Yoshida,et al.  Feasibility of self-correcting quantum memory and thermal stability of topological order , 2011, 1103.1885.

[45]  J. Ignacio Cirac,et al.  Entanglement spectrum and boundary theories with projected entangled-pair states , 2011, 1103.3427.

[46]  Jeongwan Haah.  Local stabilizer codes in three dimensions without string logical operators , 2011, 1101.1962.

[47]  Bruno Nachtergaele,et al.  A Multi-Dimensional Lieb-Schultz-Mattis Theorem , 2006, math-ph/0608046.

[48]  H. Araki.  Gibbs states of a one dimensional quantum lattice , 1969 .

[49]  E. Lieb.  Convex trace functions and the Wigner-Yanase-Dyson conjecture , 1973 .

[50]  I. Klich.  On the stability of topological phases on a lattice , 2009, 0912.0945.

[51]  Claudio Chamon,et al.  Quantum glassiness in strongly correlated clean systems: an example of topological overprotection. , 2004, Physical review letters.

[52]  Umesh Vazirani,et al.  An area law and sub-exponential algorithm for 1D systems , 2013, 1301.1162.

[53]  Xiao-Gang Wen,et al.  Topological entanglement Rényi entropy and reduced density matrix structure. , 2009, Physical review letters.

[54]  H.,et al.  Convex Trace Functions and the Wigner-Yanase-Dyson Conjecture , 2022 .

[55]  Hui Li,et al.  Entanglement spectrum as a generalization of entanglement entropy: identification of topological order in non-Abelian fractional quantum Hall effect states. , 2008, Physical review letters.

[56]  Tobias J Osborne,et al.  Bounds on the speed of information propagation in disordered quantum spin chains. , 2007, Physical review letters.

[57]  J. Eisert,et al.  Area laws for the entanglement entropy - a review , 2008, 0808.3773.

[58]  David Poulin,et al.  Operator quantum error correction , 2006, Quantum Inf. Comput..

[59]  Luigi Accardi,et al.  Conditional expectations in von Neumann algebras and a theorem of Takesaki , 1982 .

[60]  H. Bombin,et al.  Topological order with a twist: Ising anyons from an Abelian model. , 2010, Physical review letters.

[61]  Matthew B. Hastings,et al.  Quantization of Hall Conductance For Interacting Electrons Without Averaging Assumptions , 2009, 0911.4706.

[62]  Paolo Zanardi,et al.  Ground state entanglement and geometric entropy in the Kitaev model , 2005 .

[63]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[64]  N. Read,et al.  Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states , 2012, 1207.7119.

[65]  M. Ruskai.  Remarks on Kim's strong subadditivity matrix inequality: Extensions and equality conditionsa) , 2012, 1211.0049.

[66]  D. Perez-Garcia,et al.  Criticality, the area law, and the computational power of PEPS , 2006, quant-ph/0601075.

[67]  Jeongwan Haah,et al.  Quantum self-correction in the 3D cubic code model. , 2011, Physical review letters.

[68]  D. Pérez-García,et al.  Classifying quantum phases using Matrix Product States and PEPS , 2010, 1010.3732.

[69]  R.Ionicioiu,et al.  Ground state entanglement and geometric entropy in the Kitaev's model , 2004, quant-ph/0406202.

[70]  Bruno Nachtergaele,et al.  Lieb-Robinson Bounds and the Exponential Clustering Theorem , 2005, math-ph/0506030.

[71]  A. Hamma,et al.  Topological order, entanglement, and quantum memory at finite temperature , 2011, 1112.0947.

[72]  Nicolas Delfosse,et al.  Tradeoffs for reliable quantum information storage in surface codes and color codes , 2013, 2013 IEEE International Symposium on Information Theory.

[73]  F. Verstraete,et al.  Lieb-Robinson bounds and the generation of correlations and topological quantum order. , 2006, Physical review letters.

[74]  Xiao-Gang Wen,et al.  String-net condensation: A physical mechanism for topological phases , 2004, cond-mat/0404617.

[75]  X. Qi,et al.  Topological Nematic States and Non-Abelian Lattice Dislocations , 2011, 1112.3311.

[76]  M. Fannes.  A continuity property of the entropy density for spin lattice systems , 1973 .

[77]  D. Petz.  Monotonicity of quantum relative entropy revisited , 2002, quant-ph/0209053.

[78]  Bruno Nachtergaele,et al.  Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems , 2011, 1102.0842.

[79]  David Poulin,et al.  Lieb-Robinson bound and locality for general markovian quantum dynamics. , 2010, Physical review letters.

[80]  David Poulin,et al.  Local topological order inhibits thermal stability in 2D. , 2012, Physical review letters.

[81]  J. Cirac,et al.  Topological and entanglement properties of resonating valence bond wave functions , 2012, 1202.0947.

