The “classical-quantum” (cq) discord of a bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT is the smallest difference between the mutual information S(ρA:B)𝑆superscript𝜌:𝐴𝐵S(\rho^A:B)italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A : italic_B end_POSTSUPERSCRIPT ) of ρ𝜌\rhoitalic_ρ and that of ρ𝜌\rhoitalic_ρ after a measurement channel is applied on the A𝐴Aitalic_A system. Relating zero discord to the strong subadditivity of the Von Neumann entropy, Datta proved that a state has zero cq discord iff and only if it can be written in the form ∑ipi|i⟩⟨i|⊗ρiBsubscript𝑖tensor-productsubscript𝑝𝑖ket𝑖bra𝑖subscriptsuperscript𝜌𝐵𝑖\sum_ip_i|i\rangle\langle i|\otimes\rho^B_i∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_i ⟩ ⟨ italic_i | ⊗ italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for pisubscript𝑝𝑖p_iitalic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT a probability distribution, |i⟩ket𝑖|i\rangle| italic_i ⟩ a basis of the A𝐴Aitalic_A system and ρiBsubscriptsuperscript𝜌𝐵𝑖\rho^B_iitalic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT states of the B𝐵Bitalic_B system. We provide a simple proof of that same result using directly a theorem of Petz on channels that leave unchanged the relative entropy of two given states.
Various measures have been proposed to quantify the classicality or equivalently the quantumness in a multipartite state. One of them, “discord”, is the theme of this note. Limiting ourselves to a bipartite system AB𝐴𝐵ABitalic_A italic_B, given an orthonormal basis 𝖠=a∈A𝖠subscriptket𝑎𝑎𝐴\mathsfA=\big\_a\in Asansserif_A = italic_a ⟩ start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT of ℋAsubscriptℋ𝐴\mathcalH_Acaligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, any state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, i.e. density operator, can be represented by a block matrix with blocks ρaa′Bsubscriptsuperscript𝜌𝐵𝑎superscript𝑎′\rho^B_aa^\primeitalic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, corresponding to ρAB=∑aa′|a⟩⟨a′|⊗ρaa′Bsuperscript𝜌𝐴𝐵subscript𝑎superscript𝑎′tensor-productket𝑎brasuperscript𝑎′subscriptsuperscript𝜌𝐵𝑎superscript𝑎′\rho^AB=\sum_aa^\prime|a\rangle\langle a^\prime|\otimes\rho^B_aa^% \primeitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | italic_a ⟩ ⟨ italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ⊗ italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT. The state is said to be “classical-quantum” (cq) if there is such a basis for which the matrix is block diagonal i.e ρaa′B=0subscriptsuperscript𝜌𝐵𝑎superscript𝑎′0\rho^B_aa^\prime=0italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 0 if a≠a′𝑎superscript𝑎′a eq a^\primeitalic_a ≠ italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. The operator 𝒟𝖠=∑a∈A(|a⟩⟨a|⊗𝟏B)ρ(|a⟩⟨a|⊗𝟏B)subscript𝒟𝖠subscript𝑎𝐴tensor-productket𝑎bra𝑎subscript1𝐵𝜌tensor-productket𝑎bra𝑎subscript1𝐵\mathcalD_\mathsfA=\sum_a\inA(|a\rangle\langle a|\otimes\mathbf1_% B)\rho(|a\rangle\langle a|\otimes\mathbf1_B)caligraphic_D start_POSTSUBSCRIPT sansserif_A end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT ( | italic_a ⟩ ⟨ italic_a | ⊗ bold_1 start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) italic_ρ ( | italic_a ⟩ ⟨ italic_a | ⊗ bold_1 start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) which replaces off diagonal blocks by a 00 block is a quantum channel that acts independently on the A𝐴Aitalic_A and the B𝐵Bitalic_B system (it is the identity on B𝐵Bitalic_B). It is known that for such channels ℰℰ\mathcalEcaligraphic_E holds the inequality I(ℰ(ρ)A:B)≤I(ρA:B)𝐼ℰsuperscript𝜌:𝐴𝐵𝐼superscript𝜌:𝐴𝐵I(\mathcalE(\rho)^A:B)\leq I(\rho^A:B)italic_I ( caligraphic_E ( italic_ρ ) start_POSTSUPERSCRIPT italic_A : italic_B end_POSTSUPERSCRIPT ) ≤ italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A : italic_B