Comment On Quantum Discord Through The Generalized Entropy In Bipartite Quantum States

In [2014 Eur.J.Phys. D 68 1], Hou, Huang, and Cheng present, using Tsallis’ entropy, possible generalizations of the quantum discord measure, finding original results. As for the mutual informations and discord, we show here that these two types of quantifiers can take negative values. In the two qubits instance we further determine in which regions they are non-negative. Additionally, we study alternative generalizations on the basis of Renyi entropies.

On an interesting recent paper, Hou et al. Hou14 introduce generalizations for two quantifiers: mutual information and quantum discords, which they use for the study of quantum correlations in two qubits systems. It is conventionally agreed that the mutual information (MI) quantifies total correlations in bipartite systems. Given a system described by the state ρabsuperscript𝜌𝑎𝑏\rho^abitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT, with subsystems a𝑎aitalic_a and b𝑏bitalic_b, the MI reads

I(a:b):=S(ρa)+S(ρb)-S(ρab),fragmentsIfragments(a:b)assignSfragments(superscript𝜌𝑎)Sfragments(superscript𝜌𝑏)Sfragments(superscript𝜌𝑎𝑏),I(a:b):=S(\rho^a)+S(\rho^b)-S(\rho^ab)\,,italic_I ( italic_a : italic_b ) := 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 ) , (1) where ρa:=Trbρabassignsuperscript𝜌𝑎subscriptTr𝑏superscript𝜌𝑎𝑏\rho^a:=\textTr_b\rho^abitalic_ρ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT := Tr start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT y ρb:=Traρabassignsuperscript𝜌𝑏subscriptTr𝑎superscript𝜌𝑎𝑏\rho^b:=\textTr_a\rho^abitalic_ρ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT := Tr start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT are reduced states associated to our subsystems. S(⋅)𝑆⋅S(\cdot)italic_S ( ⋅ ) is von Neumann’s entropy for a state σ𝜎\sigmaitalic_σ:

S(σ):=-Tr(σlogσ).assign𝑆𝜎Tr𝜎𝜎S(\sigma):=-\textTr(\sigma\log\sigma)\,.italic_S ( italic_σ ) := - Tr ( italic_σ roman_log italic_σ ) . (2) If one wishes to quantify non-classical correlations, these should be appropriately discriminated from the total ones. A possibility is to compute classical correlations via a classical information measure (CI)

Cb(a:b):=S(ρa)-minΠi∑kpkS(ρka),fragmentssuperscript𝐶𝑏fragments(a:b)assignSfragments(superscript𝜌𝑎)subscriptsubscriptΠ𝑖subscript𝑘subscript𝑝𝑘Sfragments(subscriptsuperscript𝜌𝑎𝑘),C^b(a:b):=S(\rho^a)-\min_\\Pi_i\\sum_kp_kS(\rho^a_k)\,,italic_C start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( italic_a : italic_b ) := italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ) - roman_min start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , (3) where ΠisubscriptΠ𝑖\\Pi_i\ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is a complete projective measure, local in b𝑏bitalic_b, and

ρka:=1pkTrb[(Ia⊗Πk)ρab(Ia⊗Πk)]assignsubscriptsuperscript𝜌𝑎𝑘1subscript𝑝𝑘subscriptTr𝑏delimited-[]tensor-productsubscript𝐼𝑎subscriptΠ𝑘superscript𝜌𝑎𝑏tensor-productsubscript𝐼𝑎subscriptΠ𝑘\rho^a_k:=\frac1p_k\textTr_b[(I_a\otimes\Pi_k)\rho^ab(I_% a\otimes\Pi_k)]italic_ρ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG Tr start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT [ ( italic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT ( italic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ] (4) is the a𝑎aitalic_a’s conditional state associated to the outcome k𝑘kitalic_k of b𝑏bitalic_b. Further,