[82]  B. Halperin.  Quantized Hall conductance, current carrying edge states, and the existence of extended states in a two-dimensional disordered potential , 1982 .

[83]  S. Furukawa,et al.  Topological entanglement entropy in the quantum dimer model on the triangular lattice , 2006, cond-mat/0612227.

[84]  L. Landau.  Fault-tolerant quantum computation by anyons , 2003 .

[85]  D. Poulin,et al.  Quantum Graphical Models and Belief Propagation , 2007, 0708.1337.

[86]  Isaac H. Kim.  Perturbative analysis of topological entanglement entropy from conditional independence , 2011, 1109.3496.

[87]  Justyna P. Zwolak,et al.  Stability of Frustration-Free Hamiltonians , 2011, 1109.1588.

[88]  E. Effros.  A matrix convexity approach to some celebrated quantum inequalities , 2008, Proceedings of the National Academy of Sciences.

[89]  Haldane,et al.  Finite-size studies of the incompressible state of the fractionally quantized Hall effect and its excitations. , 1985, Physical review letters.

[90]  R. Laughlin.  Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations , 1983 .

[91]  J K Slingerland,et al.  Evaluation of ranks of real space and particle entanglement spectra for large systems. , 2012, Physical review letters.

[92]  B. Terhal,et al.  Topological order in an exactly solvable 3D spin model , 2010, 1006.4871.

[93]  F. Verstraete,et al.  Criticality, the area law, and the computational power of projected entangled pair states. , 2006, Physical review letters.

[94]  N. Read,et al.  Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect , 1999, cond-mat/9906453.

[95]  M. Nussbaum,et al.  THE CHERNOFF LOWER BOUND FOR SYMMETRIC QUANTUM HYPOTHESIS TESTING , 2006, quant-ph/0607216.

[96]  G. Vidal,et al.  Exact entanglement renormalization for string-net models , 2008, 0806.4583.

[97]  David Pérez-García,et al.  Classifying quantum phases using matrix product states and projected entangled pair states , 2011 .

[98]  H. Bombin,et al.  Topological quantum distillation. , 2006, Physical review letters.

[99]  Claudio Castelnovo,et al.  Topological order in a three-dimensional toric code at finite temperature , 2008, 0804.3591.

[100]  F. Verstraete,et al.  Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions , 2004, cond-mat/0407066.

[101]  D. Ivanov.  Non-Abelian statistics of half-quantum vortices in p-wave superconductors. , 2000, Physical review letters.

[102]  Frank Pollmann,et al.  Topological Phases of One-Dimensional Fermions: An Entanglement Point of View , 2010, 1008.4346.

[103]  D. Petz.  Quasi-entropies for finite quantum systems , 1986 .

[104]  I. Klich,et al.  Lieb-Robinson bounds for commutator-bounded operators , 2009, 0912.4544.

[105]  Alexei Kitaev,et al.  Topological phases of fermions in one dimension , 2010, 1008.4138.

[106]  G. Vidal,et al.  Entanglement renormalization and topological order. , 2007, Physical review letters.

[107]  N. Mermin,et al.  Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models , 1966 .

[108]  A. O. Pittenger,et al.  Separability and Fourier representations of density matrices , 2000, quant-ph/0001014.

[109]  K. Wilson.  Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior , 1971 .

[110]  Tomohiro Ogawa,et al.  Strong converse and Stein's lemma in quantum hypothesis testing , 2000, IEEE Trans. Inf. Theory.

[111]  M. Hastings,et al.  Markov entropy decomposition: a variational dual for quantum belief propagation. , 2010, Physical review letters.

[112]  Alexei Kitaev,et al.  Anyons in an exactly solved model and beyond , 2005, cond-mat/0506438.

[113]  L. Cincio,et al.  Characterizing topological order by studying the ground States on an infinite cylinder. , 2012, Physical review letters.

[114]  Schumacher,et al.  Quantum coding. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[115]  Wen,et al.  Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces. , 1990, Physical review. B, Condensed matter.

[116]  Robert B. Laughlin,et al.  Quantized Hall conductivity in two-dimensions , 1981 .

[117]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[118]  David Pérez-García,et al.  Gapless Hamiltonians for the toric code using the projected entangled pair state formalism. , 2011, Physical review letters.

[119]  Peter W. Shor,et al.  Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem , 2001, IEEE Trans. Inf. Theory.

[120]  N. S. Barnett,et al.  Private communication , 1969 .

[121]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[122]  Michael A. Nielsen,et al.  A simple proof of the strong subadditivity inequality , 2005, Quantum Inf. Comput..

[123]  S. Sondhi.  An RVB Phase in the Triangular Lattice Quantum Dimer Model , 2001 .

