This Paper Is Organized As Follows

Discord and entanglement characterize two kinds of quantum correlations, and discord captures more correlation than entanglement in the sense that even separable states may have nonzero discord. In this paper, we propose a new kind of quantum correlation we call it oblique discord. A zero-discord state corresponds to an orthonormal basis, while a zero-oblique-discord state corresponds to a basis which is not necessarily orthogonal. Under this definition, the set of zero-discord states is properly contained inside the set of zero-oblique-discord states, and the set of zero-oblique-discord states is properly contained inside the set of separable states. We give a characterization of zero-oblique-discord states via quantum operation, provide a geometric measure for oblique discord, and raise a conjecture with it holds we can define an information-theoretic measure for oblique discord. Also, we point out that, the definition of oblique discord can be properly extended to some different versions just as the case of quantum discord.

pacs: 03.65.Ud, 03.67.Mn, 03.65.Aa

Quantum correlation is one of the most striking features of quantum physics, and leads to powerful applications in quantum information science Horodecki2009 ; Modi2012 . Discord and entanglement characterize two kinds of quantum correlations, manifest complex structures and achieved fruitful results Horodecki2009 ; Modi2012 . Discord captures more correlation than entanglement in the sense that even separable states may have nonzero discord, although for certain measures discord not necessarily is larger than entanglement.

This paper asks the question: are there other kinds of correlation between entanglement and discord. To this aim, we properly generalize the definition of discord, we call the generalized version oblique discord. Under this definition, the set of zero-discord states is properly contained inside the set of zero-oblique-discord states, and the set of zero-oblique-discord states is properly contained inside the set of separable states. Moreover, we provide the information-theoretic measure and geometric measure for oblique discord compared to the case of discord, and propose the definition of global oblique discord compared to global discord.

This paper is organized as follows. In section 2, as preparations, we review the definitions of entanglement, discord, geometric discord, global discord and geometric global discord. In section 3, we provide the definition of oblique discord, give a characterization of zero-oblique-discord states via quantum operation, provide a geometric measure for oblique discord. Also, we raise a conjecture, if it holds we can define an information-theoretic measure for oblique discord. In section 4, we point out that, the definition of oblique discord can be properly extended to some different versions just as the case of quantum discord. In section 5, we give a summary.

II Entanglement and discord

Suppose the quantum systems A, B are described by the complex Hilbert spaces HAsuperscript𝐻𝐴H^Aitalic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT and HBsuperscript𝐻𝐵H^Bitalic_H start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT, nA=dimHAsubscript𝑛𝐴dimensionsuperscript𝐻𝐴n_A=\dim H^Aitalic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = roman_dim italic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT and nB=dimHBsubscript𝑛𝐵dimensionsuperscript𝐻𝐵n_B=\dim H^Bitalic_n start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = roman_dim italic_H start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT are finite. The bipartite system AB𝐴𝐵ABitalic_A italic_B is then described by the Hilbert space HAB=HA⊗HBsuperscript𝐻𝐴𝐵tensor-productsuperscript𝐻𝐴superscript𝐻𝐵H^AB=H^A\otimes H^Bitalic_H start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = italic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_H start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT with dimHAB=nAnB.dimensionsuperscript𝐻𝐴𝐵subscript𝑛𝐴subscript𝑛𝐵\dim H^AB=n_An_B.roman_dim italic_H start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT . Let IA,IBsubscript𝐼𝐴subscript𝐼𝐵I_A,I_Bitalic_I start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT , italic_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT be the identity operators of A and B, then the identity operator of AB is IAB=IA⊗IBsubscript𝐼𝐴𝐵tensor-productsubscript𝐼𝐴subscript𝐼𝐵I_AB=I_A\otimes I_Bitalic_I start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = italic_I start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. When we consider an N𝑁Nitalic_N-partite system A1A2…ANsubscript𝐴1subscript𝐴2…subscript𝐴𝑁A_1A_2...A_Nitalic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT, we use Aii=1Nsuperscriptsubscriptsubscript𝐴𝑖𝑖1𝑁\A_i\_i=1^N italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT to denote each subsystem and their Hilbert spaces are HAi,superscript𝐻subscript𝐴𝑖H^A_i,italic_H start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , the dimension nAi,subscript𝑛subscript𝐴𝑖n_A_i,italic_n start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT , the identity IAisubscript𝐼subscript𝐴𝑖I_A_iitalic_I start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT. We often omit the identity operator, for example we write ρA⊗IBtensor-productsuperscript𝜌𝐴subscript𝐼𝐵\rho^A\otimes I_Bitalic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT as ρAsuperscript𝜌𝐴\rho^Aitalic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT by omitting IBsubscript𝐼𝐵I_Bitalic_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, without any confusion.

A quantum state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT is called a separable state if it can be written in the form

ρAB=∑ipiρiA⊗ρiB,superscript𝜌𝐴𝐵subscript𝑖tensor-productsubscript𝑝𝑖superscriptsubscript𝜌𝑖𝐴superscriptsubscript𝜌𝑖𝐵\displaystyle\rho^AB=\sum_ip_i\rho_i^A\otimes\rho_i^B,italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , (1) where ∑ipi=1,pi≥0,ρiAiformulae-sequencesubscript𝑖subscript𝑝𝑖1subscript𝑝𝑖0subscriptsuperscriptsubscript𝜌𝑖𝐴𝑖\sum_ip_i=1,p_i\geq 0,\\rho_i^A\_i∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 , italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ 0 , italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are states on HAsuperscript𝐻𝐴H^Aitalic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT and ρiBisubscriptsuperscriptsubscript𝜌𝑖𝐵𝑖\\rho_i^B\_i italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are states on HBsuperscript𝐻𝐵H^Bitalic_H start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT. ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT is called an entangled state or disentangled state if it is not separable. By far many entanglement measures have been proposed Horodecki2009 .

