Conditional mutual information and self-commutator Lin Zhang ∗ 3 Institute of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, PRChina 1 0 2 n a J 2 Abstract 2 ] Asimplerapproachtothecharacterizationofvanishingconditionalmutualinfor- h p mation is presented. Some remarks are given as well. More specifically, relating the - t n conditional mutual information to a commutator is a very promising approach to- a u wardstheapproximate versionofSSA.Thatis, it isconjectured thatsmall conditional q mutual information implies small perturbation of quantum Markov chain. [ 2 v 3 2 0 5 . 2 1 Introduction 1 2 1 : To begin with, we fix some notations that will be used in this context. Let be a finite v H i X dimensional complex Hilbert space. A quantum state ρ on is a positive semi-definite H r a operator of trace one, in particular, for each unit vector ψ , the operator ρ = ψ ψ | i ∈ H | ih | is said to be a pure state. The set of all quantum states on is denoted by D( ). For H H each quantum state ρ D( ), its von Neumann entropy is defined by ∈ H S(ρ) =def Tr(ρlogρ). − The relative entropy of two mixed states ρ and σ is defined by Tr(ρ(logρ logσ)), if supp(ρ) supp(σ), S(ρ σ) =def − ⊆ || ( +∞, otherwise. E-mail: [email protected] ∗ 1 A quantum channel Φ on is a trace-preserving completely positive linear map defined H over the set D( ). It follows that there exists linear operators K on such that µ µ H { } H ∑ K†K = 1 and Φ = ∑ Ad , that is, for each quantum state ρ, we have the Kraus µ µ µ µ Kµ representation Φ(ρ) = ∑K ρK†. µ µ µ The celebrated strong subadditivity (SSA) inequality of quantum entropy, proved by Lie and Ruskai in [1], S(ρ )+S(ρ ) 6 S(ρ )+S(ρ ), (1.1) ABC B AB BC is a very powerful tool in quantum information theory. Recently, the operator extension of SSA is obtained by Kim in [2]. Following the line of Kim, Ruskai gives a family of new operator inequalities in [3]. Conditional mutual information, measuringthe correlationsof twoquantum systems relative to a third, is defined as follows: Given a tripartite state ρ , it is defined as ABC I(A : C B) =def S(ρ )+S(ρ ) S(ρ ) S(ρ ). (1.2) ρ AB BC ABC B | − − Clearly conditional mutual information is nonnegative by SSA. Ruskai is the first one to discuss the equality condition of SSA. By analyzing the equality condition of Golden-Thompson inequality, she obtained the following charac- terization [4]: I(A : C B) = 0 logρ +logρ = logρ +logρ . (1.3) ρ ABC B AB BC | ⇐⇒ Lateron,usingtherelativemodularapproachestablishedbyAraki,Petzgaveanother characterization of the equality condition of SSA [5]: I(A : C B) = 0 ρit ρ it = ρit ρ it ( t R), (1.4) | ρ ⇐⇒ ABC −BC AB −B ∀ ∈ where i = √ 1 is the imaginary unit. − Haydenetal. in[6]showedthat I(A : C B) = 0ifandonlyifthefollowingconditions ρ | hold: (i) = , HB kHbkL ⊗HbkR L 2 (ii) ρ = p ρ ρ , where ρ D ,ρ D for ABC k k AbkL ⊗ bkRC AbkL ∈ HA ⊗HbkL bkRC ∈ HbkR ⊗HC each indLex k; and pk is a probability distr(cid:16)ibution. (cid:17) (cid:16) (cid:17) { } In [7], Brandão et al. first obtained the following lower bound for I(A : C B) : ρ | 1 I(A : C B) > min ρ σ 2 , (1.5) | ρ 8ln2 σAC∈SEPk AC − ACk1−LOCC where ρ σ 2 =def sup (ρ ) (σ ) . k AC − ACk1 