ar X iv : q ua nt - p h / 06 04 06 1 v 1 10 A pr 2 00 6 ( 4 , 1 ) - Quantum Random Access Coding Does Not Exist

An (n, 1, p)-Quantum Random Access (QRA) coding, introduced by Ambainis, Nayak, Ta-shma and Vazirani in ACM Symp. on Theory of Computing 1999, is the following communication system: The sender which has n-bit information encodes his/her information into one qubit, which is sent to the receiver. The receiver can recover any one bit of the original n bits correctly with probability at least p, through a certain decoding process based on positive operator-valued measures. Actually, Ambainis et al. shows the existence of a (2, 1, 0.85)-QRA coding and also proves the impossibility of its classical counterpart. Chuang immediately extends it to a (3, 1, 0.79)-QRA coding and whether or not a (4, 1, p)-QRA coding such that p > 1/2 exists has been open since then. This paper gives a negative answer to this open question.