A state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT is called a zero-discord state with respect to A if it can be written in the form

ρAB=∑αpα|α⟩⟨α|⊗ραB,superscript𝜌𝐴𝐵subscript𝛼tensor-productsubscript𝑝𝛼ket𝛼bra𝛼superscriptsubscript𝜌𝛼𝐵\displaystyle\rho^AB=\sum_\alphap_\alpha|\alpha\rangle\langle\alpha|% \otimes\rho_\alpha^B,italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT | italic_α ⟩ ⟨ italic_α | ⊗ italic_ρ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , (2) where ∑αpα=1,pα≥0,α⟩αformulae-sequencesubscript𝛼subscript𝑝𝛼1subscript𝑝𝛼0subscriptket𝛼𝛼\sum_\alphap_\alpha=1,p_\alpha\geq 0,\_\alpha∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT = 1 , italic_p start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ≥ 0 , start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT is an orthonormal basis of HAsuperscript𝐻𝐴H^Aitalic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT, ραBαsubscriptsuperscriptsubscript𝜌𝛼𝐵𝛼\\rho_\alpha^B\_\alpha italic_ρ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT are states on HBsuperscript𝐻𝐵H^Bitalic_H start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT. ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT is called a discordant state if it is not a zero-discord state. The basic measure of discord is the information-theoretic measure proposed by Ollivier2001 ; Henderson2001 , that is

DA(ρAB)=infΠA[I(ρAB)-I(ΠAρAB)],superscript𝐷𝐴superscript𝜌𝐴𝐵subscriptinfimumsubscriptΠ𝐴delimited-[]𝐼superscript𝜌𝐴𝐵𝐼subscriptΠ𝐴superscript𝜌𝐴𝐵\displaystyle D^A(\rho^AB)=\inf_\Pi_A[I(\rho^AB)-I(\Pi_A\rho^AB)],italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = roman_inf start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - italic_I ( roman_Π start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ] , (3) where, I(ρAB)=S(ρA)+S(ρB)-S(ρAB)𝐼superscript𝜌𝐴𝐵𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵𝑆superscript𝜌𝐴𝐵I(\rho^AB)=S(\rho^A)+S(\rho^B)-S(\rho^AB)italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) is mutual information, ρA=trBρABsuperscript𝜌𝐴𝑡subscript𝑟𝐵superscript𝜌𝐴𝐵\rho^A=tr_B\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT = italic_t italic_r start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT is reduced state, S(ρAB)=-tr[ρABlog2ρAB]𝑆superscript𝜌𝐴𝐵𝑡𝑟delimited-[]superscript𝜌𝐴𝐵subscript2superscript𝜌𝐴𝐵S(\rho^AB)=-tr[\rho^AB\log_2\rho^AB]italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = - italic_t italic_r [ italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ] is entropy function, ΠAsubscriptΠ𝐴\Pi_Aroman_Π start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT denotes any projective measurement on A. It is shown Ollivier2001 that

DA(ρAB)≥0,superscript𝐷𝐴superscript𝜌𝐴𝐵0\displaystyle D^A(\rho^AB)\geq 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ % \ \ \ \ \ \ \ \ \ \ \ italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≥ 0 , (4)

DA(ρAB)=0⇔ρAB=∑αpα|α⟩⟨α|⊗ραB,⇔superscript𝐷𝐴superscript𝜌𝐴𝐵0superscript𝜌𝐴𝐵subscript𝛼tensor-productsubscript𝑝𝛼ket𝛼bra𝛼superscriptsubscript𝜌𝛼𝐵\displaystyle D^A(\rho^AB)=0\Leftrightarrow\rho^AB=\sum_\alphap_% \alpha|\alpha\rangle\langle\alpha|\otimes\rho_\alpha^B,italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = 0 ⇔ italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT | italic_α ⟩ ⟨ italic_α | ⊗ italic_ρ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , (5) where ∑αpα=1,pα≥0,αformulae-sequencesubscript𝛼subscript𝑝𝛼1subscript𝑝𝛼0subscriptket𝛼𝛼\sum_\alphap_\alpha=1,p_\alpha\geq 0,\\alpha\rangle\_\alpha∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT = 1 , italic_p start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ≥ 0 , italic_α ⟩ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT is an orthonormal basis of HAsuperscript𝐻𝐴H^Aitalic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT, ραBαsubscriptsuperscriptsubscript𝜌𝛼𝐵𝛼\\rho_\alpha^B\_\alpha italic_ρ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT are states on HBsuperscript𝐻𝐵H^Bitalic_H start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT. The intuitive meaning of DA(ρAB)superscript𝐷𝐴superscript𝜌𝐴𝐵D^A(\rho^AB)italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) is that it is the minimal loss of mutual information of ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT over all projective measurement on A.