pk:=Tr[(Ia⊗Πk)ρab(Ia⊗Πk)]assignsubscript𝑝𝑘Trdelimited-[]tensor-productsubscript𝐼𝑎subscriptΠ𝑘superscript𝜌𝑎𝑏tensor-productsubscript𝐼𝑎subscriptΠ𝑘p_k:=\textTr[(I_a\otimes\Pi_k)\rho^ab(I_a\otimes\Pi_k)]italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := Tr [ ( italic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT ( italic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ] (5) is the corresponding probability. Iasubscript𝐼𝑎I_aitalic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT is the identity operator for a𝑎aitalic_a. Eq. (3) quantifies the classical correlations from a b𝑏bitalic_b-perspective and, analogously, one defines Casuperscript𝐶𝑎C^aitalic_C start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT. Given that (1) and (3) compute quantum and classical correlations, respectively, the discord measure is given by Zur01

Db(a:b):=I(a:b)-Cb(a:b).fragmentssuperscript𝐷𝑏fragments(a:b)assignIfragments(a:b)superscript𝐶𝑏fragments(a:b).D^b(a:b):=I(a:b)-C^b(a:b)\,.italic_D start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( italic_a : italic_b ) := italic_I ( italic_a : italic_b ) - italic_C start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( italic_a : italic_b ) . (6) Hou et al. generalized these measures replacing von Neumann’s entropy by Tsallis’ and Renyi’s ones (Ren61 ; Gel04 ; Tsa09 , and references therein). The α𝛼\alphaitalic_α-Renyi quantifier is Ren61

Sα(σ):=logTrσα1-α,assignsubscript𝑆𝛼𝜎Trsuperscript𝜎𝛼1𝛼S_\alpha(\sigma):=\frac\log\textTr\sigma^\alpha1-\alpha\,,italic_S start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_σ ) := divide start_ARG roman_log Tr italic_σ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT end_ARG start_ARG 1 - italic_α end_ARG , (7) while Tsallis’ counterpart reads Tsa09

Sq(σ):=1-Trσq(q-1)ln2.assignsubscript𝑆𝑞𝜎1Trsuperscript𝜎𝑞𝑞12S_q(\sigma):=\frac1-\textTr\sigma^q(q-1)\ln 2\,.italic_S start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_σ ) := divide start_ARG 1 - Tr italic_σ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT end_ARG start_ARG ( italic_q - 1 ) roman_ln 2 end_ARG . (8) Both quantifiers converge to von Neumann’s in the limit α→1→𝛼1\alpha\rightarrow 1italic_α → 1 (q→1→𝑞1q\rightarrow 1italic_q → 1). Note that we use always basis-2 logarithms, which slightly modifies the usual definition of Sqsubscript𝑆𝑞S_qitalic_S start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT. Hou et al. replace then S𝑆Sitalic_S by Sαsubscript𝑆𝛼S_\alphaitalic_S start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT or Sqsubscript𝑆𝑞S_qitalic_S start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT in (1) and (3), obtaining generalized mutual information measure Iαsubscript𝐼𝛼I_\alphaitalic_I start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT (RMI) and Iqsubscript𝐼𝑞I_qitalic_I start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT (TMI). We consequently have generalized classical correlations (Cαbsubscriptsuperscript𝐶𝑏𝛼C^b_\alphaitalic_C start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT, Cqbsubscriptsuperscript𝐶𝑏𝑞C^b_qitalic_C start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT) and discords (Dαbsubscriptsuperscript𝐷𝑏𝛼D^b_\alphaitalic_D start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT, Dqbsubscriptsuperscript𝐷𝑏𝑞D^b_qitalic_D start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT).

We show below that these last correlation-quantifiers can take negative values, refuting what is conjectured by Hou et al. Even more, the discord can be different from zero, and even negative, for classical states.