LOCC kM AC −M AC k1 − 1 LOCC M∈ − Based on this result, he cracked a long-standing open problem in quantum information theory. That is, the squashed entanglement is faithful. Later, Li in [8] gave another approach to study the same problem and improved the lower bound for I(A : C B) : ρ | 1 I(A : C B) > min ρ σ 2 . (1.6) | ρ 2ln2 σAC∈SEPk AC − ACk1−LOCC Along with the above line, Ibinson et al. in [9] studied the robustness of quantum Markov chains, i.e. the perturbation of states of vanishing conditional mutual informa- tion. In order to study it further, We need to employ the following famous characteriza- tion of saturation of monotonicity inequality of relative entropy. Theorem 1.1 (Petz, [10, 11]). Let ρ,σ D( ), Φ be a quantum channel defined over . If ∈ H H supp(ρ) supp(σ), then ⊆ S(ρ σ) = S(Φ(ρ) Φ(σ)) if and only if Φ† Φ(ρ) = ρ, (1.7) σ || || ◦ where Φ† = Ad Φ† Ad . σ σ1/2 ◦ ◦ Φ(σ)−1/2 Noting that the equivalence between monotonicity of relative entropy and SSA, the above theorem, in fact, gives another characterization of vanishing conditional mutual information of quantum states. 2 Main result Inthissection, wegiveanothercharacterizationofsaturationofSSAfromtheperspective commutativity. 3 Theorem 2.1. Let ρ D( ). Denote ABC A B C ∈ H ⊗H ⊗H M =def (ρ1/2 1 )(1 ρ 1/2 1 )(1 ρ1/2) AB ⊗ C A ⊗ −B ⊗ C A ⊗ BC ρ1/2ρ 1/2ρ1/2. − ≡ AB B BC Then the following conditions are equivalent: (i) The conditional mutual information is vanished, i.e. I(A : C B) = 0; ρ | (ii) ρ = MM† = ρ1/2ρ 1/2ρ ρ 1/2ρ1/2; ABC AB −B BC −B AB (iii) ρ = M†M = ρ1/2ρ 1/2ρ ρ 1/2ρ1/2; ABC BC −B AB −B BC Proof. Clearly, the conditional mutual information is vanished, i.e. I(A : C B) = 0, if ρ | and only if S(ρ )+S(ρ ) = S(ρ )+S(ρ ). (2.1) ABC B AB BC Hence we have that S(ρ ρ ρ ) = S(ρ ρ ρ ), (2.2) AB A B ABC A BC || ⊗ || ⊗ S(ρ ρ ρ ) = S(ρ ρ ρ ). (2.3) BC B C ABC AB C || ⊗ || ⊗ Now let Φ = Tr and Ψ = Tr , it follows that C A S(ρ ρ ρ ) = S(Φ(ρ ) Φ(ρ ρ )), (2.4) ABC A BC ABC A BC || ⊗ || ⊗ S(ρ ρ ρ ) = S(Ψ(ρ ) Ψ(ρ ρ )). (2.5) ABC AB C ABC AB C || ⊗ || ⊗ By Theorem 1.1, we see that both Eq. (2.4) and Eq. (2.5) hold if and only if ρ = Φ† Φ(ρ ) and ρ = Ψ† Ψ(ρ ), (2.6) ABC ρ ρ ABC ABC ρ ρ ABC A⊗ BC ◦ AB⊗ C ◦ i.e. ρ = ρ1/2ρ 1/2ρ ρ 1/2ρ1/2 = ρ1/2ρ 1/2ρ ρ 1/2ρ1/2. (2.7) ABC AB −B BC −B AB BC −B AB −B BC This amounts to say that I(A : C B) = 0 if and only if ρ = MM† = M†M. ρ ABC | 4 Remark 2.2. In [12], Leifer and Poulin gave a condition which is equivalent to our re- sult. There they mainly focus on the characterization of conditional independence in terms of noncommutative probabilistic language by analogy with classical conditional independence. By combining the Lie-Trotter product formula: n exp(A+B) = lim [exp(A/n)exp(B/n)] (2.8) n ∞ → n = lim [exp(A/2n)exp(B/n)exp(A/2n)] , (2.9) n ∞ → where both A and B are square matrices of the same order, a characterization of van- ishing conditional mutual information was obtained. Clearly, the Lie-Trotter product formula is not easy to deal with. In fact, our proof is more naturally and much simpler than that of theirs. In the following, we denote by X,X† the self-commutator of an operator or a matrix X. (cid:2) (cid:3) Corollary 2.3. With