DA(ρAB)superscript𝐷𝐴superscript𝜌𝐴𝐵D^A(\rho^AB)italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) is difficult to get the analytical expressions except for few special cases Luo2008 . Gaming News Another measure called geometric discord Dakic2010 is defined as

DGA(ρAB)=inftr[(ρAB-χAB)2]:DA(χAB)=0.superscriptsubscript𝐷𝐺𝐴superscript𝜌𝐴𝐵infimumconditional-set𝑡𝑟delimited-[]superscriptsuperscript𝜌𝐴𝐵superscript𝜒𝐴𝐵2superscript𝐷𝐴superscript𝜒𝐴𝐵0\displaystyle D_G^A(\rho^AB)=\inf\tr[(\rho^AB-\chi^AB)^2]:D^A(% \chi^AB)=0\.italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = roman_inf italic_t italic_r [ ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT - italic_χ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] : italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_χ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = 0 . (6) It is obvious that

DGA(ρAB)≥0,superscriptsubscript𝐷𝐺𝐴superscript𝜌𝐴𝐵0\displaystyle D_G^A(\rho^AB)\geq 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≥ 0 , (7)

DGA(ρAB)⇔DA(ρAB)=0.⇔superscriptsubscript𝐷𝐺𝐴superscript𝜌𝐴𝐵superscript𝐷𝐴superscript𝜌𝐴𝐵0\displaystyle D_G^A(\rho^AB)\Leftrightarrow D^A(\rho^AB)=0.italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ⇔ italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = 0 . (8) For many cases, DGA(ρAB)superscriptsubscript𝐷𝐺𝐴superscript𝜌𝐴𝐵D_G^A(\rho^AB)italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) is easier to calculate than DA(ρAB)superscript𝐷𝐴superscript𝜌𝐴𝐵D^A(\rho^AB)italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) since DGA(ρAB)superscriptsubscript𝐷𝐺𝐴superscript𝜌𝐴𝐵D_G^A(\rho^AB)italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) avoids the complicated entropy function. For instance, DGA(ρAB)superscriptsubscript𝐷𝐺𝐴superscript𝜌𝐴𝐵D_G^A(\rho^AB)italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) allows analytical expressions for all 2×d2𝑑2\times d2 × italic_d (2≤d<∞2𝑑2\leq d2 ≤ italic_d <∞) states Dakic2010 ; Vinjanampathy2012 .

Discord with respect to A, DA(ρAB)superscript𝐷𝐴superscript𝜌𝐴𝐵D^A(\rho^AB)italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ), can be extended to the definition of global discord Rulli2011 that (here we use the equivalent expression in Xu2013

D(ρA1A2…AN)=infΠA1ΠA2…ΠAN[I(ρA1A2…AN) fragmentsDfragments(superscript𝜌subscript𝐴1subscript𝐴2…subscript𝐴𝑁)subscriptinfimumsubscriptΠsubscript𝐴1subscriptΠsubscript𝐴2…subscriptΠsubscript𝐴𝑁fragments[Ifragments(superscript𝜌subscript𝐴1subscript𝐴2…subscript𝐴𝑁)italic- \displaystyle D(\rho^A_1A_2...A_N)=\inf_\Pi_A_1\Pi_A_2...\Pi_% A_N[I(\rho^A_1A_2...A_N)\ \ \ \ \ \ \ \ italic_D ( italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) = roman_inf start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT … roman_Π start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT )

-I(ΠA1ΠA2…ΠANρA1A2…AN)],fragmentsIfragments(subscriptΠsubscript𝐴1subscriptΠsubscript𝐴2…subscriptΠsubscript𝐴𝑁superscript𝜌subscript𝐴1subscript𝐴2…subscript𝐴𝑁)],\displaystyle-I(\Pi_A_1\Pi_A_2...\Pi_A_N\rho^A_1A_2...A_N)],- italic_I ( roman_Π start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT … roman_Π start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) ] , (9) where I(ρA1A2…AN)=∑i=1NS(ρAi)-S(ρA1A2…AN)𝐼superscript𝜌subscript𝐴1subscript𝐴2…subscript𝐴𝑁superscriptsubscript𝑖1𝑁𝑆superscript𝜌subscript𝐴𝑖𝑆superscript𝜌subscript𝐴1subscript𝐴2…subscript𝐴𝑁I(\rho^A_1A_2...A_N)=\sum_i=1^NS(\rho^A_i)-S(\rho^A_1A_2.% ..A_N)italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) is the mutual information of ρA1A2…ANsuperscript𝜌subscript𝐴1subscript𝐴2…subscript𝐴𝑁\rho^A_1A_2...A_Nitalic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT. D(ρA1A2…AN)𝐷superscript𝜌subscript𝐴1subscript𝐴2…subscript𝐴𝑁D(\rho^A_1A_2...A_N)italic_D ( italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) has the property

D(ρA1A2…AN)=0 𝐷superscript𝜌subscript𝐴1subscript𝐴2…subscript𝐴𝑁0\displaystyle D(\rho^A_1A_2...A_N)=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ % \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ italic_D ( italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) = 0

⇔ρA1A2…AN=∑i1=1n1∑i2=1n2…∑iN=1nNpi1i2…iN|αi1⟩⟨αi1|⇔absentsuperscript𝜌subscript𝐴1subscript𝐴2…subscript𝐴𝑁superscriptsubscriptsubscript𝑖11subscript𝑛1superscriptsubscriptsubscript𝑖21subscript𝑛2…superscriptsubscriptsubscript𝑖𝑁1subscript𝑛𝑁subscript𝑝subscript𝑖1subscript𝑖2…subscript𝑖𝑁ketsubscript𝛼subscript𝑖1brasubscript𝛼subscript𝑖1\displaystyle\Leftrightarrow\rho^A_1A_2...A_N=\sum_i_1=1^n_1% \sum_i_2=1^n_2...\sum_i_N=1^n_Np_i_1i_2...i_N|\alpha_% i_1\rangle\langle\alpha_i_1|⇔ italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT … ∑ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_α start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟩ ⟨ italic_α start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT |

