Linear entropy and Bell inequalities

Entropy inequalities in clasical and quantum informati have been studied since long ago @1#, but recently they are receiving increasing attention @2# in view of the application to quantum computers and quantum communication. In cent times, information-theoretic Bell-type inequalities @3,2# have been derived, and a relation between the entropy equality and the standard Bell-CHSH~Clauser-HorneShimony-Holt ! @4# inequality has been established @5#. In this paper we prove, using the so-called linear entropy, that mixed states of a pair of spin-1/2 particles, the positivity the sum of the conditional entropies is a sufficient condit for the Bell-CHSH inequality. The difference between classical and quantum inform tion may be formalized in terms of the entropy. If S12 is the entropy of the composite system and S1 ,S2 those of the subsystems, the classical statement that we cannot have m information about the whole than about each part may represented by the inequality

Entropy inequalities in clasical and quantum information have been studied since long ago ͓1͔, but recently they are receiving increasing attention ͓2͔ in view of the application to quantum computers and quantum communication.In recent times, information-theoretic Bell-type inequalities ͓3,2͔ have been derived, and a relation between the entropy inequality and the standard Bell-CHSH ͑Clauser-Horne-Shimony-Holt͒ ͓4͔ inequality has been established ͓5͔.In this paper we prove, using the so-called linear entropy, that for mixed states of a pair of spin-1/2 particles, the positivity of the sum of the conditional entropies is a sufficient condition for the Bell-CHSH inequality.
The difference between classical and quantum information may be formalized in terms of the entropy.If S 12 is the entropy of the composite system and S 1 ,S 2 those of the subsystems, the classical statement that we cannot have more information about the whole than about each part may be represented by the inequality S 2/1 ϩS 1/2 у0, S i/ j ϭS 12 ϪS j , jϭ1,2, ͑1͒ where S 2/1 and S 1/2 are called conditional entropies.Actually, the stronger condition holds that each conditional entropy must be non-negative, but here we shall use only Eq. ͑1͒.
The inequality is violated by quantum mechanics in some cases, for instance in the singlet spin state, where S 12 ϭ0, but both S 1 and S 2 are positive.In this paper we shall not use the standard von Neumann ͓6͔ entropy, but the more simple one defined by where is the 4ϫ4 density matrix representing the state of the two-particle system and 1 ( 2 ) the 2ϫ2 reduced density matrix of the first ͑second͒ particle.The classical counterpart of Eq. ͑2͒, called Tsallis entropy ͓7͔ of order 2, fulfills Eq. ͑1͒.
A relevant question is whether the violation of Eq. ͑1͒ is merely a theoretical feature derived from the definition of quantum entropy, or instead has empirical consequences.
Horodecki et al. ͓5͔ proved that if both S 2/1 у0 and S 1/2 у0, then no Bell inequality can be violated.Here we prove a stronger statement: Theorem.For any state ͑mixed in general͒ of a system consisting of two spin-1/2 particles, the entropy inequality ͑1͒ is a sufficient condition for the nonviolation of the Bell-CHSH ͓4͔ inequalities.
Local realism implies constraints on the statistics of two widely separated systems.These constraints are generically known as Bell inequalities and arise from the assumption of the existence of an underlying joint probability for a set of measurable quantities.The CHSH Bell-type inequality applies to a pair of two-state systems and constrains the value of a linear combination of four correlation functions between the two systems.As is well known, the CHSH inequality can be violated by the statistical predictions of quantum mechanics.
In order to prove the theorem we consider two dichotomic observables, that is, Hermitian and traceless operators, a 1 and b 1 ͑a 2 and b 2 ͒, for the first ͑second͒ particle fulfilling the conditions and define the ''Bell operator'' ͓8͔ by The density matrix fulfils the equalities Tr ϭ1, Tr͑B ͒ϭ␤, ͑5͒ ␤ being a real number in the interval ͓Ϫ2&,2&͔.The Bell-CHSH inequality is violated in the state if ͉␤͉Ͼ2.We shall prove ͑see below͒ that which shows that all possible Bell-CHSH inequalities are fulfilled if the entropy inequality ͑1͒ holds true, which is the statement of the theorem.
In order to prove Eq. ͑6͒, we first define a different density matrix: Ј also fulfills it with the same ␤, but each reduced density matrix of Ј is one-half the unit matrix.The relevant result is that S 1/2 ϩS 2/1 is the same for as it is for Ј, which allows one to write it in terms of the entropy S 12 Ј of Ј; that is,

͑8͒
The second Eq.͑8͒, may be derived easily from Eqs. ͑1͒ and ͑2͒, taking into account that j Јϭ 1 2 I j ( jϭ1,2).The proof of the first equality is as follows.From Eq. ͑7͒ we get After some algebra, this gives which easily leads to the first Eq.͑8͒.Now we search for the density matrix Љ, which makes S 12 Љ a maximum constrained by Eqs.͑5͒.This puts forth a straightforward variational problem, and the solution is In fact, it is easy to prove that any density matrix fulfilling Eqs.͑5͒ may be written in the form Љϩ␦ with Љ given by Eq. ͑11͒ and ␦ Hermitian, traceless, and such that Tr(␦B)ϭ0.The entropy associated with such a density matrix fulfills

͑12͒
the inequality coming from the property that indicates that the trace of the square of a Hermitian matrix is non-negative.
The entropy S 12 Љ may be calculated by inserting Eq. ͑11͒ in to Eq. ͑2͒.We get where we have taken into account the relations Tr Iϭ4, Tr Bϭ0, Tr B 2 ϭTr͑4Iϩ͓a 1 ,b 1 ͔͓a 2 ,b 2 ͔ ͒ϭ16.͑14͒ Equation ͑6͒ follows immediately from Eqs. ͑8͒ and ͑13͒, if we take into account that S 12 Љ , being a maximum, is not smaller than S 12 Ј .This completes the proof.
Our result include previous ones obtained within the information-theoretic approach, in particular the one obtained in Ref. ͓5͔, where the authors showed that from S͑a,b ͒уmax͕S͑ a ͒,S͑ b ͖͒, ͑15͒ the Bell-CHSH inequality can be derived, while to obtain the same result we have started from S͑a,b ͒у 1 2 ͓S͑ a ͒ϩS͑ b ͔͒, ͑16͒ It is easy to see that Eq. ͑15͒ implies Eq. ͑16͒, so our proof is less restrictive than theirs.
Some other interesting questions still open are whether a similar result could be obtained by using the standard von Neumann and Shannon entropies, instead of Eqs.͑2͒, and if it would be possible to generalize these results to systems with more than two degrees of freedom.A particular example investigated in this respect ͓9͔ suggests that this would be the case.