the notation mentioned above in Theorem 2.1, the following statement is true: I(A : C B) = 0 implies M,M† = 0. In other words, M,M† = 0 implies ρ | 6 I(A : C B) = 0. ρ (cid:2) (cid:3) (cid:2) (cid:3) | 6 Proof. We assume that I(A : C B) = 0. From Theorem 2.1, we know that ρ = ρ ABC | MM† = M†M, implying M,M† = 0. (cid:2) (cid:3) 3 Discussion A natural question arises: Can we derive I(A : C B) = 0 from M,M† = 0? The ρ | answer is no! Indeed, we know from the discussion in [13] that, if the operators ρ ,ρ (cid:2) (cid:3) AB BC and ρ all commute, then B I(A : C B) = S(ρ MM†). (3.1) ρ ABC | || Now let ρ = ∑ p ijk ijk with p being an arbitrary joint probability distri- ABC i,j,k ijk ijk | ih | { } bution. Thus MM† = M†M = ∑ pijpjk ijk ijk , pj | ih | i,j,k 5 where p = ∑ p ,p = ∑ p and p = ∑ p are corresponding marginal distribu- ij k ijk jk i ijk j i,k ijk p p tions, respectively. But in general, p = ij jk. Therefore we have a specific example in ijk 6 pj which [M,M†] = 0, and ρ = MM†, i.e. I(A : C B) > 0. By employing the Pinsker’s ABC ρ 6 | inequality to Eq. (3.1), it follows in this special case that 1 2 I(A : C B) > ρ MM† . ρ ABC | 2ln2 − 1 (cid:13) (cid:13) (cid:13) (cid:13) Along with the above line, all tripartite sta(cid:13)tes can be clas(cid:13)sified into three categories: D( ) = D D D , A B C 1 2 3 H ⊗H ⊗H ∪ ∪ where (i) D =def ρ : ρ = MM†,[M,M†] = 0 . 1 ABC ABC (cid:8) (cid:9) (ii) D =def ρ : ρ = MM†,[M,M†] = 0 . 2 ABC ABC 6 (cid:8) (cid:9) (iii) D =def ρ : [M,M†] = 0 . 3 ABC 6 (cid:8) (cid:9) For related topics please refer to [14, 15]. The result obtained in Corollary 2.3 can be employed to discuss a small conditional mutual information. I. Kim [16] tries to give a universal proof of the following inequality: 1 2 I(A : C B) > ρ MM† . ρ ABC | 2ln2 − 1 (cid:13) (cid:13) (cid:13) (cid:13) As a matter of fact, if the above inequality holds, then a similar inequality holds: (cid:13) (cid:13) 1 2 I(A : C B) > ρ M†M . ρ ABC | 2ln2 − 1 (cid:13) (cid:13) (cid:13) (cid:13) The validity or non-validity of both inequalities can be guaranteed by Theorem 2.1. (cid:13) (cid:13) According to the numerical computation byKim, up to now, there are nostates violating these inequalities. Therefore we have the following conjecture: 1 2 2 I(A : C B) > max ρ MM† , ρ M†M . (3.2) ρ ABC ABC | 2ln2 − 1 − 1 (cid:26) (cid:27) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) We can connect the total amount of(cid:13)conditional m(cid:13)utu(cid:13)al information(cid:13)contained in the tripartite state ρ with the trace-norm of the commutator M,M† as follows: if the ABC above conjecture holds, then we have (cid:2) (cid:3) 1 2 I(A : C B) > M,M† , (3.3) ρ | 8ln2 1 (cid:13)h i(cid:13) (cid:13) (cid:13) 6 (cid:13) (cid:13) but not vice versa. Even though the above conjecture is false, it is still possible that this inequality is true. In [13], the authors proposed the following question: For a given quantum channel Φ T( , ) and states ρ,σ D( ), does there exist a quantum channel Ψ A B A ∈ H H ∈ H ∈ T( , ) with Ψ Φ(σ) = σ and B A H H ◦ S(ρ σ) > S(Φ(ρ) Φ(σ)) +S(ρ Ψ Φ(ρ))? (3.4) || || || ◦ The authors affirmatively answer this question in the classical