⊗|αi1⟩⟨αi1|⊗…⊗|αiN⟩⟨αiN|,tensor-productabsenttensor-productketsubscript𝛼subscript𝑖1brasubscript𝛼subscript𝑖1…ketsubscript𝛼subscript𝑖𝑁brasubscript𝛼subscript𝑖𝑁\displaystyle\otimes|\alpha_i_1\rangle\langle\alpha_i_1|\otimes...% \otimes|\alpha_i_N\rangle\langle\alpha_i_N|,⊗ | italic_α start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟩ ⟨ italic_α start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | ⊗ … ⊗ | italic_α start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟩ ⟨ italic_α start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT | , (10) where pi1i2…iN≥0,∑i1i2…iNpi1i2…iN=1,ij=1njfragmentssubscript𝑝subscript𝑖1subscript𝑖2…subscript𝑖𝑁0,subscriptsubscript𝑖1subscript𝑖2…subscript𝑖𝑁subscript𝑝subscript𝑖1subscript𝑖2…subscript𝑖𝑁1,fragmentssubscript𝑖𝑗1subscript𝑛𝑗p_i_1i_2...i_N\geq 0,\sum_i_1i_2...i_Np_i_1i_2...i_N=1% ,\\alpha_i_j\rangle\_i_j=1^n_jitalic_p start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ 0 , ∑ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 1 , italic_α start_FLOATSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_FLOATSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT is an orthonormal basis of HAj.superscript𝐻subscript𝐴𝑗H^A_j.italic_H start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT . For certain special states, D(ρA1A2…AN)𝐷superscript𝜌subscript𝐴1subscript𝐴2…subscript𝐴𝑁D(\rho^A_1A_2...A_N)italic_D ( italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) possess analytical expressions Rulli2011 ; Xu2013 .

Geometric discord with respect to A, DGA(ρAB)superscriptsubscript𝐷𝐺𝐴superscript𝜌𝐴𝐵D_G^A(\rho^AB)italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ), can be extended to the definition of geometric global discord Xu2012 that

DG(ρA1A2…AN)=inftr[(ρA1A2…AN-χA1A2…AN)2]:fragmentssubscript𝐷𝐺fragments(superscript𝜌subscript𝐴1subscript𝐴2…subscript𝐴𝑁)infimumfragmentstrfragments[superscriptfragments(superscript𝜌subscript𝐴1subscript𝐴2…subscript𝐴𝑁superscript𝜒subscript𝐴1subscript𝐴2…subscript𝐴𝑁)2]:\displaystyle D_G(\rho^A_1A_2...A_N)=\inf\tr[(\rho^A_1A_2...A_% N-\chi^A_1A_2...A_N)^2]:italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) = roman_inf italic_t italic_r [ ( italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - italic_χ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] :

D(ρA1A2…AN)=0. fragmentsDfragments(superscript𝜌subscript𝐴1subscript𝐴2…subscript𝐴𝑁)0.italic- \displaystyle D(\rho^A_1A_2...A_N)=0\.\ \ \ \ \ italic_D ( italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) = 0 . (11) For certain special states, DG(ρA1A2…AN)subscript𝐷𝐺superscript𝜌subscript𝐴1subscript𝐴2…subscript𝐴𝑁D_G(\rho^A_1A_2...A_N)italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) possess analytical expressions Xu2012 .

The definitions of different kinds of discord above are all associated with the projective measurements. Gaming News Projective measurements are a kind of most important quantum operations, but not all quantum operations. There are many important problems, such as the optimal scheme to distinguish a set of quantum states, involve other quantum operations, rather than a projective measurement. A quantum operation is a map which maps a quantum state into another quantum state Nielsen2000 . The familiar examples are projective measurement, general measurement, amplitude damping and phase damping of qubit, etc. In this paper, we relax the constraint of projective measurement, and seek a more general definition other than quantum discord.

III Oblique discord and its measures

Definition 1. We call the bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT a zero-oblique-discord state with respect to A, if ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT can be written in the form

ρAB=∑i=1nApi|i⟩⟨i|⊗ρiB,superscript𝜌𝐴𝐵superscriptsubscript𝑖1subscript𝑛𝐴tensor-productsubscript𝑝𝑖ket𝑖bra𝑖superscriptsubscript𝜌𝑖𝐵\displaystyle\rho^AB=\sum_i=1^n_Ap_i|i\rangle\langle i|\otimes\rho_% i^B,italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_i ⟩ ⟨ italic_i | ⊗ italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , (12) where ∑ipi=1,pi≥0,i=1nAformulae-sequencesubscript𝑖subscript𝑝𝑖1subscript𝑝𝑖0superscriptsubscriptket𝑖𝑖1subscript𝑛𝐴\sum_ip_i=1,p_i\geq 0,\i\rangle\_i=1^n_A∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 , italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ 0 , italic_i ⟩ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT is a normalized basis of HAsuperscript𝐻𝐴H^Aitalic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT, ρiBisubscriptsuperscriptsubscript𝜌𝑖𝐵𝑖\\rho_i^B\_i italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are states on HBsuperscript𝐻𝐵H^Bitalic_H start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT. Notice that i⟩i=1nAsuperscriptsubscriptket𝑖𝑖1subscript𝑛𝐴\i\rangle\_i=1^n_A italic_i ⟩ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT is not necessarily orthogonal.

Under this definition, combining Eqs.(1,2,12), we see that, the set of zero-discord states is properly contained inside the set of zero-oblique-discord, and the set of zero-oblique-discord states is properly contained inside the set of separable states, see Fig.1.

