Quantum Discord of 2ⁿ-Dimensional Bell-Diagonal States

Abstract

In this study, using the concept of relative entropy as a distance measure of correlations we investigate the important issue of evaluating quantum correlations such as entanglement, dissonance and classical correlations for 2ⁿ-dimensional Bell-diagonal states. We provide an analytical technique, which describes how we find the closest classical states(CCS) and the closest separable states(CSS) for these states. Then analytical results are obtained for quantum discord of 2ⁿ-dimensional Bell-diagonal states. As illustration, some special cases are examined. Finally, we investigate the additivity relation between the different correlations for the separable generalized Bloch sphere states.

Appendices

Appendix I:

Convex Optimization Review: An optimization problem [51], has the standard form

$$ \left\{ \begin{array}{l} \mathrm{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~maximize} \ f_{0}(x) .\\ \mathrm{subject\; to}\quad \quad f_{i}(x)\leq b_{i}, i=1,...,m \quad h_{i}(x)=0, i=1,...,p\\ \end{array}\right. $$

(I-1)

Where the vector x = (x ₁,...,x _n ) is the optimization variable of the problem, the function f ₀:R ⁿ→R is the objective function, the functions f _i :R ⁿ→R,i = 1,...,m are the (inequality) constraint functions, and the constants b ₁,...,b _m are the limits, or bounds, for the constraints. A convex optimization problem, is an optimization problem where the objective and the constraint functions are convex functions which means they satisfy inequality f _i (αx+βy)≤αf _i (x)+βf _i (y), for all (x,y,α,β)∈R with α+β = 1,α≥0,β≥0 and the equality constraint functions h _i (x)=0 must be affine (A set C∈R ⁿ is affine if the line through any two distinct points in C lies in C). One can solve this convex optimization problem using Lagrangian duality. The basic idea in the Lagrangian duality is to take the constraints in convex optimization problem into account by augmenting the objective function with a weighted sum of the constraint functions. The Lagrangian L:R ⁿ×R ^m×R ^p→R associated with the problem is defined as

$$ L(x,\lambda,\nu)=f_{0}(x)+\sum\limits_{i=1}^{m}\lambda_{i}f_{i}(x)+\sum\limits_{i=1}^{p}\nu_{i}h_{i}(x). $$

(I-2)

The Lagrange dual function g:R ⁿ×R ^m×R ^p→R is defined as the minimum value of the Lagrangian over x: for λ∈R ^m,ν∈R ^p,

$$ g(\lambda,\nu)=inf_{x\in D}L(x,\lambda,\nu). $$

(I-3)

The dual function yields lower bounds on the optimal value p ^⋆ of the convex optimization problem, i.e for any λ≥0 and any ν we have

$$ g(\lambda,\nu)\leq p^{\star}. $$

(I-4)

The optimal value of the Lagrange dual problem, which we denote d ^⋆, is, by definition, the best lower bound on d ^⋆ that can be obtained from the Lagrange dual function. In particular, we have the simple but important inequality

$$ d^{\star}\leq p^{\star}. $$

(I-5)

This property is called weak duality. If the equality d ^⋆ = p ^⋆ holds, i.e., the optimal duality gap is zero, then we say that strong duality holds. If strong duality holds and a dual optimal solution (λ ^⋆,ν ^⋆) exists, then any primal optimal point is also a minimizer of L(x,λ ^⋆,ν ^⋆). This fact sometimes allows us to compute a primal optimal solution from a dual optimal solution. For the best lower bound that can be obtained from the Lagrange dual function one can solve the following optimization problem

$$ \left\{ \begin{array}{l} \text{maximize} \ \ g(\lambda,\nu) .\\ \mathrm{~subject\; to} \ \ \lambda\geq 0\\ \end{array}\right. $$

(I-6)

This problem is called the Lagrange dual problem associated with the main problem. Conditions for the optimality of a convex problem is called Karush-Kuhn-Tucker (KKT) conditions. If f _i are convex and h _i are affine, and $\tilde {x},\tilde {\lambda },\tilde {\nu }$ are any points that satisfy the KKT conditions

$$h_{i}(\tilde{x})=0, \ i=1,...,p, $$

$$f_{i}(\tilde{x})\leq0, \ i=1,...,m, $$

$$\tilde{\lambda}_{i}\geq 0 \ \ \tilde{\lambda}_{i}f_{i}(\tilde{x})=0, \ i=1,...,m, $$

$$ \bigtriangledown f_{0}(\tilde{x})+{\sum\limits_{i}^{m}} \tilde{\lambda}_{i} \bigtriangledown f_{i}(\tilde{x})+{{\sum}_{i}^{p}}\tilde{\nu}_{i} \bigtriangledown h_{i}(\tilde{x})=0. $$

