aa r X i v : . [ qu a n t - ph ] A ug Shortnote on local hidden Grassmann variables vs. quantum correlations
Lajos Di´osi ∗ Research Institute for Particle and Nuclear PhysicsH-1525 Budapest 114, POB 49, Hungary (Dated: December 2, 2018)Grassmannian local hidden variables are shown to generate all possible quantum correlations in abipartite quantum system. Grassmann representation of fermions, common in field theory, opens arelated perspective. Although Grassmann hidden variables can not challange Bell’s locality theorem,they can become an interesting mathematical tool to investigate entanglement.
Given two uncorrelated quantum systems A and B with density matrices ˆ ρ A and ˆ ρ B , respectively, the stateof the bipartite composite system is the tensor productˆ ρ A ˆ ρ B . Let the states of both A and B depend on a certainvariable λ so that the composite state were ˆ ρ B ( λ )ˆ ρ B ( λ )had we known the value of λ . However, we suppose that λ is hidden variable in the sense that we only know thestatistics of it. Therefore, the emerging composite stateis the statistical mean value of ˆ ρ B ( λ )ˆ ρ B ( λ ):ˆ ρ AB = M[ˆ ρ A ( λ )ˆ ρ A ( λ )] (1)defined through the normalized probability p of the hid-den variable: M[ . . . ] = Z . . . p ( λ )d λ . (2)After Werner [1], we consider eq. (1) the separability con-dition for the state ˆ ρ AB . If A and B were classical sys-tems their composite states would always be separable,i.e., each composite classical density is a weighted mix-ture of uncorrelated densities. We say classical correla-tions emerge from ignorance regarding some hidden vari-ables. This is not so in quantum theory. The separablequantum states which are mixtures of uncorrelated stateswill be called classically correlated quantum states. Thenon-separable quantum states ˆ ρ AB , for which the expan-sions (1) do not exist, are called quantum correlated or,equivalently, entangled. The existence of non-classicalcorrelations is a principal difference of quantum theoryfrom the classical one.The lack of separability (1) shows up for two Paulispins ˆ σ A and ˆ σ B already. The most general forms of thetwo spin states, respectively, read:ˆ ρ A ( a ) = 12 (ˆ1 A + aˆ σ A ) , ˆ ρ A ( b ) = 12 (ˆ1 A + bˆ σ B ) , (3)where a = ( a , a , a ) and b = ( b , b , b ) are real spatialpolarization vectors satisfying a , b ≤
1. Without re-stricting generality, the hidden variable is the pair of thepolarization vectors, λ = ( a , b ), with the probability dis-tribution p ( a , b ). The composite state ˆ ρ AB is separableif the probability distribution p ( a , b ) exists such thatˆ ρ AB = M[ˆ ρ A ( a )ˆ ρ B ( b )] . (4) Let us apply this condition to the rotational invariantspecial case where M[ a ] = M[ b ] = 0 and, in particular,M[ a i b j ] = ηδ ij (5)for i, j = 1 , ,
3. Most importantly, the correlation η isconstrained by | η | ≤ / a and b wereconstrained by a , b ≤
1. If we substitute (3) and (5)into the separability condition (4), we get the followingform: ˆ ρ AB = 14 (ˆ1 A ˆ1 B + η ˆ σ A ˆ σ B ) , (6)which is thus separable if | η | ≤ / ρ AB is non-negative for η ∈ [ − , /
3] hence the states are indeed non-separable(quantum correlated, entangled) for η ∈ [ − , − / a , b ≤ p ( a , b ) thatcould provide the (anti-)correlation stronger than η = − /
3. Quantum mechanics can achieve η = − η = − / p ( a , b ) λ is Grassmann variable. As for eq. (2),the theory of Grassmann variables contains the notion ofintegral [2] and of the corresponding measure p ( λ ).In particualar, we are going to discuss the case of thetwo correlated Pauli spins. Suppose a , b are Grassmannvariables: a i a j + a j a i = b i b j + b j b i = a i b j + b j a i = 0 , (7)for all i, j = 1 , ,
3. Let us choose the following normal-ized rotation invariant Gaussian distribution [2] of theGrasmannian hidden variables: p ( a , b )d b d a = η exp (cid:0) η − ab (cid:1) d b d a , (8)satisfying M[ a ] = M[ b ] = 0 and the correlation equation(5). Since η is not constrained at all, we conclude that theGrassmann hidden variables a , b can generate all possiblecorrelations that two Pauli spins may have in quantummechanics.It is plausible to conjecture that quantum correlationscan universally be reproduced by Grassmann hidden vari-ables. Namely, a multipartite composite state can beexpressed in the formˆ ρ AB...K... = M[ˆ ρ A ( λ )ˆ ρ B ( λ ) . . . ˆ ρ K ( λ ) . . . ] (9)where λ is the hidden variable. If the state is separablethen λ can be chosen real valued; if the state is entan-gled then λ must be a combination of real and Grass-mann numbers. Perhaps the real numbers generate theclassical while the Grassmann ones generate the quan-tum correlations, respectively, although the existence ofsuch separation is an open issue itself. The combinationof real number and Grassmann algebras might give somenew insight into the generic entanglement structure.I got the hint of Grassmann hidden variables fromquantum field theory where the fermionic quantum fieldsˆ ψ ( t, r ) can equivalently be represented by the correspond-ing local Grassmann fields ψ ( t, r ) that satisfy a certainnormalized distribution [2]. This means, at least for equi-librium states, that all fermion-mediated quantum cor-relations (entanglements) emerge from the local Grass-mann variables ψ ( t, r ) which play the role of hidden vari-ables.Another motivation comes from the recent work [4] byChristian, who suggested Clifford algebra valued hiddenvariables to violate Bell inequalities. In both his andmy proposal the non-commutative hidden variables are able to generate entanglement of the composite state.Grassmann might have some theoretical advantage overClifford numbers because of the mentioned universalGrassmann-fermion correspondence. Therefore I see acertain mathematical perspective to treat entanglementin the language of Grassmann hidden variables. Thepresent work has no intention to challange the Bell the-orem [3] since Grassmann (or Clifford) numbers can notparametrize individual measurement results [5], only realvalued hidden variables can, cf. [6]. Yet, the proposedhidden Grassmann variables λ are local in the spirit closeto Bell’s: they are by construction independent of the oc-casional local experimental settings at sides A or B .This work was supported by the Hungarian OTKAGrant No. 49384. ∗ Electronic address: [email protected];URL: [1] R. F. Werner, Phys. Rev. A , 4277 (1989).[2] F. A. Berezin: The Method of Second Quantization , (Aca-demic Press, 1966).[3] J. S. Bell, Physics 1, 195 (1964).[4] J. Christian, arXiv:quant-ph/0703244.[5] E.g., to the measurement of ˆ σ A the eq. (3) would assignthe value tr(ˆ σ A ˆ ρ A ) which is the Grassmannian a3