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(4,1)-Quantum random access coding does not exist—one qubit is not enough to recover one of four bits

M Hayashi, K Iwama, H Nishimura, R Raymond, S Yamashita

2006
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New Journal of Physics
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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.
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... ows 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.

doi:10.1088/1367-2630/8/8/129
fatcat:6abfbsiqlzcrlj6pmooozf44wy