Introduction Generalized Relative Entropy Conclusion A generalization of Quantum Relative Entropy Luiza H.F. Andrade 1 Rui F. Vigelis 2 Charles C. Cavalcante 3 1 Department of Natural Science, Mathematics and Statistics Federal Rural University of Semi-arid Region-UFERSA 2 Computer Engineering, Campus Sobral, Federal University of Ceará,Sobral-CE, 3 Department of Teleinformatics Engineering Federal University of Ceará Latin American Week on Coding and Information, 2018 Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative Entropy Conclusion Outline Introduction 1 Generalized Relative Entropy 2 Generalized Relative entropy Properties of the generalized quantum relative entropy Conclusion 3 Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative Entropy Conclusion Introduction Vigelis & Cavalcante, 2013 : Exponential families are generalized, replacing exponential function by a deformed exponential function ϕ . The ϕ -divergence was provided, where for ϕ ( x ) = exp( x ) reduces to Kullback-Leibler divergence; de Souza et al., 2016 : A generalization of the Rényi relative entropy was given from a deformed exponential ϕ . Umegaki, 1962 : Quantum relative entropy was introduced. Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative entropy Generalized Relative Entropy Properties of the generalized quantum relative entropy Conclusion Quantum relative entropy Let ρ and σ be density matrices, the quantum relative entropy is given by: S ( ρ � σ ) = Tr[ ρ (log ρ − log σ )] where ρ and σ belong to L ( H ) , which is the set of Hilbert space operators, with Tr( ρ ) = 1 and Tr( σ ) = 1. Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative entropy Generalized Relative Entropy Properties of the generalized quantum relative entropy Conclusion Quantum relative entropy There exists an embedding of P into A , defined as l : P → A ρ �→ log ρ where A is a subspace of L ( H ) of self-adjoint operators and P is the set of all invertible density matrices. Thus, P is a manifold which has the exponential with a natural path. Let A ρ be a subspace of A and ρ ∈ P given as A ρ = { A ∈ A ; Tr( A ρ ) = 0 } is the tangent bundle of P . Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative entropy Generalized Relative Entropy Properties of the generalized quantum relative entropy Conclusion Quantum relative entropy Let H be the self-adjoint matrix such that ρ = exp( H ) and ω belong to P . The e -geodesic is given as exp( H + tA ) γ e ( t ) = ( t ∈ [ 0 , 1 ]) , Tr(exp( H + tA )) , where A = log ω − log ρ , so A ∈ A . We can rewrite γ e ( t ) as γ e ( t ) = exp( H + tA − log(Tr(exp( H + tA ))) I ) , ( t ∈ [ 0 , 1 ]) . Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative entropy Generalized Relative Entropy Properties of the generalized quantum relative entropy Conclusion Quantum relative entropy Let H be the self-adjoint matrix such that ρ = exp( H ) and ω belong to P . The e -geodesic is given as exp( H + tA ) γ e ( t ) = ( t ∈ [ 0 , 1 ]) , Tr(exp( H + tA )) , where A = log ω − log ρ , so A ∈ A . We can rewrite γ e ( t ) as γ e ( t ) = exp( H + tA − log(Tr(exp( H + tA ))) I ) , ( t ∈ [ 0 , 1 ]) . Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative entropy Generalized Relative Entropy Properties of the generalized quantum relative entropy Conclusion Quantum relative entropy Defining a function ψ H ( tA ) := log(Tr(exp( H + tA ))) , ( t ∈ [ 0 , 1 ]) , the e -geodesic can be rewritten as γ e ( t ) = exp( H + tA − ψ H ( tA ) I ) , ( t ∈ [ 0 , 1 ]) . We need to verify if γ e ( t ) ∈ P . Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative entropy Generalized Relative Entropy Properties of the generalized quantum relative entropy Conclusion Quantum relative entropy By the convexity of the exponential function, we have Tr(exp( H + A )) ≥ Tr(exp( H ))+Tr( A exp( H )) , where ρ = exp( H ) . And, by the fact Tr( ρ ) = 1 and A ∈ A ρ , we have Tr(exp( H + A )) ≥ 1 . Thus, there exists a unique ψ H ( A ) such that γ e ( t ) ∈ P . Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative entropy Generalized Relative Entropy Properties of the generalized quantum relative entropy Conclusion Quantum relative entropy Now, take another density matrix σ = exp( H + A − ψ ( A ) I ) , where A ∈ A ρ . As a consequence we have Tr( ψ H ( A ) ρ ) = Tr( ρ (log ρ − log σ )) = S ( ρ � σ ) and therefore, S ( ρ � σ ) = ψ H ( A ) . Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative entropy Generalized Relative Entropy Properties of the generalized quantum relative entropy Conclusion Outline Introduction 1 Generalized Relative Entropy 2 Generalized Relative entropy Properties of the generalized quantum relative entropy Conclusion 3 Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative entropy Generalized Relative Entropy Properties of the generalized quantum relative entropy Conclusion deformed exponential ϕ Definition A deformed exponential is a function ϕ : R → ( 0 , ∞ ) that satisfies the following properties: (a1) ϕ ( · ) is convex and injective; (a2) lim u →− ∞ ϕ ( u ) = 0 and lim u → ∞ ϕ ( u ) = ∞ ; In this work we consider that a deformed exponential function is continuously differentiable with its inverse function also continuously differentiable. Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative entropy Generalized Relative Entropy Properties of the generalized quantum relative entropy Conclusion Generalized relative entropy By the convexity of the deformed exponential ϕ , we have: � � A ϕ ′ ( H ) Tr( ϕ ( H + A )) ≥ Tr( ϕ ( H ))+Tr , where H and A are self-adjoint matrices and ρ = ϕ ( H ) is a density matrix. We will consider the subspace of A � � A ∈ A ; Tr A ϕ ′ ( H ) = 0 A ϕ ρ = , which is the equivalent to A ρ . Equivalently, there exists a unique ψ H ( A ) ≥ 0 such that � � � − 1 �� � ϕ ′ ( H ) Tr H + A − ψ H ( A ) = 1 . ϕ Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative entropy Generalized Relative Entropy Properties of the generalized quantum relative entropy Conclusion Generalized relative entropy Now, taking a density matrix σ given as � � − 1 � � ϕ ′ ( H ) σ = ϕ H + A − ψ H ( A ) ∈ P , where A ∈ A ϕ ρ . We obtain � � − 1 = S ϕ ( ρ � σ ) , ϕ ) ′ ( ρ ) ψ H ( A ) = Tr[ � ϕ ( ρ ) − � ϕ ( σ )] ( � where � ϕ ( · ) is the inverse of the deformed exponential ϕ ( · ) . Generalized relative entropy � � − 1 [ � ϕ ) ′ ( ρ ) ( � ϕ ( ρ ) − � S ϕ ( ρ � σ ) =Tr ϕ ( σ )] . Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative entropy Generalized Relative Entropy Properties of the generalized quantum relative entropy Conclusion Non-negativity of the generalized relative entropy By the concavity of � ϕ , we have �� � � − 1 [ � ϕ ) ′ ( ρ ) ( � ϕ ( ρ ) − � S ϕ ( ρ � σ ) =Tr ϕ ( σ )] ≥ Tr( ρ − σ ) = 0 ; It is clear that S ϕ ( ρ � σ ) = 0 if ρ = σ ; If � ϕ is strictly concave S ϕ ( ρ � σ ) = 0 iff ρ = σ . Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative entropy Generalized Relative Entropy Properties of the generalized quantum relative entropy Conclusion Outline Introduction 1 Generalized Relative Entropy 2 Generalized Relative entropy Properties of the generalized quantum relative entropy Conclusion 3 Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
Introduction Generalized Relative entropy Generalized Relative Entropy Properties of the generalized quantum relative entropy Conclusion Properties 1 (Non negativity) S ϕ ( ρ || σ ) ≥ 0. 2 S ϕ is invariant under the unitary transformation U , it means: S ϕ ( U ρ U ∗ || U σ U ∗ ) = S ϕ ( ρ || σ ) � � � ρ � ρ ⊗ I n || σ ⊗ I 3 S ϕ n || σ = n S ϕ , where n is the dimension of n n Hilbert space of the density matrix. Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy
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