Tensors: From Entanglement to Computational Complexity Matthias Christandl (Copenhagen & MIT) Peter Vrana (Budapest) and Jeroen Zuiddam (Amsterdam->IAS) arXiv:1709.0781, Proc. STOC’18
Outline • Two motivations • Resource theory of tensors • Entanglement polytopes • Tensor tensor …. tensor ⊗ ⊗ ⊗ • Quantum functionals
Two motivations
Quantum states 1 1 1 State of a classical 0 0 0 system 1 1 1 or (3 bits) 0 0 0 State of a 1 1 1 quantum 0 0 0 system (3 qubits) 1 1 1 + ✓ 1 ◆ 0 0 0 e 0 = 0 ✓ 0 t = e 0 ⊗ e 0 ⊗ e 0 + e 1 ⊗ e 1 ⊗ e 1 ◆ e 1 = 1
Quantum state=tensor t ∈ C d ⊗ C d ⊗ C d d X t = t ijk e i ⊗ e j ⊗ e k i,j,k =1
GHZ state = unit tensor Greenberger-Horne-Zeilinger r r X h r i = e i ⌦ e i ⌦ e i i =1 1 1
Local operations 1 1 1 Local trans- 0 0 0 formation: Flip first bit 1 1 1 Local 0 0 0 trans- formation: 1 1 1 + Flip first 0 0 0 qubit ✓ ◆ 0 1 1 0 t = e 1 ⊗ e 0 ⊗ e 0 + e 0 ⊗ e 1 ⊗ e 1 t = e 0 ⊗ e 0 ⊗ e 0 + e 1 ⊗ e 1 ⊗ e 1
Local operations=restrictions t ≥ t 0 if ( a ⊗ b ⊗ c ) t = t 0 for some matrices a, b, c Linear combination of slices
3 qubits Greenberger-Horne-Zeilinger GHZ-state e 0 ⊗ e 0 ⊗ e 0 + e 1 ⊗ e 1 ⊗ e 1 Einstein-Podolsky-Rosen (EPR)-state W-state e 0 ⊗ e 0 ⊗ e 1 + e 0 ⊗ e 1 ⊗ e 0 + e 1 ⊗ e 0 ⊗ e 0 ≈ e 0 ⊗ e 0 ⊗ e 0 + e 1 ⊗ e 1 ⊗ e 0 e 0 ⊗ e 0 ⊗ e 0 + e 0 ⊗ e 1 ⊗ e 1 e 0 ⊗ e 0 ⊗ e 0 + e 1 ⊗ e 0 ⊗ e 1 e 0 ⊗ e 0 ⊗ e 0 unentangled state
Algebraic Complexity M ( d ) = algebra of d × d complex matrices Mamu ( d ) : M ( d ) × M ( d ) → M ( d ) bilinear ( A, B ) 7! A · B d x d 3 multiplications = d
Bilinear maps=tensors Mamu ( d ) : M ( d ) × M ( d ) × M ( d ) ∗ → C ( A, B, C ) 7! trA · B · C d e ij = e i ⊗ e j X Mamu ( d ) = e ij ⊗ e jk ⊗ e ki i,j,k =1 d X = ( e i ⊗ e j ) ⊗ ( e j ⊗ e k ) ⊗ ( e k ⊗ e i ) i,j,k =1 d EPR states
Complexity=Tensor rank Strassen: # elementary multiplications = tensor rank Do you like Strassen’s 2 7 decomposition? ≥ Then you might want to look at some tensor surgery next! Ch. & Zuiddam, e 00 ⊗ e 00 ⊗ e 00 + e 11 ⊗ e 11 ⊗ e 11 Comp. Compl. 2018 e 01 ⊗ e 10 ⊗ e 00 + e 10 ⊗ e 01 ⊗ e 11 e ± := e 0 ± e 1 arXiv:1606.04085 e 01 ⊗ e 11 ⊗ e 10 + e 10 ⊗ e 00 ⊗ e 01 e 00 ⊗ e 01 ⊗ e 10 + e 11 ⊗ e 10 ⊗ e 01 = e − 1 ⊗ e 1+ ⊗ e 00 + e 1+ ⊗ e 00 ⊗ e − 1 + e 00 ⊗ e − 1 ⊗ e 1+ − e − 0 ⊗ e 0+ ⊗ e 11 − e 0+ ⊗ e 11 ⊗ e − 0 − e 11 ⊗ e − 0 ⊗ e 0+ + ( e 00 + e 11 ) ⊗ ( e 00 + e 11 ) ⊗ ( e 00 + e 11 )
Resource theory of tensors
Resource theory of tensors free operations valuable resource t ≥ t 0 if ( a ⊗ b ⊗ c ) t = t 0 • Restriction for some matrices a, b, c r • Unit X h r i = e i ⌦ e i ⌦ e i i =1 • Rank R ( t ) = min { r : h r i � t } r X = min { r : t = α i ⊗ β i ⊗ γ i } i =1 • Subrank Q ( t ) = max { r : t � h r i}
