- ρ≥0 whileρptis not a positive matrix.
- Bothρ,ρpt≥0 butρis not separable.
In the case thatρis a pure state or^14 N 1 N 2 ≤6, only the first type exists [16].
The non-positivity ofρpt, uncovers entangled state of the first type. This is
known as Peres test^15 [27]. These states can be distilled to Bell pairs by local oper-
ations [28], whereas entangled states of the second type, aka “bound entangled”,
can not [28, 29].
Figure 13: The green triangle represent a high dimensional simplex of states and
the blue triangle and its partial transposition. The green triangles that stick
out describe entangled states that are discoverable by partial transposition. The
intersection may or may not contain bound entangled states. It is separable iff its
vertices are separable.
7.5 The largest ball of bi-partite separable states
The Gurvits-Barnum ball was introduced in section 3.4 as the largest inscribed
ball inDN:
Bgb={
x∣
∣
∣|x|≤r^20}
⊆DN (7.12)
Since partial transposition is a reflection in theσα coordinates, and any sphere
centered at the origin is invariant under reflection, we have that
Bgb⊆DN∩DptN (7.13)Bgbtherefore does not contain entangled states that are discoverable by the Peres
test.
Gurvits and Barnum replace partial transposition by contracting positive maps
of the form 1 ⊗φ(D) to show [6], thatBgbis a ball of bi-partite separable states
Bgb⊆SN 1 ,N 2 (7.14)(^14) Simple geometric proofs for two qubits are given in [3, 4].
(^15) Gurvits and Barnum attribute the test to an older 1976 paper of Woronowitz. Apparently,
nothing is ever discovered for the first time (M. Berry’s law).