Separability Criterion for Density Matrices
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
A. M. Khorunzhy, B. A. Khoruzhenko, L. A. Pastur
Journal of Mathematical Physics, Vol. 37, (1996), pp. 5033-5060
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
Gravity coupled with matter and foundations of non-commutative
Communications in Mathematical Physics, Vol. 182, (1996), pp. 155-176
Connes argues that modern QFT is closely related to non-commutative
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
Connes uses a Hochshield cycle as a substitute for volume element and in-
troduces several axioms for the non-commutative geometry.