A generalization of Quantum Relative Entropy Luiza H.F. Andrade 1 Rui - - PowerPoint PPT Presentation

Introduction Generalized Relative Entropy Conclusion

A generalization of Quantum Relative Entropy

Luiza H.F. Andrade1 Rui F. Vigelis2 Charles C. Cavalcante3

1Department of Natural Science, Mathematics and Statistics

Federal Rural University of Semi-arid Region-UFERSA

2Computer Engineering, Campus Sobral,

Federal University of Ceará,Sobral-CE,

3Department 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

1

Introduction

2

Generalized Relative Entropy Generalized Relative entropy Properties of the generalized quantum relative entropy

3

Conclusion

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 Conclusion Generalized Relative entropy Properties of the generalized quantum relative entropy

Quantum relative entropy

Let ρ and σ be density matrices, the quantum relative entropy is given by: S(ρ σ) = Tr[ρ(logρ −logσ)] where ρ and σ belong to L (H ), which is the set of Hilbert space

• perators, 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 Conclusion Generalized Relative entropy Properties of the generalized quantum relative entropy

Quantum relative entropy

There exists an embedding of P into A , defined as l : P → A ρ → logρ 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 Aρ be a subspace of A and ρ ∈ P given as Aρ = {A ∈ A ; Tr(Aρ) = 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 Conclusion Generalized Relative entropy Properties of the generalized quantum relative entropy

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 γe(t) = exp(H +tA) Tr(exp(H +tA)), (t ∈ [0,1]), where A = logω −logρ, so A ∈ A . We can rewrite γe(t) as γe(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 Conclusion Generalized Relative entropy Properties of the generalized quantum relative entropy

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 γe(t) = exp(H +tA) Tr(exp(H +tA)), (t ∈ [0,1]), where A = logω −logρ, so A ∈ A . We can rewrite γe(t) as γe(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 Conclusion Generalized Relative entropy Properties of the generalized quantum relative entropy

Quantum relative entropy

Defining a function ψH(tA) := log(Tr(exp(H +tA))), (t ∈ [0,1]), the e-geodesic can be rewritten as γe(t) = exp(H +tA−ψH(tA)I), (t ∈ [0,1]). We need to verify if γe(t) ∈ P.

Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy

Introduction Generalized Relative Entropy Conclusion Generalized Relative entropy Properties of the generalized quantum relative entropy

Quantum relative entropy

By the convexity of the exponential function, we have Tr(exp(H +A)) ≥ Tr(exp(H))+Tr(Aexp(H)), where ρ = exp(H). And, by the fact Tr(ρ) = 1 and A ∈ Aρ, we have Tr(exp(H +A)) ≥ 1. Thus, there exists a unique ψH(A) such that γe(t) ∈ P.

Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy

Introduction Generalized Relative Entropy Conclusion Generalized Relative entropy Properties of the generalized quantum relative entropy

Quantum relative entropy

Now, take another density matrix σ = exp(H +A−ψ(A)I), where A ∈ Aρ. As a consequence we have Tr(ψH(A)ρ) = Tr(ρ(logρ −logσ)) = S(ρ σ) and therefore, S(ρ σ) = ψH(A).

Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy

Introduction Generalized Relative Entropy Conclusion Generalized Relative entropy Properties of the generalized quantum relative entropy

Outline

1

Introduction

2

Generalized Relative Entropy Generalized Relative entropy Properties of the generalized quantum relative entropy

3

Conclusion

Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy

Introduction Generalized Relative Entropy Conclusion Generalized Relative entropy Properties of the generalized quantum relative entropy

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) limu→−∞ ϕ(u) = 0 and limu→∞ 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 Conclusion Generalized Relative entropy Properties of the generalized quantum relative entropy

Generalized relative entropy

By the convexity of the deformed exponential ϕ, we have: Tr(ϕ(H +A)) ≥ Tr(ϕ(H))+Tr

• Aϕ′(H)
• ,

where H and A are self-adjoint matrices and ρ = ϕ(H) is a density matrix. We will consider the subspace of A A ϕ

ρ =

• A ∈ A ; TrAϕ′(H) = 0
• ,

which is the equivalent to Aρ. Equivalently, there exists a unique ψH(A) ≥ 0 such that Tr

• ϕ
• H +A−ψH(A)
• ϕ′(H)

−1 = 1.

Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy

Introduction Generalized Relative Entropy Conclusion Generalized Relative entropy Properties of the generalized quantum relative entropy

Generalized relative entropy

Now, taking a density matrix σ given as σ = ϕ

• H +A−ψH(A)
• ϕ′(H)

−1 ∈ P, where A ∈ A ϕ

ρ .

We obtain ψH(A) = Tr[ ϕ(ρ)− ϕ(σ)]

• (

ϕ)′(ρ) −1 = Sϕ(ρ σ), where ϕ(·) is the inverse of the deformed exponential ϕ(·). Generalized relative entropy Sϕ(ρ σ) =Tr

• (

ϕ)′(ρ) −1 [ ϕ(ρ)− ϕ(σ)].

Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy

Introduction Generalized Relative Entropy Conclusion Generalized Relative entropy Properties of the generalized quantum relative entropy

Non-negativity of the generalized relative entropy

By the concavity of ϕ, we have Sϕ(ρ σ) =Tr

• (

ϕ)′(ρ) −1 [ ϕ(ρ)− ϕ(σ)]

• ≥ Tr(ρ −σ) = 0;

It is clear that Sϕ(ρ σ) = 0 if ρ = σ; If ϕ is strictly concave Sϕ(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 Conclusion Generalized Relative entropy Properties of the generalized quantum relative entropy

Outline

1

Introduction

2

Generalized Relative Entropy Generalized Relative entropy Properties of the generalized quantum relative entropy

3

Conclusion

Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy

Introduction Generalized Relative Entropy Conclusion Generalized Relative entropy Properties of the generalized quantum relative entropy

Properties

1 (Non negativity) Sϕ(ρ||σ) ≥ 0. 2 Sϕ is invariant under the unitary transformation U, it means:

Sϕ(UρU∗||UσU∗) = Sϕ(ρ||σ)

3 Sϕ

• ρ ⊗ I

n||σ ⊗ I n

• = nSϕ

ρ

n|| σ n

• , where n is the dimension of

Hilbert space of the density matrix.

Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy

Introduction Generalized Relative Entropy Conclusion

Conclusions

We obtain a more general form of the relative entropy of Von Neumann by replacing the classical exponential function by a deformed one; We prove some properties of this generalized relative entropy. A future work is to use the generalized relative entropy to compare different quantum sign and verify how random they are, in order to maximize the difference between them.

Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy

Introduction Generalized Relative Entropy Conclusion

Thank You!

Luiza H.F. Andrade, Rui F. Vigelis, Charles C. Cavalcante A generalization of Quantum Relative Entropy