end_POSTSUPERSCRIPT ) where I(ρA:B)𝐼superscript𝜌:𝐴𝐵I(\rho^A:B)italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A : italic_B end_POSTSUPERSCRIPT ), the mutual information, is S(ρA)+S(ρB)-S(ρAB)𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵𝑆superscript𝜌𝐴𝐵S(\rho^A)+S(\rho^B)-S(\rho^AB)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 ) and S(ρ)=-tr[ρlg(ρ)]𝑆𝜌trdelimited-[]𝜌lg𝜌S(\rho)=-\mathrmtr\big[\rho\lg(\rho)\big]italic_S ( italic_ρ ) = - roman_tr [ italic_ρ roman_lg ( italic_ρ ) ]. It is the infimum of I(ρA:B)-I(𝒟𝖠(ρ)A:B)𝐼superscript𝜌:𝐴𝐵𝐼subscript𝒟𝖠superscript𝜌:𝐴𝐵I(\rho^A:B)-I(\mathcalD_\mathsfA(\rho)^A:B)italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A : italic_B end_POSTSUPERSCRIPT ) - italic_I ( caligraphic_D start_POSTSUBSCRIPT sansserif_A end_POSTSUBSCRIPT ( italic_ρ ) start_POSTSUPERSCRIPT italic_A : italic_B end_POSTSUPERSCRIPT ) over all bases 𝖠𝖠\mathsfAsansserif_A that was called “discord” by Ollivier and Zurek Ollivier and Zurek (2001) and that we shall call qc “classical-quantum” discord (provided the dimension of ℋAsubscriptℋ𝐴\mathcalH_Acaligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT is properly chosen). It is clear that if ρ𝜌\rhoitalic_ρ is classical-quantum, then the qc discord is 00. The converse is not obvious. The argument in Ollivier and Zurek (2001) appears to lead nowhere (cf appendix). Datta Datta (2010) gave a proof that zero discord implies classical-quantum using a result of Hayden et al. Hayden et al. (2004) on the structure of states which satisfy strong subadditivity of quantum entropy with equality. His definition of discord however looks more general than the above since he optimizes over all channels defined by rank 1111 POVM’s on the A𝐴Aitalic_A system instead of complete projective measurements. Nevertheless a rank 1111 POVM with outputs in M𝑀Mitalic_M is equivalent to a unitary embedding of ℋAsubscriptℋ𝐴\mathcalH_Acaligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT into ℋMsubscriptℋ𝑀\mathcalH_Mcaligraphic_H start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT with basis the |m⟩ket𝑚|m\rangle| italic_m ⟩ and we can simply apply a projective measurements in ℋMsubscriptℋ𝑀\mathcalH_Mcaligraphic_H start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT. In fact, we can always choose M𝑀Mitalic_M of size at most dA2superscriptsubscript𝑑𝐴2d_A^2italic_d start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT where dA=dimsupp(ρA)subscript𝑑𝐴dimensionsuppsuperscript𝜌𝐴d_A=\dim\mathrmsupp(\rho^A)italic_d start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = roman_dim roman_supp ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) so that we need only choose ℋAsubscriptℋ𝐴\mathcalH_Acaligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT of dimension dA2superscriptsubscript𝑑𝐴2d_A^2italic_d start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and work with block matrices. Details are to be found in the appendix where it is also shown that there is always a basis corresponding to the discord. The following theorem thus implies that if the cq discord is 00, the state is classical-quantum. The approach is similar to that of Piani et al Hayden et al. (2004) for “classical-classical” states.
Theorem Let 𝒟=𝒟𝖠𝒟subscript𝒟𝖠\mathcalD=\mathcalD_\mathsfAcaligraphic_D = caligraphic_D start_POSTSUBSCRIPT sansserif_A end_POSTSUBSCRIPT. If I(𝒟(ρ)A:B)=I(ρA:B)𝐼𝒟superscript𝜌normal-:𝐴𝐵𝐼superscript𝜌normal-:𝐴𝐵I(\mathcalD(\rho)^A:B)=I(\rho^A:B)italic_I ( caligraphic_D ( italic_ρ ) start_POSTSUPERSCRIPT italic_A : italic_B end_POSTSUPERSCRIPT ) = italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A : italic_B end_POSTSUPERSCRIPT ) then ρ𝜌\rhoitalic_ρ can be block diagonalized in some basis of ℋAsubscriptℋ𝐴\mathcalH_Acaligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT.