Rank-three classical states of two qubits. von Neumann’s entropy properties guarantee the positivity of I𝐼Iitalic_I, Cbsuperscript𝐶𝑏C^bitalic_C start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT, and Dbsuperscript𝐷𝑏D^bitalic_D start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT. We will see that generalized quantifiers do not, in general, share such positivity property.

As an example consider the family of states given below. We focus attention on classical states of range 3 (standard basis).

ρuvab=(u0000v00001-u-v00000),subscriptsuperscript𝜌𝑎𝑏𝑢𝑣𝑢0000𝑣00001𝑢𝑣00000\rho^ab_uv=\left(\beginarray[]ccccu&0&0&0\\ 0&v&0&0\\ 0&0&1-u-v&0\\ 0&0&0&0\endarray\right)\,,italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_u italic_v end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL italic_u end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_v end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 1 - italic_u - italic_v end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW end_ARRAY ) , (9) with u,v≥0𝑢𝑣0u,v\geq 0italic_u , italic_v ≥ 0 and u+v≤1𝑢𝑣1u+v\leq 1italic_u + italic_v ≤ 1. There exists a non-perturbative, complete and local projective measurement given by the projectors basis ket0bra0ket1bra1\\ket0\bra0,\ket1\bra1\ start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG for the two subsystems. Thus, for the family ρuvabsubscriptsuperscript𝜌𝑎𝑏𝑢𝑣\rho^ab_uvitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_u italic_v end_POSTSUBSCRIPT one has Db(a:b)=0fragmentssuperscript𝐷𝑏fragments(a:b)0D^b(a:b)=0italic_D start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( italic_a : italic_b ) = 0 (and Da(a:b)=0fragmentssuperscript𝐷𝑎fragments(a:b)0D^a(a:b)=0italic_D start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ( italic_a : italic_b ) = 0). In Figs. 1-2 we note that generalized measures can be negative even for classical states.Consequently, the ensuing generalized discords cannot discriminate classical in the sense discussed above.

CI turns out to be positive [all (α𝛼\alphaitalic_α, q𝑞qitalic_q)] for the family ρuvabsubscriptsuperscript𝜌𝑎𝑏𝑢𝑣\rho^ab_uvitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_u italic_v end_POSTSUBSCRIPT. It would seem that, fora q>1𝑞1q>1italic_q >1, Tsallis’ discord works better, since it is always positive. In the case (q<1𝑞1q<1italic_q <1, α<1𝛼1\alpha<1italic_α <1), neither Renyi nor Tsallis measures behaves os one would expect for classical states.

Random states of two qubits. In this case we compute generalized MI, CI, and discord for different pairs (α𝛼\alphaitalic_α, q𝑞qitalic_q) so as to estimate the range, in such a plane, for positivity. We considered 105superscript10510^510 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT random states for each of these parameters. Fig. 3 plots minima of MI and discord for a given α𝛼\alphaitalic_α or q𝑞qitalic_q.

For 2- qubits states, generalized CI’s turned out to be positive for all our states-sample, with α𝛼\alphaitalic_α and q𝑞qitalic_q ranging in (0,1000)01000(0,1000)( 0 , 1000 ). This makes it credible that the quantifier is positive for all (α𝛼\alphaitalic_α, q𝑞qitalic_q). Instead, minima for generalized MI and discord reach negative values for all α≠1𝛼1\alpha eq 1italic_α ≠ 1 in the Renyi instance, while they are positive in the Tsallis case for q≥1𝑞1q\geq 1italic_q ≥ 1. This would indicate that Tsallis’ entropy is strongly sub-additive (see below). Regretfully enough, the negativity of these discords does not signal classicality. As an example, states that are known to be of a non-classical nature display negative Renyi discord for α=2𝛼2\alpha=2italic_α = 2 (Fig. 4).