(I-7)

then $\tilde {x}$ and $(\tilde {\lambda },\tilde {\nu })$ are primal and dual optimal, with zero duality gap. In other words, for any convex optimization problem with differentiable objective and constraint functions, any points that satisfy the KKT conditions are primal and dual optimal, and have zero duality gap. Hence, $f_{0}(\tilde {x})=g(\tilde {\lambda },\tilde {\nu })$. The condition $\tilde {\lambda }_{i}f_{i}(\tilde {x})=0, \ i=1,...,m,$ is known as complementary slackness; it holds for any primal optimal $x\tilde {}$ and any dual optimal $(\tilde {\lambda },\tilde {\nu })$ (when strong duality holds)

Appendix II:

Throughout the paper, we have used the formalism of Dirac γ matrices. Therefore, in this appendix we define the algebra of Dirac γ matrices and exhibit matrices which realize the algebra in the Euclidean representation and explain our notations and conventions. To do this, let γ _μ ,μ = 1,...,d, be a set of d matrices satisfying the anticommuting relations:

$$ \gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2\delta_{\mu\nu}I, $$

(I-1)

in which I is the identity matrix. These matrices are the generatores of a Clifford algebra similar to the algebra of operators acting on Grassmann algebras. It follows from relations (I-1) that the γ matrices generate an algebra which, as a vector space, has a dimension 2^d. In the following, we will give an inductive construction (d→?d+2) of hermitian matrices satisfying (I-1). In the algebra one element plays a special role, the product of all γ matrices. The matrix γ _s :

$$ \gamma_{s}=i^{\frac{-d}{2}}\gamma_{1}\gamma_{2}...\gamma_{d}, $$

(I-2)

anticommutes, because d is even, with all other γ matrices and ${\gamma _{s}^{2}}= I$.

In calculations involving γ matrices, it is not always necessary to distinguish γ _s from other γ matrices. Identifying thus γ _s with γ _d+1, we have:

$$ \gamma_{i}\gamma_{j}+\gamma_{j}\gamma_{i}=2\delta_{ij}I, i,j=1,...,d,d+1. $$

(I-3)

The Greek letters μν... are usually used to indicate that the value d+1 for the index has been excluded.

1.1 An Explicit Construction of $\gamma_{i}^{(d)}$

It is sometimes useful to have an explicit realization of the algebra of γ matrices. For d = 2, the standard Pauli matrices realize the algebra:

$$\gamma_{1}^{(d=2)}=\sigma_{1}=\left( \begin{array}{ll} 0 & 1 \\ 1 & 0 \\ \end{array} \right) , \gamma_{2}^{(d=2)}=\sigma_{2}=\left( \begin{array}{ll} 0 & -i \\ i & 0 \\ \end{array} \right), $$

$$ \gamma_{s}^{(d=2)}= \gamma_{3}^{(d=2)}=\sigma_{3}=\left( \begin{array}{ll} 1 & 0 \\ 0 & -1 \\ \end{array} \right) $$

(I-4)

The three matrices are hermitian, i.e., $\gamma _{i}=\gamma _{i}^{\dagger }$. The matrices γ ₁ and γ ₃ are symmetric and γ ₂ is antisymmetric, i.e., $\gamma _{1}={\gamma _{1}^{t}}$, $\gamma _{3}={\gamma _{3}^{t}}$ and $\gamma _{2}=-{\gamma _{2}^{t}}$. To construct the ? matrices for higher even dimensions, we then proceed by induction, setting:

$$\gamma_{i}^{(d+2)}=\sigma_{1}\otimes \gamma_{i}^{(d)}=\left( \begin{array}{cc} 0 & \gamma_{i}^{(d)} \\ \gamma_{i}^{(d)} & 0 \\ \end{array} \right), i=1,2,...,d+1, $$

$$ \gamma_{d+2}=\sigma_{2}\otimes I^{(d)}=\left( \begin{array}{ll} 0 & -iI_{d} \\ iI_{d} & 0 \\ \end{array} \right), $$

(I-5)

where, I _d is the unit matrix in $2^{\frac {d}{2}}$ dimensions. As a consequence $\gamma _{s}^{(d+2)}$ has the form:

$$ \gamma_{s}^{(d+2)}=\gamma_{d+3}^{(d+2)}=\sigma_{3}\otimes I_{d}=\left( \begin{array}{ll} I_{d} & 0 \\ 0 & -I_{d} \\ \end{array} \right) $$

(I-6)

A straightforward calculation shows that if the matrices $\gamma _{i}^{(d)}$ satisfy relations (I-3), the $\gamma _{i}^{(d+2)}$ matrices satisfy the same relations. By induction we see that the γ matrices are all hermitian. from (I-5), it is seen that, if $\gamma _{i}^{(d)}$ is symmetric or antisymmetric, $\gamma _{i}^{(d+2)}$ has the same property. The matrix $\gamma _{d+2}^{(d+2)}$ is antisymmetric and the matrix $\gamma _{d+3}^{d+2}$ is symmetric. It follows immediately that, in this representation, all γ matrices with odd index are symmetric and all matrices with even index are antisymmetric, i.e.,

$$ {\gamma_{i}^{t}}=(-1)^{i+1}\gamma_{i}. $$

(I-7)