[124]  B. Terhal,et al.  A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes , 2008, 0810.1983.

[125]  Isaac H. Kim.  Determining the structure of the real-space entanglement spectrum from approximate conditional independence , 2013 .

[126]  F. Verstraete,et al.  Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems , 2008, 0907.2796.

[127]  Michael Reed,et al.  Methods of modern mathematical physics (vol.) I : functional analysis / Reed Michael, Barry Simon , 1980 .

[128]  Kareljan Schoutens,et al.  Entanglement entropy in fermionic Laughlin states. , 2007, Physical review letters.

[129]  John B. Kogut,et al.  An introduction to lattice gauge theory and spin systems , 1979 .

[130]  Isaac H. Kim.  Long-range entanglement is necessary for a topological storage of quantum information. , 2013, Physical review letters.

[131]  White,et al.  Density matrix formulation for quantum renormalization groups. , 1992, Physical review letters.

[132]  M. Fannes,et al.  On thermalization in Kitaev's 2D model , 2008, 0810.4584.

[133]  Michal Horodecki,et al.  On Thermal Stability of Topological Qubit in Kitaev's 4D Model , 2008, Open Syst. Inf. Dyn..

[134]  Zohar Nussinov,et al.  Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems , 2007, 0709.2717.

[135]  M. Lewenstein,et al.  Quantum Entanglement , 2020, Quantum Mechanics.

[136]  Matthew B. Hastings,et al.  Spectral Gap and Exponential Decay of Correlations , 2005 .

[137]  B. Nachtergaele,et al.  Approximating the ground state of gapped quantum spin systems , 2009, 0904.4642.

[138]  M. Fannes,et al.  Finitely correlated states on quantum spin chains , 1992 .

[139]  M. Hastings.  Quantum belief propagation: An algorithm for thermal quantum systems , 2007, 0706.4094.

[140]  C. Castelnovo,et al.  Entanglement and topological entropy of the toric code at finite temperature , 2007, 0704.3616.

[141]  Tobias J. Osborne.  Simulating adiabatic evolution of gapped spin systems , 2007 .

[142]  Yi Zhang,et al.  Topological entanglement entropy of Z 2 spin liquids and lattice Laughlin states , 2011, 1106.0015.

[143]  M. Reed,et al.  Fourier Analysis, Self-Adjointness , 1975 .

[144]  Frank Verstraete,et al.  Peps as unique ground states of local hamiltonians , 2007, Quantum Inf. Comput..

[145]  J. Preskill,et al.  Topological quantum memory , 2001, quant-ph/0110143.

[146]  X. Qi,et al.  Entanglement entropy and entanglement spectrum of the Kitaev model. , 2010, Physical review letters.

[147]  F. Hiai,et al.  The proper formula for relative entropy and its asymptotics in quantum probability , 1991 .

[148]  Leon Balents,et al.  Identifying topological order by entanglement entropy , 2012, Nature Physics.

[149]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[150]  S. Bravyi,et al.  Quantum self-correction in the 3D cubic code model. , 2013, Physical review letters.

[151]  Bruno Nachtergaele,et al.  Propagation of Correlations in Quantum Lattice Systems , 2006, math-ph/0603064.

[152]  Mikhail N. Vyalyi,et al.  Classical and Quantum Computation , 2002, Graduate studies in mathematics.

[153]  D. Perez-Garcia,et al.  Thermal states of anyonic systems , 2008, 0812.4975.

[154]  David Poulin,et al.  Tradeoffs for reliable quantum information storage in 2D systems , 2010, Quantum Cryptography and Computing.

[155]  K. Wilson.  Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture , 1971 .

[156]  A. Winter,et al.  Communications in Mathematical Physics Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality , 2022 .

[157]  E. Rezayi,et al.  Bipartite entanglement entropy in fractional quantum Hall states , 2007, 0705.4176.

[158]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[159]  B. Nachtergaele,et al.  ON THE EXISTENCE OF THE DYNAMICS FOR ANHARMONIC QUANTUM OSCILLATOR SYSTEMS , 2009, 0909.2249.

[160]  M. Ruskai.  Another Short and Elementary Proof of Strong Subadditivity of Quantum Entropy , 2006, quant-ph/0604206.

[161]  D. M. Appleby.  SIC-POVMs and the Extended Clifford Group , 2004 .

[162]  L. Erdős,et al.  Quantum Dynamics with Mean Field Interactions: a New Approach , 2008, 0804.3774.

[163]  D. Poulin,et al.  Coarse-grained belief propagation for simulation of interacting quantum systems at all temperatures , 2009, 0910.2299.

[164]  Matthew B. Hastings,et al.  Topological order at nonzero temperature. , 2011, Physical review letters.

[165]  M. B. Hastings,et al.  Locality in Quantum Systems , 2010, 1008.5137.