Alternative generalizations. It is easy to see that the von Neumann-sub-aditivity (SA) of S(ρab)𝑆superscript𝜌𝑎𝑏S(\rho^ab)italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT ):

S(ρab)≤S(ρa)+S(ρb),𝑆superscript𝜌𝑎𝑏𝑆superscript𝜌𝑎𝑆superscript𝜌𝑏S(\rho^ab)\leq S(\rho^a)+S(\rho^b)\,,italic_S ( 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 ) , (10) is tantamount to MI-positivity and that the concavity:

S(∑ipiρi)≥∑ipiS(ρi),𝑆subscript𝑖subscript𝑝𝑖subscript𝜌𝑖subscript𝑖subscript𝑝𝑖𝑆subscript𝜌𝑖S(\sum_ip_i\rho_i)\geq\sum_ip_iS(\rho_i)\,,italic_S ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≥ ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , (11) implies that Cb(a:b)fragmentssuperscript𝐶𝑏fragments(a:b)C^b(a:b)italic_C start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( italic_a : italic_b ) and Ca(a:b)fragmentssuperscript𝐶𝑎fragments(a:b)C^a(a:b)italic_C start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ( italic_a : italic_b ) are positive measures. Discord positivity is deduced from strong subadditivity (SSA) Zur01 ; Dat10 :

S(ρabc)+S(ρb)≤S(ρab)+S(ρbc),𝑆superscript𝜌𝑎𝑏𝑐𝑆superscript𝜌𝑏𝑆superscript𝜌𝑎𝑏𝑆superscript𝜌𝑏𝑐S(\rho^abc)+S(\rho^b)\leq S(\rho^ab)+S(\rho^bc)\,,italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ) ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) , (12) equivalent to the concavity of the conditional entropy S(a|b):=S(ρab)-S(ρb)assign𝑆conditional𝑎𝑏𝑆superscript𝜌𝑎𝑏𝑆superscript𝜌𝑏S(a|b):=S(\rho^ab)-S(\rho^b)italic_S ( italic_a | italic_b ) := italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ). In general, generalized entropies do not share these properties for arbitrary values of α𝛼\alphaitalic_α and q𝑞qitalic_q. Renyi ones are concave in the interval (0,α*)0superscript𝛼(0,\alpha^*)( 0 , italic_α start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ), with α*=1+log4log(N-1)superscript𝛼14𝑁1\alpha^*=1+\frac\log 4\log(N-1)italic_α start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT = 1 + divide start_ARG roman_log 4 end_ARG start_ARG roman_log ( italic_N - 1 ) end_ARG, N𝑁Nitalic_N being the density matrix range XuE10 ; BeZ06 . For α≥α*𝛼superscript𝛼\alpha\geq\alpha^*italic_α ≥ italic_α start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT, Sαsubscript𝑆𝛼S_\alphaitalic_S start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT is neither convex nor concave. Given q>1𝑞1q>1italic_q >1, Tsallis’ entropy is sub-additive so that the associated mutual information is positive as well, i.e., Iq≥0subscript𝐼𝑞0I_q\geq 0italic_I start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ≥ 0 for q>1𝑞1q>1italic_q >1 Aud07 . However, for 0<10𝑞10<0

<italic_q>



<1, Tsallis’s measure is super-additive for product states while for general states it is neither sub- nor super-additive Rag95 . Thus, Iqsubscript𝐼𝑞I_qitalic_I start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT can adopt negative values for 0







<10𝑞10<0

<italic_q>



<1. Renyi’s entropies are sub-additive for α=0𝛼0\alpha=0italic_α = 0 and α=1𝛼1\alpha=1italic_α = 1 Acz75 . Discord server For all other α-limit-from𝛼\alpha-italic_α -values one can find states for which the associated MI is negative. SSA does not hold in general, save for the von Neumann instance PeV14 . For classical states, Sqsubscript𝑆𝑞S_qitalic_S start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT displays SSA if q≥1𝑞1q\geq 1italic_q ≥ 1 Fur06 . (There exist particular cases in which Sαsubscript𝑆𝛼S_\alphaitalic_S start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT also displays SSA, as, for instance, Gaussian states with α=2𝛼2\alpha=2italic_α = 2 Add12 .) Table 1 details ‘properties of the different entropies. Discord server