We give a characterization of zero-oblique-discord states via quantum operations. Suppose i=1nAsuperscriptsubscriptket𝑖𝑖1subscript𝑛𝐴\_i=1^n_A start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT is a normalized basis of HAsuperscript𝐻𝐴H^Aitalic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT which not necessarily orthogonal to each other. There exists an unique basis i~⟩i=1nsuperscriptsubscriptket~𝑖𝑖1𝑛\\widetildei\rangle\_i=1^n over~ start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT of HAsuperscript𝐻𝐴H^Aitalic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT such that ⟨i|j~⟩=δijinner-product𝑖~𝑗subscript𝛿𝑖𝑗\langle i|\widetildej\rangle=\delta_ij⟨ italic_i | over~ start_ARG italic_j end_ARG ⟩ = italic_δ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT, note that i=1nsuperscriptsubscriptket~𝑖𝑖1𝑛\\widetildei\rangle\_i=1^n over~ start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT not necessarily orthogonal and not necessarily normalized. i=1nAsuperscriptsubscriptket~𝑖𝑖1subscript𝑛𝐴\\widetildei\rangle\_i=1^n_A start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT is called the dual basis of i=1nAsuperscriptsubscriptket𝑖𝑖1subscript𝑛𝐴\_i=1^n_A italic_i ⟩ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT. We define the quantum operation ΦA=i⟩⟨i~i=1nAsubscriptΦ𝐴superscriptsubscriptket𝑖bra~𝑖𝑖1subscript𝑛𝐴\Phi_A=\_i=1^n_Aroman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_i ⟩ ⟨ over~ start_ARG italic_i end_ARG start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT which operates the bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT as

ΦAρAB=∑i=1nA|i⟩⟨i~|ρAB|i~⟩⟨i|tr[∑i=1nA⟨i~|ρAB|i~⟩].subscriptΦ𝐴superscript𝜌𝐴𝐵superscriptsubscript𝑖1subscript𝑛𝐴ket𝑖quantum-operator-product~𝑖superscript𝜌𝐴𝐵~𝑖bra𝑖𝑡𝑟delimited-[]superscriptsubscript𝑖1subscript𝑛𝐴quantum-operator-product~𝑖superscript𝜌𝐴𝐵~𝑖\displaystyle\Phi_A\rho^AB=\frac\widetildei\rangle\langle i\widetildei\rangle].roman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT | italic_i ⟩ ⟨ over~ start_ARG italic_i end_ARG | italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT | over~ start_ARG italic_i end_ARG ⟩ ⟨ italic_i | end_ARG start_ARG italic_t italic_r [ ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⟨ over~ start_ARG italic_i end_ARG | italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT | over~ start_ARG italic_i end_ARG ⟩ ] end_ARG . (13) With this definition, we have Theorem 1 below.

Theorem 1. A bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT is a zero-oblique-discord state with respect to A, iff there exists an operation ΦA=i⟩⟨i~i=1nAsubscriptΦ𝐴superscriptsubscriptket𝑖bra~𝑖𝑖1subscript𝑛𝐴\Phi_A=\i\rangle\langle\widetildei_i=1^n_Aroman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT defined as in Eq.(13) such that

ΦAρAB=ρAB.subscriptΦ𝐴superscript𝜌𝐴𝐵superscript𝜌𝐴𝐵\displaystyle\Phi_A\rho^AB=\rho^AB.roman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT . (14) Proof. Suppose there exists an operation ΦA=i=1nAsubscriptΦ𝐴superscriptsubscriptket𝑖bra~𝑖𝑖1subscript𝑛𝐴\Phi_A=\i\rangle\langle\widetildei_i=1^n_Aroman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT such that ΦAρAB=ρABfragmentssubscriptΦ𝐴superscript𝜌𝐴𝐵ρ𝐴𝐵\Phi_A\rho^AB=\rho^ABroman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = italic_ρ start_FLOATSUPERSCRIPT italic_A italic_B end_FLOATSUPERSCRIPT, we expand ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT as

ρAB=∑jk=1nA∑λμ=1nBρjk,λμ|j⟩⟨k|⊗|λ⟩⟨μ|,superscript𝜌𝐴𝐵superscriptsubscript𝑗𝑘1subscript𝑛𝐴superscriptsubscript𝜆𝜇1subscript𝑛𝐵tensor-productsubscript𝜌𝑗𝑘𝜆𝜇ket𝑗bra𝑘ket𝜆bra𝜇\displaystyle\rho^AB=\sum_jk=1^n_A\sum_\lambda\mu=1^n_B\rho_jk,% \lambda\mu|j\rangle\langle k|\otimes|\lambda\rangle\langle\mu|,italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_j italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_λ italic_μ = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT italic_j italic_k , italic_λ italic_μ end_POSTSUBSCRIPT | italic_j ⟩ ⟨ italic_k | ⊗ | italic_λ ⟩ ⟨ italic_μ | , (15) where j⟩j=1nA=k=1nA=i⟩i=1nAsuperscriptsubscriptket𝑗𝑗1subscript𝑛𝐴superscriptsubscriptket𝑘𝑘1subscript𝑛𝐴superscriptsubscriptket𝑖𝑖1subscript𝑛𝐴\_j=1^n_A=\k\rangle\_k=1^n_A=\_i=1^n% _A italic_j ⟩ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, λ⟩λ=1nB=μ=1nBsuperscriptsubscriptket𝜆𝜆1subscript𝑛𝐵superscriptsubscriptket𝜇𝜇1subscript𝑛𝐵\\lambda\rangle\_\lambda=1^n_B=\\mu\rangle\_\mu=1^n_B italic_λ ⟩ start_POSTSUBSCRIPT italic_λ = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = start_POSTSUBSCRIPT italic_μ = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUPERSCRIPT is an orthonormal basis of HBsuperscript𝐻𝐵H^Bitalic_H start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT, ρjk,λμ=⟨j~λ|ρAB|k~μ⟩subscript𝜌𝑗𝑘𝜆𝜇quantum-operator-product~𝑗𝜆superscript𝜌𝐴𝐵~𝑘𝜇\rho_jk,\lambda\mu=\langle\widetildej\lambda|\rho^AB|\widetildek\mu\rangleitalic_ρ start_POSTSUBSCRIPT italic_j italic_k , italic_λ italic_μ end_POSTSUBSCRIPT = ⟨ over~ start_ARG italic_j end_ARG italic_λ | italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT | over~ start_ARG italic_k end_ARG italic_μ ⟩. Eq.(14) then reads

