arXiv:1303.5570v2 [quant-ph] 19 May 2013 Tight Lower Bound on the Geometric Discord: A Measure of the Quantumness S. Javad Akhtarshenas, 1, 2, 3 Hamidreza Mohammadi, 1, 2 Saman Karimi, 1 and Zahra Azmi 1 1 Department of Physics, University of Isfahan, Isfahan, Iran 2 Quantum Optics Group, University of Isfahan, Isfahan, Iran 3 Department of Physics, Ferdowsi University of Mashhad, Mashhad, Iran A general state of an m ⊗ n system is a classical-quantum state if and only if its associated left- correlation matrix has rank no larger than m - 1. Based on this condition, a computable measure of quantum discord is presented, which coincides with the tight lower bound on the geometric measure of discord. Therefore such obtained tight lower bound fully captures the quantum correlation of a bipartite system, so it can be used as a measure of discord in its own right. Accordingly, a vanishing tight lower bound on the geometric discord is a necessary and sufficient condition for a state to be zero-discord. We present an alternative form for the geometric discord and provide a comparison between the geometric discord and our computable measure of quantum correlation. PACS numbers: 03.67.-a, 03.65.Ta, 03.65.Ud I. INTRODUCTION Quantum discord represents a new type of quantum correlation which looks at the correlations from a new perspective, i.e. measurement theory, different than the entanglement-separability paradigm [1, 2]. Hence, there exist separable (not-entangled) states which have non- zero discord such that one can employ these separable states as a resource to enhance the quality of quantum in- formation and computation processing [3, 4]. Nowadays, quantum discord became a subject of intensive study in different contexts [5] and different versions of quantum discord and their measures have been introduced and an- alyzed [5, 6]. Since the evaluation of quantum discord involves an optimization procedure, almost all quantum discord measures are very difficult to calculate analyt- ically and quantum discord was analytically computed only for a few families of two-qubit states [7] and for some reduced two-qubit states of pure three-qubit states and also a class of rank-2 mixed state of 4 ⊗ 2 systems [8]. Among the various measures of quantum discord, the geometric discord, has been firstly proposed by Dakic et al., is a simple and intuitive quantifier of general non- classical correlations [9]. Geometric discord is defined as the squared Hilbert-Schmidt distance between the state of the quantum system and the closest zero-discord state. For a bipartite state ρ on H A ⊗H B , with dim H A = m and dim H B = n, the geometric discord is defined by [9] D G (ρ) = min χ∈Ω0 ‖ρ − χ‖ 2 , (1) where Ω 0 denotes the set of all zero-discord states and ‖X − Y ‖ 2 = Tr(X − Y ) 2 is the 2-norm or square norm in the Hilbert-Schmidt space. This quantity vanishes on the classical-quantum states. Dakic et al. also obtained a closed formula for the geometric discord of an arbi- trary two-qubit state in terms of coherence vectors and correlation matrix of the state. Furthermore, an exact expression for the pure m ⊗ m states and arbitrary 2 ⊗ n states are obtained [10, 11]. An alternative form for the geometric discord is intro- duced by Luo and Fu [10] D G (ρ) = min Π A ‖ρ − Π A (ρ)‖ 2 , (2) where the minimum is taken over all von Neumann measurements Π A = {Π A k } m k=1 on H A , and Π A (ρ)= ∑ m k=1 (Π A k ⊗ I)ρ(Π A k ⊗ I) with I as the identity opera- tor on the appropriate space. They have also shown that Eq. (2) is equivalent to [10] D G (ρ) = Tr(CC t ) − max A Tr(ACC t A t ), (3) where t denotes transpose, and C =(c ij ) is an m 2 × n 2 - dimensional matrix defined by ρ = m 2 −1  i=0 n 2 −1  j=0 c ij X i ⊗ Y j , (4) with {X i } m 2 −1 i=1 and {Y j } n 2 −1 j=1 as the sets of Hermitian operators which constitute orthonormal basis for SU (m) and SU (n) algebra, respectively, i.e. Tr(X i X i ′ )= δ ii ′ , Tr(Y j Y j ′ )= δ jj ′ . (5) In Eq. (3), the maximum is taken over all m × m 2 - dimensional matrices A =(a ki ) such that a ki = Tr(|k〉〈k|X i )= 〈k|X i |k〉, (6) where {|k〉} m k=1 is any orthonormal base for H A . Based on the definition (3), Rana et al. [12] and Hassan et al. [13] have obtained a tight lower bound on the geometric discord. Let { ˆ λ A i } m 2 −1 i=1 and { ˆ λ B j } n 2 −1 j=1 be generators of SU (m) and SU (n), respectively, fulfilling the following relations Tr ˆ λ s i =0, Tr( ˆ λ s i ˆ λ s j )=2δ ij , s = A,B. (7) Then a general bipartite state ρ on H A ⊗H B can be written in this basis as ρ = 1 mn  I ⊗ I + x · ˆ λ A ⊗ I + I ⊗  y · ˆ λ B + ∑ m 2 −1 i=1 ∑ n 2 −1 j=1 t ij ˆ λ A i ⊗ ˆ λ B j  . (8)