arXiv:quant-ph/0004071v1 18 Apr 2000 Does Anti-Parallel Spin contain more Information ? Sibasish Ghosh a * , Anirban Roy a † , Ujjwal Sen b ‡ May 20, 2019 a Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Calcutta -700035, India b Dept. of Physics, Bose Institute, 93/1 A.P.C. Road, Calcutta - 700009, India Abstract We show that the Bloch vectors lying on any great circle is the largest set SL for which the parallel states | − → n, − → n 〉 can always be exactly transformed into the anti-parallel states | − → n, − − → n 〉. Thus more information about − → n is not extractable from | − → n, − − → n 〉 than from | − → n, − → n 〉 by any measuring strategy, for − → n ∈ SL. Surprisingly, the largest set of Bloch vectors for which the corresponding qubits can be flipped is again SL. We then show that probabilistic exact parallel to anti-parallel transformation is not possible if the corresponding anti-parallels span the whole Hilbert space of the two qubits. These considerations allow us to generalise a conjecture of Gisin and Popescu (Phys. Rev. Lett. 83 432 (1999)). Recently, Gisin and Popescu [1] revealed that for qubits, there exists a measuring strategy (see [2], [3]) on the anti-parallel state | − → n, − − → n 〉 that extracts more information about an arbitrary Bloch vector − → n than that can be extracted from the parallel state | − → n, − → n 〉 [4]. In this paper, we ask whether there exists any proper subset S of the unit ball B 3 = { − → n : − → n ∈ R 3 , | − → n | =1} for which an anti-parallel state selected at random from the set A S = {| − → n, − − → n 〉 : − → n ∈ S} carry more information about the Bloch vector − → n than a parallel state from the set P S = {| − → n, − → n 〉 : − → n ∈ S}. We provide a partial answer to this query. Specifically, we find the largest set S L for which there exists a unitary operator U = U (S L ) such that U | − → n, − → n 〉|M 〉 = e iθ( − → n ) | − → n, − − → n 〉|N 〉 (1) for all − → n ∈ S L where |M 〉 and |N 〉 are the states of a possible ancilla. |N 〉 must be independent of − → n except possibly in a phase (which has been taken care of in (1)) to satisfy the unitarity of U . Consequently an anti-parallel state chosen at random from A SL cannot contain more information about − → n than in the corresponding parallel state from P SL . A related query is how far we can go by just flipping the second state in order to transform | − → n, − → n 〉 to | − → n, − − → n 〉. Here one may think that the largest set of Bloch vectors would in this case be a very small subset of S L . Surprisingly, as we would show here, this conjecture is not true: the largest set is again S L . We shall show here that universal exact machines for the transformations | − → n, − → n 〉 to | − → n, − − → n 〉 and | − → n, − − → n 〉 to | − → n, − → n 〉 do not exist. Therefore, in the same vein as one considered deterministic inexact [5] and probabilistic exact cloning [6,7] when faced with the no-cloning theorem [8], one * res9603@isical.ac.in † res9708@isical.ac.in ‡ dhom@bosemain.boseinst.ernet.in 1