The equality I(𝒟(ρ)A:B)=I(ρA:B)𝐼𝒟superscript𝜌:𝐴𝐵𝐼superscript𝜌:𝐴𝐵I(\mathcalD(\rho)^A:B)=I(\rho^A:B)italic_I ( caligraphic_D ( italic_ρ ) start_POSTSUPERSCRIPT italic_A : italic_B end_POSTSUPERSCRIPT ) = italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A : italic_B end_POSTSUPERSCRIPT ) is equivalent to S(𝒟(ρAB)||𝒟(ρA⊗ρB))=S(ρAB||ρA⊗ρB)fragmentsSfragments(Dfragments(superscript𝜌𝐴𝐵)||Dfragments(superscript𝜌𝐴tensor-productsuperscript𝜌𝐵))Sfragments(superscript𝜌𝐴𝐵||superscript𝜌𝐴tensor-productsuperscript𝜌𝐵)S\left(\mathcalD(\rho^AB)\,|\kern-0.50003pt|\,\mathcalD(\rho^A% \otimes\rho^B)\right)=S\left(\rho^AB\,|\kern-0.50003pt|\,\rho^A% \otimes\rho^B\right)italic_S ( caligraphic_D ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) | | caligraphic_D ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT | | italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ). A theorem of Petz states that if a channel ℰℰ\mathcalEcaligraphic_E is such that S(ℰ(ρ)||ℰ(σ))=S(ρ||σ)fragmentsSfragments(Efragments(ρ)||Efragments(σ))Sfragments(ρ||σ)S\left(\mathcalE(\rho)\,|\kern-0.50003pt|\,\mathcalE(\sigma)\right)=S% \left(\rho\,|\kern-0.50003pt|\,\sigma\right)italic_S ( caligraphic_E ( italic_ρ ) | | caligraphic_E ( italic_σ ) ) = italic_S ( italic_ρ | | italic_σ ) then there exists ℰ^^ℰ\widehat\mathcalEover^ start_ARG caligraphic_E end_ARG such that ℰ^ℰ(ρ)=ρ^ℰℰ𝜌𝜌\widehat\mathcalE\mathcalE(\rho)=\rhoover^ start_ARG caligraphic_E end_ARG caligraphic_E ( italic_ρ ) = italic_ρ and moreover ℰ^(Y)=σ1/2ℰ*((ℰ(σ))-1/2Y(ℰ(σ))-1/2)σ1/2^ℰ𝑌superscript𝜎12superscriptℰsuperscriptℰ𝜎12𝑌superscriptℰ𝜎12superscript𝜎12\widehat\mathcalE(Y)=\sigma^1/2\mathcalE^*\left(\big(\mathcalE(% \sigma)\big)^-1/2Y\big(\mathcalE(\sigma)\big)^-1/2\right)\sigma^1% /2over^ start_ARG caligraphic_E end_ARG ( italic_Y ) = italic_σ start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT caligraphic_E start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( ( caligraphic_E ( italic_σ ) ) start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT italic_Y ( caligraphic_E ( italic_σ ) ) start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT ) italic_σ start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT where ℰ*superscriptℰ\mathcalE^*caligraphic_E start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT is the adjoint of ℰℰ\mathcalEcaligraphic_E. Letting ρAB=∑aa′|a⟩⟨a′|⊗ρaa′Bsuperscript𝜌𝐴𝐵subscript𝑎superscript𝑎′tensor-productket𝑎brasuperscript𝑎′subscriptsuperscript𝜌𝐵𝑎superscript𝑎′\rho^AB=\sum_aa^\prime|a\rangle\langle a^\prime|\otimes\rho^B_aa^% \primeitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | italic_a ⟩ ⟨ italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ⊗ italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, pa=tr[ρaaB]subscript𝑝𝑎trdelimited-[]subscriptsuperscript𝜌𝐵𝑎𝑎p_a=\mathrmtr\big[\rho^B_aa\big]italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = roman_tr [ italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a end_POSTSUBSCRIPT ], paρaB=ρaaBsubscript𝑝𝑎subscriptsuperscript𝜌𝐵𝑎subscriptsuperscript𝜌𝐵𝑎𝑎p_a\rho^B_a=\rho^B_aaitalic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a end_POSTSUBSCRIPT and σ=ρA⊗ρB𝜎tensor-productsuperscript𝜌𝐴superscript𝜌𝐵\sigma=\rho^A\otimes\rho^Bitalic_σ = italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT gives 𝒟(σ)=∑apa|a⟩⟨a|⊗ρB𝒟𝜎subscript𝑎tensor-productsubscript𝑝𝑎ket𝑎bra𝑎superscript𝜌𝐵\mathcalD(\sigma)=\sum_ap_a|a\rangle\langle a|\otimes\rho^Bcaligraphic_D ( italic_σ ) = ∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT | italic_a ⟩ ⟨ italic_a | ⊗ italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT and 𝒟(ρAB)=∑apa|a⟩⟨a|⊗ρaB𝒟superscript𝜌𝐴𝐵subscript𝑎tensor-productsubscript𝑝𝑎ket𝑎bra𝑎subscriptsuperscript𝜌𝐵𝑎\mathcalD(\rho^AB)=\sum_ap_a|a\rangle\langle a|\otimes\rho^B_acaligraphic_D ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT | italic_a ⟩ ⟨ italic_a | ⊗ italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT. It follows that (𝒟σ)-1/2=∑a∈Apa-1/2|a⟩⟨a|⊗ρB-1/2superscript𝒟𝜎12subscript𝑎𝐴tensor-productsuperscriptsubscript𝑝𝑎12ket𝑎bra𝑎superscriptsubscript𝜌𝐵12(\mathcalD\sigma)^-1/2=\sum_a\in Ap_a^-1/2|a\rangle\langle a|\otimes% \rho_B^-1/2( caligraphic_D italic_σ ) start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT | italic_a ⟩ ⟨ italic_a | ⊗ italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT and
ρAB=𝒟^(𝒟(ρAB))=∑a∈AρA1/2|a⟩⟨a|ρA1/2⊗ρaBsuperscript𝜌𝐴𝐵^𝒟𝒟superscript𝜌𝐴𝐵subscript𝑎𝐴tensor-productsuperscriptsubscript𝜌𝐴12ket𝑎bra𝑎superscriptsubscript𝜌𝐴12subscriptsuperscript𝜌𝐵𝑎\rho^AB=\widehat\mathcalD(\mathcalD(\rho^AB))=\sum_a\in A\rho_A^% 1/2|a\rangle\langle a|\rho_A^1/2\otimes\rho^B_aitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = over^ start_ARG caligraphic_D end_ARG ( caligraphic_D ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ) = ∑ start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT | italic_a ⟩ ⟨ italic_a | italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT (1)
and (|a⟩⟨a|⊗𝟏B)ρAB(|a⟩⟨a|⊗𝟏B)=∑a′∈A|⟨a|ρA1/2|a′⟩|2ρa′B=ρaaB=paρaBtensor-productket𝑎bra𝑎subscript1𝐵superscript𝜌𝐴𝐵tensor-productket𝑎bra𝑎subscript1𝐵subscriptsuperscript𝑎′𝐴superscriptquantum-operator-product𝑎superscriptsubscript𝜌𝐴12superscript𝑎′2subscriptsuperscript𝜌𝐵superscript𝑎′subscriptsuperscript𝜌𝐵𝑎𝑎subscript𝑝𝑎subscriptsuperscript𝜌𝐵𝑎(|a\rangle\langle a|\otimes\mathbf1_B)\rho^AB(|a\rangle\langle a|\otimes% \mathbf1_B)=\sum_a^\prime\in A\big\langle a|\rho_A^1/2|a^% \prime\rangle\big^2\rho^B_a^\prime=\rho^B_aa=p_a\rho^B_a( | italic_a ⟩ ⟨ italic_a | ⊗ bold_1 start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ( | italic_a ⟩ ⟨ italic_a | ⊗ bold_1 start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_A end_POSTSUBSCRIPT | ⟨ italic_a | italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT | italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a end_POSTSUBSCRIPT = italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT; if pa≠0subscript𝑝𝑎0p_a eq 0italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ≠ 0 then
ρaBsubscriptsuperscript𝜌𝐵𝑎\displaystyle\rho^B_aitalic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT =∑a′≠apa′ρa′Babsentsubscriptsuperscript𝑎′𝑎subscript𝑝superscript𝑎′subscriptsuperscript𝜌𝐵superscript𝑎′\displaystyle=\sum_a^\prime eq ap_a^\prime\rho^B_a^\prime= ∑ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≠ italic_a end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT pa′=|⟨a|ρA1/2|a′⟩|2pa-|⟨a|ρA1/2|a⟩|2subscript𝑝superscript𝑎′superscriptquantum-operator-product𝑎superscriptsubscript𝜌𝐴12superscript𝑎′2subscript𝑝𝑎superscriptquantum-operator-product𝑎superscriptsubscript𝜌𝐴12𝑎2\displaystyle p_a^\prime=\frac\rho_A^1/2a\rangle\big^2italic_p start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = divide start_ARG | ⟨ italic_a | italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT | italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT - | ⟨ italic_a | italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT | italic_a ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG (2)