Restriction t ≥ t 0 if ( a ⊗ b ⊗ c ) t = t 0 for some matrices a, b, c = t 0 if t ≥ t 0 and t 0 ≥ t t ∼ i ff ( a ⊗ b ⊗ c ) t = t 0 for invertible a, b, c G = GL ( d ) × GL ( d ) × GL ( d ) i ff G.t = G.t 0 Classifying orbits Deciding restriction and their relations
GHZ state Degeneration W state ( e 0 + ✏ e 1 ) ⊗ 3 − e ⊗ 3 0 = ✏ ( e 0 ⊗ e 0 ⊗ e 1 + e 0 ⊗ e 1 ⊗ e 0 + e 1 ⊗ e 0 ⊗ e 0 ) + O ( ✏ 2 ) t D t 0 if t ✏ → t 0 , t ≥ t ✏ ✏ 7! 0 Classifying orbit closures and Deciding degeneration their relations
Deciding degeneration • Orbit closures are G-invariant algebraic varieties t 6 D t 0 i ff there exists G � covariant polynomial f : f ( t ) 6 = f ( t 0 ) f ( t ) = 0 , but f ( t 0 ) 6 = 0 • Example: e 0 ⊗ e 0 ⊗ e 0 + e 1 ⊗ e 1 ⊗ e 1 f=Cayley hyperdeterminant e 0 ⊗ e 0 ⊗ e 1 + e 0 ⊗ e 1 ⊗ e 0 + e 1 ⊗ e 0 ⊗ e 0 ≈
be happy with partial information Entanglement polytopes
Local spectra A ∈ C d ⊗ C d ⊗ C d � � t 0 A ) 2 λ A = singular values ( t 0 B ) 2 λ B = singular values ( t 0 normalised t 0 ∈ C d ⊗ C d ⊗ C d ordered probability distribution B ∈ · · · =spectrum of reduced density operator t 0 C d ⊗ C d � ⊗ C d � t 0 C ∈ C ) 2 λ C = singular values ( t 0
Entanglement polytopes Reduced density matrices EPR GHZ product =all W marginal polytope 14 Ch-Mitchison, Klyachko, Walter-Doran-Gross-Ch, Daftuar-Hayden (2004) Sawicki-Oszmaniec-Kus (2010) based on Brion based in part on Kirwan
Experimental Detection 1 ρ λ (1) λ (2) max max ψ 1 λ (2) λ (3) max max λ (3) 0 . 5 λ (1) C max max 1 Science THE INTERNATIONAL WEEKLY JOURNAL OF SCIENCE • if measured value XX XXXX 2013 I www.nature.com/nature I E10 Headline headline Subline sublkdjfksdfdf – not in W-polytope lkdsjfdkjfdf – Then must be in GHZ-class! • easy test for entanglement!
A little more partial information? • Orbit closures are G-invariant algebraic varieties t 6 D t 0 i ff there exists G � covariant polynomial f : f ( t ) 6 = f ( t 0 ) f ( t ) = 0 , but f ( t 0 ) 6 = 0 • f’s come in types indexed by 3 Young diagrams . λ A = # boxes=degree
Weyl’s construction S n acts GL ( d ) acts • Schur-Weyl duality ( C d ) ⊗ n ∼ M [ λ ] ⊗ V λ = λ orthogonal projector onto component • P λ A λ A t ⊗ n ( P λ A ⊗ P λ B ⊗ P λ C ) | {z } =: P λ X ! t ⊗ n = X v i v ∗ v ∗ = i f i ( t ) i i i
Relaxation • Orbit closures are G-invariant algebraic varieties t 6 D t 0 i ff there exists G � covariant polynomial f : f ( t ) 6 = f ( t 0 ) f ( t ) = 0 , but f ( t 0 ) 6 = 0 if there is λ s.th. P λ t ⌦ n = 0 but P λ t 0⌦ n 6 = 0 occurrence obstructions (Geometric Complexity Theory) Mulmuley-Sohoni, Strassen, Bürgisser-Ikenmeyer, …
Entanglement polytopes Invariant-theoretic ✓ 4 ◆ 8 , 3 8 , 1 . 8 EPR GHZ product =all W g λ 6 = 0 Kronecker P λ t ⊗ n 6 = 0 = marginal polytope 14
tensor tensor ... tensor ⊗ ⊗ ⊗
(Quantum) information theory Source Encoder Storage Decoder Shannon: storage cost= all bits Source Source Encoder Decoder Storage . . . . . . Source Shannon: storage cost= H(X) bits/symbol
