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					   1996
   Separability Criterion for Density Matrices
   Asher Peres
   Physical Review Letters, Vol. 77, (1996), pp. 1413-1415

    A density matrix in B (H1 H2 ) is called separable if it can be represented
as a convex combination of tensor products of density matrices in B (H1 ) and
B (H2 ) : Some density matrices are non-separable. This paper proposed a simple
necessary condition which allows to check if a density matrix is separable. The
condition is that all eigenvalues of the partial transpose of a separable must be
positive. As was shown by Horodecki, this condition is su¢ cient if dim H1 = 2
and dim H2 is 2 or 3: Horodecki also showed that a su¢ cient condition is that all
eigenvalues remain positive under transformation I        ; where is any positive
operator. Unfortunately, characterization of positive operators is also di¢ cult.
Since this paper a couple of other criterions appeared due mainly to Horodecki.
However, it seems that at the time of writing there are no convenient su¢ cient
and necessary conditions.

   Asymptotic properties of large random matrices with independent
entries
   A. M. Khorunzhy, B. A. Khoruzhenko, L. A. Pastur
   Journal of Mathematical Physics, Vol. 37, (1996), pp. 5033-5060

                                                                      1
    This paper studies the behavior of gn (z) = n 1 tr (H zI) for n n matri-
ces H; as n ! 1: They …nd asymptotic formulas for Egn (z) and E [gn (z1 ) gn (z2 )]
Egn (z1 ) Egn (z2 ) provided that the imaginary part of z is su¢ ciently large.
However, this restriction does not allows us to understand how the eigenvalues
behave.

   Gravity coupled with matter and foundations of non-commutative
geometry
   Alain Connes
   Communications in Mathematical Physics, Vol. 182, (1996), pp. 155-176
   Connes argues that modern QFT is closely related to non-commutative
geometry.
   In this geometry, we work not with manifolds but with algebras. The length
element ds is an operator (say, the inverse of the Dirac operator). In the com-
mutative case of functions on manifold, the distance between points x and y can
be recovered by the following formula:

        d (x; y) = sup jf (x)   f (y)j ; f 2 C 1 (M ) ;   f; ds   1
                                                                          1 :

   Connes uses a Hochshield cycle as a substitute for volume element and in-
troduces several axioms for the non-commutative geometry.




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