ρAB=∑i=1nA∑λμ=1nBρii,λμ|i⟩⟨i|⊗|λ⟩⟨μ|∑i=1nA∑λ=1nBρii,λλ,superscript𝜌𝐴𝐵superscriptsubscript𝑖1subscript𝑛𝐴superscriptsubscript𝜆𝜇1subscript𝑛𝐵tensor-productsubscript𝜌𝑖𝑖𝜆𝜇ket𝑖bra𝑖ket𝜆bra𝜇superscriptsubscript𝑖1subscript𝑛𝐴superscriptsubscript𝜆1subscript𝑛𝐵subscript𝜌𝑖𝑖𝜆𝜆\displaystyle\rho^AB=\fraci\rangle\langle i\sum_i=% 1^n_A\sum_\lambda=1^n_B\rho_ii,\lambda\lambda,italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_λ italic_μ = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT italic_i italic_i , italic_λ italic_μ end_POSTSUBSCRIPT | italic_i ⟩ ⟨ italic_i | ⊗ | italic_λ ⟩ ⟨ italic_μ | end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_λ = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT italic_i italic_i , italic_λ italic_λ end_POSTSUBSCRIPT end_ARG , (16) it is of the form in Eq.(12).

Conversely, suppose ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT can be expressed by Eq.(12), then ΦA=i=1nAsubscriptΦ𝐴superscriptsubscriptket𝑖bra~𝑖𝑖1subscript𝑛𝐴\Phi_A=\i\rangle\langle\widetildei_i=1^n_Aroman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_i ⟩ ⟨ over~ start_ARG italic_i end_ARG start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT fulfils ΦAρAB=ρAB.subscriptΦ𝐴superscript𝜌𝐴𝐵superscript𝜌𝐴𝐵\Phi_A\rho^AB=\rho^AB.roman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT . □□\Box□

Compared to the geometric measure of discord in Eq.(6), we propose the definition of geometric oblique discord as follows.

Definition 2. We define the geometric oblique discord of the bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT with respect to A𝐴Aitalic_A as

DGOA(ρAB)=infχABd(ρAB,χAB): fragmentssuperscriptsubscript𝐷𝐺𝑂𝐴fragments(superscript𝜌𝐴𝐵)subscriptinfimumsuperscript𝜒𝐴𝐵fragmentsdfragments(superscript𝜌𝐴𝐵,superscript𝜒𝐴𝐵):italic- \displaystyle D_GO^A(\rho^AB)=\inf_\chi^AB\d(\rho^AB,\chi^AB):% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ italic_D start_POSTSUBSCRIPT italic_G italic_O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = roman_inf start_POSTSUBSCRIPT italic_χ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_d ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT , italic_χ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) :

χAB is a zero-oblique-discord state,fragmentssuperscript𝜒𝐴𝐵 is a zero-oblique-discord state,\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \chi^AB\text is a zero-oblique% -discord state\,italic_χ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT is a zero-oblique-discord state , (17) where, inf𝑖𝑛𝑓infitalic_i italic_n italic_f runs over all zero-oblique-discord states χABsuperscript𝜒𝐴𝐵\chi^ABitalic_χ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, d𝑑ditalic_d is a distance, for example,

d(ρAB,χAB)=tr[(ρAB-χAB)2].𝑑superscript𝜌𝐴𝐵superscript𝜒𝐴𝐵𝑡𝑟delimited-[]superscriptsuperscript𝜌𝐴𝐵superscript𝜒𝐴𝐵2\displaystyle d(\rho^AB,\chi^AB)=tr[(\rho^AB-\chi^AB)^2].italic_d ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT , italic_χ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_t italic_r [ ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT - italic_χ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] . (18)

Definition 3. In the same spirit of Ref. Luo2010 , we can also define another geometric oblique discord as

DGO1A(ρAB)=infΦAd[ρAB,ΦA(ρAB)]: fragmentssuperscriptsubscript𝐷𝐺𝑂1𝐴fragments(superscript𝜌𝐴𝐵)subscriptinfimumsubscriptΦ𝐴fragmentsdfragments[superscript𝜌𝐴𝐵,subscriptΦ𝐴fragments(superscript𝜌𝐴𝐵)]:italic- \displaystyle D_GO1^A(\rho^AB)=\inf_\Phi_A\d[\rho^AB,\Phi_A(% \rho^AB)]:\ \ \ \ \ \ \ \ \ \ \ \ \ italic_D start_POSTSUBSCRIPT italic_G italic_O 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = roman_inf start_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_d [ italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT , roman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ] :

ΦA is defined in Eq.(13),fragmentssubscriptΦ𝐴 is defined in Eq.(13),\displaystyle\ \ \ \ \ \ \Phi_A\text is defined in Eq.(\refeq.13)\,roman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT is defined in Eq.( ) , (19) once more for example,

d[ρAB,ΦA(ρAB)]=tr[ρAB-ΦA(ρAB)]2.𝑑superscript𝜌𝐴𝐵subscriptΦ𝐴superscript𝜌𝐴𝐵𝑡𝑟superscriptdelimited-[]superscript𝜌𝐴𝐵subscriptΦ𝐴superscript𝜌𝐴𝐵2\displaystyle d[\rho^AB,\Phi_A(\rho^AB)]=tr\[\rho^AB-\Phi_A(\rho^% AB)]^2\.italic_d [ italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT , roman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ] = italic_t italic_r [ italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT - roman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . (20)

Definition 4. Compared to the information-theoretic measure of discord in Eq.(3), it is very desirable to define an information-theoretic measure of oblique discord as

DOA(ρAB)=infΦA[I(ρ)-I(ΦAρ)],superscriptsubscript𝐷𝑂𝐴superscript𝜌𝐴𝐵subscriptinfimumsubscriptΦ𝐴delimited-[]𝐼𝜌𝐼subscriptΦ𝐴𝜌\displaystyle D_O^A(\rho^AB)=\inf_\Phi_A[I(\rho)-I(\Phi_A\rho)],italic_D start_POSTSUBSCRIPT italic_O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = roman_inf start_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_I ( italic_ρ ) - italic_I ( roman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_ρ ) ] , (21) where I(ρ)=S(ρA)+S(ρB)-S(ρ)𝐼𝜌𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵𝑆𝜌I(\rho)=S(\rho^A)+S(\rho^B)-S(\rho)italic_I ( italic_ρ ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ ) is the mutual information.

However, we do not know whether DOA(ρAB)superscriptsubscript𝐷𝑂𝐴superscript𝜌𝐴𝐵D_O^A(\rho^AB)italic_D start_POSTSUBSCRIPT italic_O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) defined above is always nonnegative. Note that DOA(ρAB)≥0superscriptsubscript𝐷𝑂𝐴superscript𝜌𝐴𝐵0D_O^A(\rho^AB)\geq 0italic_D start_POSTSUBSCRIPT italic_O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≥ 0 iff I(ρAB)≥I(ΦAρAB)𝐼superscript𝜌𝐴𝐵𝐼subscriptΦ𝐴superscript𝜌𝐴𝐵I(\rho^AB)\geq I(\Phi_A\rho^AB)italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≥ italic_I ( roman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) for any ΦAsubscriptΦ𝐴\Phi_Aroman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT. We raise the conjecture below.

Conjecture:

I(ρAB)≥I(ΦAρAB) for any ΦA and any ρAB,𝐼superscript𝜌𝐴𝐵𝐼subscriptΦ𝐴superscript𝜌𝐴𝐵 for any subscriptΦ𝐴 and any superscript𝜌𝐴𝐵\displaystyle I(\rho^AB)\geq I(\Phi_A\rho^AB)\text for any \Phi_A% \text and any \rho^AB,italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≥ italic_I ( roman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) for any roman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and any italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT , (22) where I(ρAB)𝐼superscript𝜌𝐴𝐵I(\rho^AB)italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) is the mutual information, ΦAsubscriptΦ𝐴\Phi_Aroman_Φ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT is defined in Eq.(13).

IV Extend oblique discord in some ways

As in the case of discord, we can extend the definition of oblique discord in many ways.

Definition 5. An N𝑁Nitalic_N-partite state ρ𝜌\rhoitalic_ρ is said to be of zero global oblique discord, if it can be written in the form

ρ=∑i1=1nA1∑i2=1nA2…∑iN=1nANpi1i2…iN|i1⟩⟨i1| 𝜌superscriptsubscriptsubscript𝑖11subscript𝑛subscript𝐴1superscriptsubscriptsubscript𝑖21subscript𝑛subscript𝐴2…superscriptsubscriptsubscript𝑖𝑁1subscript𝑛subscript𝐴𝑁subscript𝑝subscript𝑖1subscript𝑖2…subscript𝑖𝑁ketsubscript𝑖1brasubscript𝑖1\displaystyle\rho=\sum_i_1=1^n_A_1\sum_i_2=1^n_A_2...\sum_% i_N=1^n_A_Np_i_1i_2...i_N|i_1\rangle\langle i_1|\ \ \ \ % \ \ \ \ \ \ \ \ \ italic_ρ = ∑ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUPERSCRIPT … ∑ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ ⟨ italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT |

⊗|i2⟩⟨i2|⊗…⊗|iN⟩⟨iN|,tensor-productabsenttensor-productketsubscript𝑖2brasubscript𝑖2…ketsubscript𝑖𝑁brasubscript𝑖𝑁\displaystyle\ \ \ \ \ \ \otimes|i_2\rangle\langle i_2|\otimes...\otimes|i% _N\rangle\langle i_N|,⊗ | italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⟩ ⟨ italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | ⊗ … ⊗ | italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ⟩ ⟨ italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT | , (23) where ∑i1=1nA1∑i2=1nA2…∑iN=1nANpi1i2…iN=1,pi1i2…iN≥0,ij=1nAjformulae-sequencesuperscriptsubscriptsubscript𝑖11subscript𝑛subscript𝐴1superscriptsubscriptsubscript𝑖21subscript𝑛subscript𝐴2…superscriptsubscriptsubscript𝑖𝑁1subscript𝑛subscript𝐴𝑁subscript𝑝subscript𝑖1subscript𝑖2…subscript𝑖𝑁1subscript𝑝subscript𝑖1subscript𝑖2…subscript𝑖𝑁0superscriptsubscriptketsubscript𝑖𝑗subscript𝑖𝑗1subscript𝑛subscript𝐴𝑗\sum_i_1=1^n_A_1\sum_i_2=1^n_A_2...\sum_i_N=1^n_A_N% p_i_1i_2...i_N=1,p_i_1i_2...i_N\geq 0,\_i_% j=1^n_A_j∑ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUPERSCRIPT … ∑ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 1 , italic_p start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_i start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ 0 , italic_i start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUPERSCRIPT is a normalized basis of HAj.superscript𝐻subscript𝐴𝑗H^A_j.italic_H start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT . Notice that ij⟩ij=1nAjsuperscriptsubscriptketsubscript𝑖𝑗subscript𝑖𝑗1subscript𝑛subscript𝐴𝑗\i_j\rangle\_i_j=1^n_A_j start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUPERSCRIPT is not necessarily orthogonal.