A small observation d = 2 n e i = e i 1 i 2 ··· i n = e i 1 ⊗ e i 2 ⊗ · · · ⊗ e i n d 2 ! 2 ! 2 ! X X X X e i ⊗ e i = e i 1 ⊗ e i 1 ⊗ e i 2 ⊗ e i 2 ⊗ · · · ⊗ e i n ⊗ e i n i 1 =1 i 2 =1 i n =1 i =1 = ( e 0 ⊗ e 0 + e 1 ⊗ e 1 ) ⊗ n d e i ⊗ e i ⊗ e i = ( e 0 ⊗ e 0 ⊗ e 0 + e 1 ⊗ e 1 ⊗ e 1 ) ⊗ n X h d i = = h 2 i ⊗ n i =1 ⊗ n 0 1 d 2 X X = Mamu (2) ⊗ n Mamu ( d ) = e ij ⊗ e jk ⊗ e ki = e ij ⊗ e jk ⊗ e ki @ A i,j,k =1 i,j,k =1
Algebraic complexity theory d x = d d 3 multiplications • Exponent of matrix multiplication O ( d ω ) 2 ≤ 2 . 38 ≤ · · · ≤ 2 . 8 ≤ 3 …, Coppersmith-Winograd Strassen ω = inf { r : h 2 i ⊗ ( nr + o ( n )) � Mamu (2) ⊗ n } h 2 i ⊗ 2 n + o ( n ) � Mamu (2) ⊗ n • Conjecture:
Asymptotic resource theory t & t 0 if t ⌦ n + o ( n ) ≥ t 0⌦ n • Asymp. restriction r • Unit X h r i = e i ⌦ e i ⌦ e i i =1 ˜ 1 n →∞ R ( t ⊗ n ) • Asymp. rank R ( t ) := lim n ˜ 1 • Asymp. subrank n →∞ Q ( t ⊗ n ) Q ( t ) := lim n ˜ R ( Mamu (2)) = 2 ω
Asymptotic analogue of completeness of invariants for degeneration Strassen’s spectral theorem t & t 0 i ff F ( t ) ≥ F ( t 0 ) for all F : under restriction F monotone F ( s ) ≥ F ( s 0 ) for all s ≥ s 0 F normalised F ( h r i ) = r F multiplicative F ( s ⊗ s 0 ) = F ( s ) · F ( s 0 ) F additive F ( s ⊕ s 0 ) = F ( s ) + F ( s 0 ) ˜ R ( t ) = max F ( t ) easy ⇒ F difficult ⇐ ˜ Q ( t ) = min F ( t ) every F is an obstruction F
What are the F’s? • Existence non-constructive – Compact space worth of them – 3 Gauge points: ranks of slicings – Construction of others open since ’80s • Theorem also true for subclasses of tensors – Oblique tensor – Strassen’s support functionals
Main Result: Quantum functionals θ = ( θ A , θ B , θ C ) probability distribution e.g. θ A = θ B = θ C = 1 3 operator scaling E θ ( t ) := max λ ∈ ∆ ( t ) { θ A H ( λ A ) + θ B H ( λ B ) + θ C H ( λ C ) } entanglement polytope F θ ( t ) := 2 E θ ( t ) quantum functionals Measures distance to origin (relative entropy distance) 14 ✓ 1 ◆ 2 E ( 1 1 0 h ≈ 0 . 92 3 , 1 3 , 1 3 ) 3 3
Main Result: Quantum functionals E θ ( t ) := max λ ∈ ∆ ( t ) { θ A H ( λ A ) + θ B H ( λ B ) + θ C H ( λ C ) } F θ ( t ) := 2 E θ ( t ) easy, since polytope gets smaller under restriction quantum functional gets smaller F θ monotone easy, since polytope of unit tensor contains uniform point F ( h r i ) = r F θ normalised F θ multiplicative similar to multiplicativity, see paper F θ additive
Multiplicativity F θ ( t ⊗ t 0 ) = F θ ( t ) · F θ ( t 0 ) F θ ( t ) := 2 E θ ( t ) E θ ( t ⊗ t 0 ) = E θ ( t ) + E θ ( t 0 ) ≥ ≤ Entanglement polytope: Entanglement polytopes: Reduced density matrices Invariant-theoretic
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