</italic_q>







</italic_q>

Concavity and SA are sufficient, but not necessary, to guarantee positivity. In the case of the range 3-classical family (ρuvabsubscriptsuperscript𝜌𝑎𝑏𝑢𝑣\rho^ab_uvitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_u italic_v end_POSTSUBSCRIPT), our numerical results show that Renyi’s CI is positive for all α𝛼\alphaitalic_α, being concave only for α<3𝛼3\alpha<3italic_α <3. As for discord’s positivity, it suffices to demand that

I(a:b)≥χ(Pa,b),fragmentsIfragments(a:b)χfragments(subscript𝑃𝑎,b),I(a:b)\geq\chi(P_a,b)\,,italic_I ( italic_a : italic_b ) ≥ italic_χ ( italic_P start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_b ) , (13) where χ(Pa,b):=S(ρa)-S(b|Pa)assign𝜒subscript𝑃𝑎𝑏𝑆superscript𝜌𝑎𝑆conditional𝑏subscript𝑃𝑎\chi(P_a,b):=S(\rho^a)-S(b|P_a)italic_χ ( italic_P start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_b ) := italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ) - italic_S ( italic_b | italic_P start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) is Holevo’s quantity associated to the b𝑏bitalic_b-state conditioned to a POVM measurement of a𝑎aitalic_a of operators Pasubscript𝑃𝑎P_aitalic_P start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT. Coles speaks here of firm subadditivity (FSA), that is less restrictive than SSA. Hierarchically: SSA⇒⇒\Rightarrow⇒FSA⇒⇒\Rightarrow⇒SA Col11 . Results for a 2 qubits random simulation (see Fig. 3) would indicate that Tsallis entropies are FSA for q≥1𝑞1q\geq 1italic_q ≥ 1, while Renyi ones are FSA for α=1𝛼1\alpha=1italic_α = 1 and, possibly, for α=0𝛼0\alpha=0italic_α = 0. Discord servers

In von Neumann’s entropic scheme, it is equivalent to define the MI as the relative entropy between the given state and the product of the concomitant reduced states, i.e.,

I(a:b):=minσa,σbS(ρab||σa⊗σb),fragmentsIfragments(a:b)assignsubscriptsuperscript𝜎𝑎superscript𝜎𝑏Sfragments(superscript𝜌𝑎𝑏||superscript𝜎𝑎tensor-productsuperscript𝜎𝑏),I(a:b):=\min_\\sigma^a,\sigma^b\S(\rho^ab||\sigma^a\otimes\sigma^% b)\,,italic_I ( italic_a : italic_b ) := roman_min start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT , italic_σ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT | | italic_σ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ⊗ italic_σ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ) , (14) where S(ρ||σ):=-S(ρ)-Tr(ρlogσ)fragmentsSfragments(ρ||σ)assignSfragments(ρ)Trfragments(ρσ)S(\rho||\sigma):=-S(\rho)-\textTr(\rho\log\sigma)italic_S ( italic_ρ | | italic_σ ) := - italic_S ( italic_ρ ) - Tr ( italic_ρ roman_log italic_σ ) is the relative entropy, and the minimization runs over the set of all completely uncorrelated states. Here, Klein’s inequality guarantees the positivity of I(a:b)fragmentsIfragments(a:b)I(a:b)italic_I ( italic_a : italic_b ). Eq. (14) offers an alternative path for generalizing the MI in terms of other entropic measures, different from the one associated to (1). This alternative was employed by different authors and is known as the quantum conditional MI Ber14 ; Tom09 . Different definitions of Rényi’s or Tsallis relative entropies determine distinct alternatives for the conditional MI.