so that each diagonal block is a convex combination of the others. Thus, for all the extremal states ρaBsubscriptsuperscript𝜌𝐵𝑎\rho^B_aitalic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT of the convex hull of the ρaBsubscriptsuperscript𝜌𝐵𝑎\rho^B_aitalic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT, |⟨a′|ρA1/2|a⟩|2=0superscriptquantum-operator-productsuperscript𝑎′superscriptsubscript𝜌𝐴12𝑎20\big\langle a^\prime|\rho_A^1/2|a\rangle\big^2=0| ⟨ italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT | italic_a ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 0 if ρa′≠ρasubscript𝜌superscript𝑎′subscript𝜌𝑎\rho_a^\prime eq\rho_aitalic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ≠ italic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT. If we consider the non extremal states, the extremal ones do not appear in their convex combination (2) and we may apply the same argument to their convex hull, and so on, eventually getting that ⟨a|ρ1/2|a′⟩=0quantum-operator-product𝑎superscript𝜌12superscript𝑎′0\langle a|\rho^1/2|a^\prime\rangle=0⟨ italic_a | italic_ρ start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT | italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⟩ = 0 if ρaB≠ρa′Bsubscriptsuperscript𝜌𝐵𝑎subscriptsuperscript𝜌𝐵superscript𝑎′\rho^B_a eq\rho^B_a^\primeitalic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ≠ italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT. Grouping together the a𝑎aitalic_a with equal ρaBsubscriptsuperscript𝜌𝐵𝑎\rho^B_aitalic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT gives a partition A1,…,Aksubscript𝐴1…subscript𝐴𝑘A_1,\ldots,A_kitalic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT of A𝐴Aitalic_A and Eq. (1) becomes
ρAB=∑i=1kPi⊗ρaisuperscript𝜌𝐴𝐵superscriptsubscript𝑖1𝑘tensor-productsubscript𝑃𝑖subscript𝜌subscript𝑎𝑖\displaystyle\rho^AB=\sum_i=1^kP_i\otimes\rho_a_iitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT Pi=ρA1/2(∑a∈Ai|a⟩⟨a|)ρA1/2subscript𝑃𝑖superscriptsubscript𝜌𝐴12subscript𝑎subscript𝐴𝑖ket𝑎bra𝑎superscriptsubscript𝜌𝐴12\displaystyle P_i=\rho_A^1/2\left(\sum_a\in A_i|a\rangle\langle a|% \right)\rho_A^1/2italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT ( ∑ start_POSTSUBSCRIPT italic_a ∈ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_a ⟩ ⟨ italic_a | ) italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT
with ⟨a′|Pi|a⟩=0quantum-operator-productsuperscript𝑎′subscript𝑃𝑖𝑎0\langle a^\prime|P_i|a\rangle=0⟨ italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_a ⟩ = 0 if either a′∈A\Aisuperscript𝑎′\𝐴subscript𝐴𝑖a^\prime\in A\backslash A_iitalic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_A \ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT or a∈A\Ai𝑎\𝐴subscript𝐴𝑖a\in A\backslash A_iitalic_a ∈ italic_A \ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. Discord-servers Letting ℋiA=Spana⟩:a∈Aifragmentssubscriptsuperscriptℋ𝐴𝑖Spanfragments\mathcalH^A_i=\mathrmSpan\big\caligraphic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = roman_Span italic_a ⟩ : italic_a ∈ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , that implies that Supp(Pi)⊆ℋiASuppsubscript𝑃𝑖subscriptsuperscriptℋ𝐴𝑖\mathrmSupp(P_i)\subseteq\mathcalH^A_iroman_Supp ( italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ⊆ caligraphic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT; the supports of the Pisubscript𝑃𝑖P_iitalic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT being pairwise orthogonal, the Pisubscript𝑃𝑖P_iitalic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT can be simultaneously diagonalized, letting ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT block diagonal. ∎
We thank Kavan Modi for useful comments.