 
Summary: Tensor products of convex sets and the volume of separable states
on N qudits
Guillaume Aubrun and Stanislaw J. Szarek
Abstract
This note deals with estimating the volume of the set of separable mixed quantum states when
the dimension of the state space grows to infinity. This has been studied recently for qubits; here we
consider larger particles and conclude that, in all cases, the proportion of the states that are separable
is superexponentially small in the dimension of the set. We also show that the partial transpose
criterion becomes imprecise when the dimension increases, and that the lower bound 6N/2
on the
(HilbertSchmidt) inradius of the set of separable states on N qubits obtained recently by Gurvits
and Barnum is essentially optimal. We employ standard tools of classical convexity, highdimensional
probability and geometry of Banach spaces. One relatively nonstandard point is a formal introduc
tion of the concept of projective tensor products of convex bodies, and an initial study of this concept.
PACS numbers: 03.65.Ud, 03.67.Mn, 03.65.Db, 02.40.Ft
1 Introduction and summary of results
An important problem in quantum information theory is to estimate quantitatively parameters related to
entanglement. This phenomenon is thought to be at the heart of quantum information processing while,
on the other hand, its experimental creation and handling are still challenging. The set of separable (i.e.,
unentangled) states has been defined in [1] under the name "classically correlated," as opposed to "EPR
