Better bound on the exponent of the radius of the multipartite separable ball
Abstract
We show that for an m -qubit quantum system, there is a ball of radius asymptotically approaching κ 2-γm in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices, for an exponent γ=0.5 (ln 3/ln 2-1) ≈0.292 481 25 much smaller in magnitude than the best previously known exponent, from our earlier work, of 1/2 . For normalized m -qubit states, we get a separable ball of radius 3m+1 / √(( 3m +3) )× 2- (1+γ) m ≡ 3m+1 / √(( 3m +3) )× 6-m/2 (note that κ=√(3) ), compared to the previous 2× 2-3m/2 . This implies that with parameters realistic for current experiments, nuclear magnetic resonance (NMR) with standard pseudopure-state preparation techniques can access only unentangled states if 36 qubits or fewer are used (compared to 23 qubits via our earlier results). We also obtain an improved exponent for m -partite systems of fixed local dimension d0 , although approaching our earlier exponent as d0 →∞ .
- Publication:
-
Physical Review A
- Pub Date:
- September 2005
- DOI:
- 10.1103/PhysRevA.72.032322
- arXiv:
- arXiv:quant-ph/0409095
- Bibcode:
- 2005PhRvA..72c2322G
- Keywords:
-
- 03.67.Mn;
- 03.65.Ud;
- Entanglement production characterization and manipulation;
- Entanglement and quantum nonlocality;
- Quantum Physics
- E-Print:
- 30 pp doublespaced, latex/revtex, v2 added discussion of Szarek's upper bound, and reference to work of Vidal, v3 fixed some errors (no effect on results), v4 involves major changes leading to an improved constant, same exponent, and adds references to and discussion of Szarek's work showing that exponent is essentially optimal for qubit case, and Hildebrand's alternative derivation for qubit case. To appear in PRA