The geometry of multi-qubit entanglement
Toshihiro Iwai · Journal of Physics A Mathematical and Theoretical · 2007
This paper, a continuation of a previous one ( Iwai 2007 J. Phys. A: Math. Theor. 40 1361 ), studies the geometry of multi-qubit entanglement with respect to bipartite partitions. An n -qubit system ( C 2 ) ⊗ n is isomorphic with , the linear space of 2 ℓ × 2 m complex matrices, where ℓ + m = n . According to the isomorphism, the local transformation group U (2 ℓ ) × U (2 m ) acts on the space of normalized states in . Let M and G denote the space of normalized states and the local transformation group acting on M , respectively. According to the orbit types, M is stratified into strata, among which a principal stratum and the sets of separable states and maximally entangled states will be identified. For , a function F ( C ) = det( I − CC *) proves to serve as a measure of entanglement, where I denotes the 2 ℓ × 2 ℓ identity matrix. The F ( C ) attains the minimal and the maximal values, respectively, on the sets of separable states and maximally entangled states, and further takes no extremal values on the principal stratum. The F ( C ) projects to a function on the factor space G \ M . A naturally defined metric on M also projects to that on G \ M , which serves to measure the distance between the separable states, F −1 (0), and the states, F −1 ( k ), of prescribed value k of measure. To be precise, one has to restrict M to the principal stratum, when projecting the metric. Three- and four-qubit systems will be studied intensively, and then multi-qubit systems discussed.