Quantum entanglement, topological order, and tensor category theory Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
Δ Local unitary trans. de fi nes gapped quantum phases Two gapped states, | Ψ (0) � and | Ψ (1) � (or more precisely, two ground state subspaces), are in the same phase i ff they are related through a local unitary (LU) evolution � 1 0 dg � H ( g � ) � � e − i | Ψ (1) � = P | Ψ (0) � where H ( g ) = � i O i ( g ) and O i ( g ) are local hermitian operators. Hastings, Wen 05; Bravyi, Hastings, Michalakis 10 • LU evolution = local unitary transformation : � 1 � 0 dg H ( g ) � e − i T | Ψ (1) � = P | Ψ (0) � − >finite gap ground − state subspace = | Ψ (0) � ε − > 0 • The local unitary transformations de fi ne an equivalence relation: Two gapped states related by a local unitary transformation are in the same phase. A gapped quantum phase is an equivalence class of local unitary transformations – a conjecture. Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ Δ ψ ψ A gapped quantum liquid phase: • A gapped quantum phase: N 1 N N N 2 3 4 H N 1 , H N 2 , H N 3 , H N 4 , · · · LU LU LU LU H � N 1 , H � N 2 , H � N 3 , H � N 4 , · · · ’ ’ ’ ’ N 1 N N N 2 3 4 N i +1 = sN i , s ∼ 2 • A gapped quantum liquid phase: gLU gLU gLU N 1 N N N H N 1 , H N 2 , H N 3 , H N 4 , · · · 2 3 4 LU LU LU LU H � N 1 , H � N 2 , H � N 3 , H � N 4 , · · · ’ ’ ’ ’ N k +1 = sN k , s ∼ 2 N 1 N N N 2 3 4 Generalized local unitary (gLU) trans. N N N k+1 k k − >finite gap ground − state gLU LU subspace ε − > 0 • 3+1D toric code model → a 3+1D gaped quantum liquid. • Stacking 2+1D FQH states and Haah cubic model → gapped quantum state, but not liquids. Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
Bosonic/fermionic gapped quantum phases Both local bosonic and fermionic systems have the following local property: V tot = ⊗ i V i | Ψ (1) � = | Ψ (0) � • Bosonic gapped phases are the equivalent classes of LU transformation: LU = � U ijk , which acts within V i ⊗ V j ⊗ V k , and [ U ijk , U i � j � k � ] = 0, e.g. U ijk = e i ( b i b j b † k + h . c . ) • Fermionic gapped phases are the equivalent classes of fermionic LU transformation: fLU = � U f ijk , which does not act within ijk = e i ( c i c j c † k c k + h . c . ) V i ⊗ V j ⊗ V k , but [ U f ijk , U f i � j � k � ] = 0, e.g. U f Gapped quantum liquids for bosons and fermions have very di ff erent mathematical structures Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
LU trans. de fi nes long-range entanglement (ie topo. order) For gapped systems with no symmetry : • According to Landau theory, no symmetry to break → all systems belong to one trivial phase Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
LU trans. de fi nes long-range entanglement (ie topo. order) For gapped systems with no symmetry : • According to Landau theory, no symmetry to break → all systems belong to one trivial phase • Thinking about entanglement: Chen-Gu-Wen 2010 - There are long range entangled (LRE) states - There are short range entangled (SRE) states | LRE � � = | product state � = | SRE � local unitary transformation LRE SRE state product state Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
LU trans. de fi nes long-range entanglement (ie topo. order) For gapped systems with no symmetry : • According to Landau theory, no symmetry to break → all systems belong to one trivial phase • Thinking about entanglement: Chen-Gu-Wen 2010 - There are long range entangled (LRE) states → many phases - There are short range entangled (SRE) states → one phase g | LRE � � = | product state � = | SRE � 2 topological order LRE 1 LRE 2 local unitary local unitary local unitary transformation transformation transformation phase transition LRE SRE SRE SRE LRE 1 LRE 2 SRE state product product product state state state • All SRE states belong to the same trivial phase g 1 • LRE states can belong to many di ff erent phases = di ff erent patterns of long-range entanglements de fi ned by the LU trans. = di ff erent topological orders Wen 1989 → A classi fi cation by tensor category theory Levin-Wen 05, Chen-Gu-Wen 2010 Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) • | ↑� ⊗ | ↑� + | ↓� ⊗ | ↓� + | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → more entangled Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) • | ↑� ⊗ | ↑� + | ↓� ⊗ | ↓� + | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� = ( | ↑� + | ↓� ) ⊗ ( | ↑� + | ↓� ) = | x � ⊗ | x � → unentangled Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) • | ↑� ⊗ | ↑� + | ↓� ⊗ | ↓� + | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� = ( | ↑� + | ↓� ) ⊗ ( | ↑� + | ↓� ) = | x � ⊗ | x � → unentangled • = | ↓� ⊗ | ↑� ⊗ | ↓� ⊗ | ↑� ⊗ | ↓� ... → unentangled Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) • | ↑� ⊗ | ↑� + | ↓� ⊗ | ↓� + | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� = ( | ↑� + | ↓� ) ⊗ ( | ↑� + | ↓� ) = | x � ⊗ | x � → unentangled • = | ↓� ⊗ | ↑� ⊗ | ↓� ⊗ | ↑� ⊗ | ↓� ... → unentangled • = ( | ↓↑� − | ↑↓� ) ⊗ ( | ↓↑� − | ↑↓� ) ⊗ ... → short-range entangled (SRE) entangled Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) • | ↑� ⊗ | ↑� + | ↓� ⊗ | ↓� + | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� = ( | ↑� + | ↓� ) ⊗ ( | ↑� + | ↓� ) = | x � ⊗ | x � → unentangled • = | ↓� ⊗ | ↑� ⊗ | ↓� ⊗ | ↑� ⊗ | ↓� ... → unentangled • = ( | ↓↑� − | ↑↓� ) ⊗ ( | ↓↑� − | ↑↓� ) ⊗ ... → short-range entangled (SRE) entangled � � • Crystal order: | Φ crystal � = = | 0 � x 1 ⊗ | 1 � x 2 ⊗ | 0 � x 3 ... � � = direct-product state → unentangled state (classical) Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
Quantum entanglements through examples • | ↑� ⊗ | ↓� = direct-product state → unentangled (classical) • | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� → entangled (quantum) • | ↑� ⊗ | ↑� + | ↓� ⊗ | ↓� + | ↑� ⊗ | ↓� + | ↓� ⊗ | ↑� = ( | ↑� + | ↓� ) ⊗ ( | ↑� + | ↓� ) = | x � ⊗ | x � → unentangled • = | ↓� ⊗ | ↑� ⊗ | ↓� ⊗ | ↑� ⊗ | ↓� ... → unentangled • = ( | ↓↑� − | ↑↓� ) ⊗ ( | ↓↑� − | ↑↓� ) ⊗ ... → short-range entangled (SRE) entangled � � • Crystal order: | Φ crystal � = = | 0 � x 1 ⊗ | 1 � x 2 ⊗ | 0 � x 3 ... � � = direct-product state → unentangled state (classical) • Particle condensation (super fl uid) � � | Φ SF � = � � all conf. � Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014 Quantum entanglement, topological order, and tensor category
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