case. The quantum case is still open. Although the authors proved that the following inequality is not valid in general: S(ρ σ) (cid:11) S(Φ(ρ) Φ(σ)) +S(ρ Φ† Φ(ρ)) || || || σ ◦ However, the following inequality may still be correct: 1 2 S(ρ σ) > S(Φ(ρ) Φ(σ)) + ρ Φ† Φ(ρ) . σ || || 2ln2 − ◦ 1 (cid:13) (cid:13) (cid:13) (cid:13) In fact, if this modified inequality holds, then Eq. (3(cid:13).2) will hold. (cid:13) A future research may be directed to establish a connection between conditional mu- tual information and almost commuting normal matrices [17]. Acknowledgement This project is supported by the Research Program of Hangzhou Dianzi University (KYS075612038). LZ acknowledges Matthew Leifer, David Poulin and Mark M. Wilde for drawing my attention to the References [12, 14]. The author also would like to thank Isaac H. Kim for valuable discussions. Minghua Lin’s remarks are useful as well. Especially, thank F. Brandão for drawing my attention to the problem concerning the approximate version of vanishing conditional mutual information. References [1] E. Lieb and M. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys. 14, 1938-1941 (1973). 7 [2] I. Kim, Operator extension of strong subadditivity of entropy, J. Math. Phys. 53, 122204 (2012). [3] M. Ruskai, Remarks on on Kim’s strong subadditivity matrix inequality: extensions and equality conditions, arXiv:1211.0049 [4] M. Ruskai, Inequalitiesfor quantum entropy: Areviewwith conditions for equality, J. Math. Phys. 43, 4358-4375 (2002); erratum 46, 019901 (2005). [5] D. Petz, Monotonicity of quantum relative entropy revisited, Rev. Math. Phys. 15, 79-91 (2003). [6] P. Hayden, R. Jozsa, D. Petz, A. Winter, Structure of states which satisfy strong subadditivity of quantum entropy with equality, Commun. Math. Phys. 246, 359-374 (2004). [7] F. Brandão, M. Christandl, J. Yard, Faithful squashed entanglement, Commun. Math. Phys. 306, 805-830 (2011). Erratum to: Faithful Squashed Entanglement, 316, 287-288 (2012). arXiv:1010.1750v5 [8] K. Li and A. Winter, Relative entropy and squahsed entanglement, arXiv:1210.3181 [9] B. Ibinson, N. Linden, A. Winter, Robustness of Quantum Markov Chains, Com- mun. Math. Phys. 277, 289-304 (2008). [10] D. Petz, Sufficiency of channels over von Neumann algebras, Quart. J. Math. 39 (1), 97-108 (1988). [11] F. Hiai, M. Mosonyi, D. Petz, C. Bény, Quantum f-divergences and error correction, Rev. Math. Phys. 23(7), 691-747 (2011). [12] M. Leifer,D. Poulin, Quantum graphical modelsandbeliefpropagation, Ann.Phys. 323(8), 1899-1946 (2008). [13] A. Winter and K. Li, A stronger subadditivity relation? With ap- plications to squashed entanglement, sharability and separability. See http://www.maths.bris.ac.uk/~csajw/stronger_subadditivity.pdf [14] D. Poulin, M. Hastings, Markov entropy decomposition: A variational dual for quantum belief propagation, Phys. Rev. Lett. 106, 080403 (2011). 8 [15] W. Brown, D. Poulin, Quantum markov networks and commuting Hamiltonians, arXiv:1206.0755 [16] I. Kim, In private communications. [17] P. Friis, M. Rördam, Almost commuting self-adjoint matrices—a short proof of Huaxin Lin’s theorem, J. reine angew. Math. 479, 121-131 (1996). 9