Theorem 2. An N𝑁Nitalic_N-partite state ρ𝜌\rhoitalic_ρ is of zero global oblique discord iff there exists an operation ΦAjj=1N=ij=1nAjj=1NsuperscriptsubscriptsubscriptΦsubscript𝐴𝑗𝑗1𝑁superscriptsubscriptsuperscriptsubscriptketsubscript𝑖𝑗bra~subscript𝑖𝑗subscript𝑖𝑗1subscript𝑛subscript𝐴𝑗𝑗1𝑁\\Phi_A_j\_j=1^N=\\\_i_j% =1^n_A_j\_j=1^N roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT = start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT such that

ΦA1A2…ANρ=ΦA1…ΦAN-1ΦρAN=ρ.subscriptΦsubscript𝐴1subscript𝐴2…subscript𝐴𝑁𝜌subscriptΦsubscript𝐴1…subscriptΦsubscript𝐴𝑁1Φsubscript𝜌subscript𝐴𝑁𝜌\displaystyle\Phi_A_1A_2...A_N\rho=\Phi_A_1...\Phi_A_N-1\Phi% _A_N\rho=\rho.roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ = roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT … roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_N - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_Φ start_FLOATSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_FLOATSUBSCRIPT italic_ρ = italic_ρ . (24)

It can be directly checked that

ΦA1(ΦA2ρ)=ΦA2(ΦA1ρ),subscriptΦsubscript𝐴1subscriptΦsubscript𝐴2𝜌subscriptΦsubscript𝐴2subscriptΦsubscript𝐴1𝜌\displaystyle\Phi_A_1(\Phi_A_2\rho)=\Phi_A_2(\Phi_A_1\rho),roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ ) = roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ ) , (25) hence ΦA1A2…ANρ=ΦA1…ΦAN-1ΦANρsubscriptΦsubscript𝐴1subscript𝐴2…subscript𝐴𝑁𝜌subscriptΦsubscript𝐴1…subscriptΦsubscript𝐴𝑁1subscriptΦsubscript𝐴𝑁𝜌\Phi_A_1A_2...A_N\rho=\Phi_A_1...\Phi_A_N-1\Phi_A_N\rhoroman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ = roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT … roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_N - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ above can be defined without any ambiguity.

Definition 6. We define the geometric global oblique discord of N𝑁Nitalic_N-partite state ρ𝜌\rhoitalic_ρ as

DGO(ρ)=infΦA1A2…ANd[ρ,ΦA1A2…AN(ρ)],subscript𝐷𝐺𝑂𝜌subscriptinfimumsubscriptΦsubscript𝐴1subscript𝐴2…subscript𝐴𝑁𝑑𝜌subscriptΦsubscript𝐴1subscript𝐴2…subscript𝐴𝑁𝜌\displaystyle D_GO(\rho)=\inf_\Phi_A_1A_2...A_Nd[\rho,\Phi_A_1A% _2...A_N(\rho)],italic_D start_POSTSUBSCRIPT italic_G italic_O end_POSTSUBSCRIPT ( italic_ρ ) = roman_inf start_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_d [ italic_ρ , roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) ] , (26) where, d is a distance as in Eq.(7).

Definition 7. We define the global oblique discord of an N-partite state ρ𝜌\rhoitalic_ρ as

DO(ρ)=infΦA1A2…AN[I(ρ)-I(ΦA1A2…ANρ)],subscript𝐷𝑂𝜌subscriptinfimumsubscriptΦsubscript𝐴1subscript𝐴2…subscript𝐴𝑁delimited-[]𝐼𝜌𝐼subscriptΦsubscript𝐴1subscript𝐴2…subscript𝐴𝑁𝜌\displaystyle D_O(\rho)=\inf_\Phi_A_1A_2...A_N[I(\rho)-I(\Phi_A_% 1A_2...A_N\rho)],italic_D start_POSTSUBSCRIPT italic_O end_POSTSUBSCRIPT ( italic_ρ ) = roman_inf start_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_I ( italic_ρ ) - italic_I ( roman_Φ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ ) ] , (27) where I(ρ)=∑j=1NS(ρAj)-S(ρ)𝐼𝜌superscriptsubscript𝑗1𝑁𝑆superscript𝜌subscript𝐴𝑗𝑆𝜌I(\rho)=\sum_j=1^NS(\rho^A_j)-S(\rho)italic_I ( italic_ρ ) = ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ ) is the mutual information. Similar to the case of Eq.(21), DO(ρ)≥0subscript𝐷𝑂𝜌0D_O(\rho)\geq 0italic_D start_POSTSUBSCRIPT italic_O end_POSTSUBSCRIPT ( italic_ρ ) ≥ 0 requires Eq.(22) holds.

V Summary and discussion

The definition of quantum discord corresponds to orthogonal basis, in this paper, we relaxed the constraint of orthogonality, and proposed the definition of oblique discord. Oblique discord characterizes a new kind of quantum correlation between entanglement and discord.

There left many open questions for future investigations. Firstly, are there physical effects which can be revealed by oblique discord? Secondly, conjecture in Eq.(22) is true or false? Thirdly, how to calculate the different measures of oblique discord analytically or efficiently numerically, especially for n𝑛nitalic_n-qubit states.

This work was supported by the National Natural Science Foundation of China (Grant No.11347213) and the Chinese Universities Scientific Fund (Grant No.2014YB029). The author thanks Kai-Liang Lin and Lin Zhang for helpful discussions.