A reasonable idea would then entail to define the generalized mutual information as in Eq. (14), using some generalized relative entropy or divergence:

I~α(a:b):=minσa,σbSα(ρab||σa⊗σb).fragmentssubscript~𝐼𝛼fragments(a:b)assignsubscriptsuperscript𝜎𝑎superscript𝜎𝑏subscript𝑆𝛼fragments(superscript𝜌𝑎𝑏||superscript𝜎𝑎tensor-productsuperscript𝜎𝑏).\tildeI_\alpha(a:b):=\min_\\sigma^a,\sigma^b\S_\alpha(\rho^ab|% |\sigma^a\otimes\sigma^b)\,.over~ start_ARG italic_I end_ARG start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_a : italic_b ) := roman_min start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT , italic_σ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT | | italic_σ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ⊗ italic_σ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ) . (15) In similar vein, the classical generalized information will be

C~αa(a:b):=maxΠiminσa,σbSα(ρab′||σa⊗σb),fragmentssubscriptsuperscript~𝐶𝑎𝛼fragments(a:b)assignsubscriptsubscriptΠ𝑖subscriptsuperscript𝜎𝑎superscript𝜎𝑏subscript𝑆𝛼fragments(superscriptsuperscript𝜌𝑎𝑏′||superscript𝜎𝑎tensor-productsuperscript𝜎𝑏),\tildeC^a_\alpha(a:b):=\max_\\Pi_i\\min_\\sigma^a,\sigma^b% \S_\alpha(\rho^ab^\prime||\sigma^a\otimes\sigma^b)\,,over~ start_ARG italic_C end_ARG start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_a : italic_b ) := roman_max start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_min start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT , italic_σ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | | italic_σ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ⊗ italic_σ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ) , (16) where ρab′:=∑k(Ia⊗Πk)ρab(Ia⊗Πk)assignsuperscriptsuperscript𝜌𝑎𝑏′subscript𝑘tensor-productsubscript𝐼𝑎subscriptΠ𝑘superscript𝜌𝑎𝑏tensor-productsubscript𝐼𝑎subscriptΠ𝑘\rho^ab^\prime:=\sum_k(I_a\otimes\Pi_k)\rho^ab(I_a\otimes\Pi_% k)italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT ( italic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) is the posterior state to the measurement of ΠisubscriptΠ𝑖\\Pi_i\ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT in b𝑏bitalic_b. The new generalized discord would be given by the difference between these two quantities

D~αa(a:b):=I~α(a:b)-C~αa(a:b).fragmentssubscriptsuperscript~𝐷𝑎𝛼fragments(a:b)assignsubscript~𝐼𝛼fragments(a:b)subscriptsuperscript~𝐶𝑎𝛼fragments(a:b).\tildeD^a_\alpha(a:b):=\tildeI_\alpha(a:b)-\tildeC^a_\alpha(a:% b)\,.over~ start_ARG italic_D end_ARG start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_a : italic_b ) := over~ start_ARG italic_I end_ARG start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_a : italic_b ) - over~ start_ARG italic_C end_ARG start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_a : italic_b ) . (17) The positivity of I~~𝐼\tildeIover~ start_ARG italic_I end_ARG and C~asuperscript~𝐶𝑎\tildeC^aover~ start_ARG italic_C end_ARG start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT will be guaranteed by the positivity of the generalized relative entropies. Noting that our relative entropies fulfill the data processing inequality, D~asuperscript~𝐷𝑎\tildeD^aover~ start_ARG italic_D end_ARG start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT will be positive as well (see, for instance, Mis14 ). The scheme being advanced here should be the subject further exploration.

Recently, the introduction of a new Renyi relative entropy, monotonous against general quantum (trace preserving) operations, in the range 1/2≤q<∞12𝑞1/2\leq q1 / 2 ≤ italic_q <∞ seem to constitute the most convenient way of computing a states’ MI and, a posteriori, to define a new generalized discord quantifier Mul13 ; Wil13 ; Lie13 ; Mis14 .