Antiparallel spin does not always contain more information
Sibasish Ghosh, Anirban Roy, Ujjwal Sen · Physical Review A · 2000
We show that the Bloch vectors lying on any great circle comprise the largest set ${S}_{L}$ for which the parallel states $|\stackrel{\ensuremath{\rightarrow}}{n},\stackrel{\ensuremath{\rightarrow}}{n}〉$ can always be exactly transformed into the antiparallel states $|\stackrel{\ensuremath{\rightarrow}}{n},\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{n}〉.$ Thus more information about $\stackrel{\ensuremath{\rightarrow}}{n}$ is not extractable from $|\stackrel{\ensuremath{\rightarrow}}{n},\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{n}〉$ than from $|\stackrel{\ensuremath{\rightarrow}}{n},\stackrel{\ensuremath{\rightarrow}}{n}〉$ by any measuring strategy, for $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\in}{S}_{L}.$ Surprisingly this most general transformation reduces to just a flip operation on the second particle. We also show here that a probabilistic exact parallel to antiparallel transformation is not possible if the corresponding antiparallel states span the whole Hilbert space of the two qubits. These considerations allow us to generalize a conjecture of Gisin and Popescu [Phys. Rev. Lett. $83,$ 432 (1999)].