We study separability properties in a 5-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension. These are the states, which can be written as linear combinations of permutation operators, or, equivalently, commute with unitaries of the form U ⊗ U ⊗ U . We compute explicitly the following subsets and their extreme points: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial transpose with respect to such a partition. Tripartite entanglement is investigated in terms of the relative entropy of tripartite entanglement and of the trace norm.03.65. Bz, 03.65.Ca, 89.70.+c

We present a general technique for hiding a classical bit in multi-partite quantum states. The hidden bit, encoded in the choice of one of two possible density operators, cannot be recovered by local operations and classical communication without quantum communication. The scheme remains secure if quantum communication is allowed between certain partners, and can be designed for any choice of quantum communication patterns to be secure, but to allow near perfect recovery for all other patterns. The maximal probability of unwanted recovery of the hidden bit, as well as the maximal error for allowed recovery operations can be chosen to be arbitrarily small, given sufficiently high dimensional systems at each site. No entanglement is needed since the hiding states can be chosen to be separable. A single ebit of prior entanglement is not sufficient to break the scheme.

We prove a conjecture by DiVincenzo, which in the terminology of Preskill et al. [quant-ph/0102043] states that "semicausal operations are semilocalizable". That is, we show that any operation on the combined system of Alice and Bob, which does not allow Bob to send messages to Alice, can be represented as an operation by Alice, transmitting a quantum particle to Bob, and a local operation by Bob. The proof is based on the uniqueness of the Stinespring representation for a completely positive map. We sketch some of the problems in transferring these concepts to the context of relativistic quantum field theory.

We show that all quantum states that do not have a positive partial transpose are distillable via channels, which preserve the positivity of the partial transpose. The question whether bound entangled states with non-positive partial transpose exist is therefore closely related to the connection between the set of separable superoperators and positive partial transpose-preserving maps.

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