Motivation Results Applications and outlook Quantum logarithmic Sobolev inequalities and rapid mixing Michael Kastoryano and Kristan Temme 1 Dahlem Center for Complex Quantum Systems, Freie Universit¨ at Berlin, 14195 Berlin, Germany 2 Center for Theoretical Physics, MIT, Cambridge, MA 02139, USA QIP 2013 Beijing January 21, 2013 Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Outline 1 Motivation Setting Convergence rates 2 Results Mixing times Mathematical results 3 Applications and outlook Quantum Expanders Liouvillian Complexity Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Setting Setting • We consider only finite dimensional state spaces. Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Setting Setting • We consider only finite dimensional state spaces. • We consider an open quantum system described by a Markovian master equation k − 1 d � L k ρ L † 2 { L † dt ρ t = L ( ρ ) = i [ H , ρ ] + k L k , ρ } (1) k Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Setting Setting • We consider only finite dimensional state spaces. • We consider an open quantum system described by a Markovian master equation k − 1 d � L k ρ L † 2 { L † dt ρ t = L ( ρ ) = i [ H , ρ ] + k L k , ρ } (1) k • We assume that the Liouvillian is primitive , meaning that L has a unique full-rank stationary state σ > 0 Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Setting Setting • We consider only finite dimensional state spaces. • We consider an open quantum system described by a Markovian master equation k − 1 d � L k ρ L † 2 { L † dt ρ t = L ( ρ ) = i [ H , ρ ] + k L k , ρ } (1) k • We assume that the Liouvillian is primitive , meaning that L has a unique full-rank stationary state σ > 0 • If Γ σ L = L ∗ Γ σ , where σ is the stationary state of L and Γ σ ( X ) = √ σ X √ σ , the L is reversible . Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Setting Setting • We consider only finite dimensional state spaces. • We consider an open quantum system described by a Markovian master equation k − 1 d � L k ρ L † 2 { L † dt ρ t = L ( ρ ) = i [ H , ρ ] + k L k , ρ } (1) k • We assume that the Liouvillian is primitive , meaning that L has a unique full-rank stationary state σ > 0 • If Γ σ L = L ∗ Γ σ , where σ is the stationary state of L and Γ σ ( X ) = √ σ X √ σ , the L is reversible . Note: we do not yet make any assumptions about locality or geometry at this point. Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Convergence rates The question Let L be the generator of a primitive reversible quantum dynamical semigroup. Given ǫ > 0, for what τ ≥ t > 0 do we have || ρ t − σ || 1 ≤ ǫ ? (2) Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Convergence rates The question Let L be the generator of a primitive reversible quantum dynamical semigroup. Given ǫ > 0, for what τ ≥ t > 0 do we have || ρ t − σ || 1 ≤ ǫ ? (2) The answer: general convergence theorem Let λ > 0 be the spectral gap of L , then for any b ≤ λ , there exists a finite A such that || ρ t − σ || 1 ≤ Ae − bt . (3) Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Convergence rates The question Let L be the generator of a primitive reversible quantum dynamical semigroup. Given ǫ > 0, for what τ ≥ t > 0 do we have || ρ t − σ || 1 ≤ ǫ ? (2) The answer: general convergence theorem Let λ > 0 be the spectral gap of L , then for any b ≤ λ , there exists a finite A such that || ρ t − σ || 1 ≤ Ae − bt . (3) What are good choices for A and b ? We will argue that the Log Sobolev machinery is the finest available to answer this question. Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Convergence rates Applications 1 Unital quantum channels and random unitary maps (the fast scrambling conjecture). Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Convergence rates Applications 1 Unital quantum channels and random unitary maps (the fast scrambling conjecture). 2 Quantum memories: Davies generators of stabilizer Hamiltonians. Rigorous no-go theorems. Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Convergence rates Applications 1 Unital quantum channels and random unitary maps (the fast scrambling conjecture). 2 Quantum memories: Davies generators of stabilizer Hamiltonians. Rigorous no-go theorems. 3 Liouvillian complexity: what can we say about systems whose Log Sobolev constant is independent of the system size? Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Convergence rates Applications 1 Unital quantum channels and random unitary maps (the fast scrambling conjecture). 2 Quantum memories: Davies generators of stabilizer Hamiltonians. Rigorous no-go theorems. 3 Liouvillian complexity: what can we say about systems whose Log Sobolev constant is independent of the system size? 4 Dissipative algorithms? Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Convergence rates Applications 1 Unital quantum channels and random unitary maps (the fast scrambling conjecture). 2 Quantum memories: Davies generators of stabilizer Hamiltonians. Rigorous no-go theorems. 3 Liouvillian complexity: what can we say about systems whose Log Sobolev constant is independent of the system size? 4 Dissipative algorithms? 5 Concentration of measure? Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook A few definitions to start with... non-commutative L p spaces • The L p inner product. For two hermitian operators f , g : � f , g � σ = tr [Γ σ ( f ) g ] ≡ tr [ σ 1 / 2 f σ 1 / 2 g ] . (4) Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook A few definitions to start with... non-commutative L p spaces • The L p inner product. For two hermitian operators f , g : � f , g � σ = tr [Γ σ ( f ) g ] ≡ tr [ σ 1 / 2 f σ 1 / 2 g ] . (4) • The L p norm. For any hermitian operator f : 1 / p || f || p ,σ = tr [ | Γ 1 / p ( f ) | p ] (5) σ Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook A few more definitions... Variance and Entropy functionals • The variance Var σ ( g ) = tr [ Γ σ ( g ) g ] − tr [ Γ σ ( g )] 2 . (6) Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook A few more definitions... Variance and Entropy functionals • The variance Var σ ( g ) = tr [ Γ σ ( g ) g ] − tr [ Γ σ ( g )] 2 . (6) • The L p relative entropies. For any hermitian operator f : Ent 1 ( f ) = tr [Γ σ ( f )(log(Γ σ ( f )) − log( σ ))] (7) − tr [Γ σ ( f )] log( tr [Γ σ ( f )]) (8) � 2 � � � Γ 1 / 2 Γ 1 / 2 Ent 2 ( f ) = tr [ σ ( f ) log σ ( f ) ] (9) − 1 � 2 � Γ 1 / 2 2 tr [ σ ( f ) log ( σ )] − 1 2 � f � 2 � f � 2 � � 2 ,σ log . 2 ,σ Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Yet more... (sorry!) Dirichlet Forms − 1 E 1 ( f ) = 2 tr [Γ σ ( L ( f ))(log(Γ σ ( f )) − log( σ ))] (10) E 2 ( f ) = − � f , L ( f ) � σ . (11) Useful identities: Var ( Γ − 1 Ent 2 ( Γ − 1 σ ( ρ )) = χ 2 ( ρ, σ ) , σ ( ρ )) = S ( ρ || σ ) (12) Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Spectral Gap and Log-Sobolev constant • The spectral gap of L : E 2 ( f ) λ = inf (13) Var σ ( f ) f � =0 Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Spectral Gap and Log-Sobolev constant • The spectral gap of L : E 2 ( f ) λ = inf (13) Var σ ( f ) f � =0 • The (1 , 2)- logarithmic Sobolev constant E 1 , 2 ( f ) α 1 , 2 = inf (14) Ent 1 , 2 ( f ) f > 0 Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Spectral Gap and Log-Sobolev constant • The spectral gap of L : E 2 ( f ) λ = inf (13) Var σ ( f ) f � =0 • The (1 , 2)- logarithmic Sobolev constant E 1 , 2 ( f ) α 1 , 2 = inf (14) Ent 1 , 2 ( f ) f > 0 Note: one can in fact define a whole family of Log Sobolev constants α p , with p ≥ 0. Quantum LogSobolev Michael Kastoryano and Kristan Temme
Motivation Results Applications and outlook Mixing times Theorem Let L denote the generator of a primitive reversible semigroup with fixed point σ . Then, 1 χ 2 bound : � χ 2 ( ρ t , σ ) || ρ t − σ || 1 ≤ (15) � χ 2 ( ρ, σ ) e − λ t ≤ � 1 /σ min e − λ t . ≤ Where σ min denotes the smallest eigenvalue of the fixed point σ . Quantum LogSobolev Michael Kastoryano and Kristan Temme
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