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arXiv:quant-ph/0604061 · New Journal of Physics

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

Masahito Hayashi, Kazuo Iwama, Harumichi Nishimura, Rudy Raymond, Shigeru Yamashita

Published Aug 4, 2006• 66 citations
arXivHTMLPDF

Abstract

An ( n ,1, p )-quantum random access (QRA) coding, introduced by Ambainis et al (1999 ACM Symp. Theory of Computing p 376), 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. Moreover, we generalize its negative answer for one-qubit encoding to the case of multiple-qubit encoding

DOI
10.1088/1367-2630/8/8/129

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An ( n ,1, p )-quantum random access (QRA) coding, introduced by Ambainis et al (1999 ACM Symp. Theory of Computing p 376), 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. Moreover, we generalize its negative answer for one-qubit encoding to the case of multiple-qubit encoding
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
We consider a communication method, where the sender encodes n classical bits into 1 qubit and sends it to the receiver who performs a certain measurement depending on which of the initial bits must be recovered. This procedure is called (n,1,p) quantum random access code (QRAC) where p > 1/2 is its success probability. It is known that (2,1,0.85) and (3,1,0.79) QRACs (with no classical counterparts) exist and that (4,1,p) QRAC with p > 1/2 is not possible. We extend this model with shared randomness (SR) that is accessible to both parties. Then (n,1,p) QRAC with SR and p > 1/2 exists for any n > 0. We give an upper bound on its success probability (the known (2,1,0.85) and (3,1,0.79) QRACs match this upper bound). We discuss some particular constructions for several small values of n. We also study the classical counterpart of this model where n bits are encoded into 1 bit instead of 1 qubit and SR is used. We give an optimal construction for such codes and find their success probability exactly--it is less than in the quantum case. Interactive 3D quantum random access codes are available on-line at http://home.lanet.lv/~sd20008/racs .
A quantum random access code (QRAC) is a map x↦ρxx\mapstoρ_xx↦ρx​ that encodes nnn-bit strings xxx into mmm-qubit quantum states ρxρ_xρx​, in a way that allows us to recover any one bit of xxx with success probability ≥p\geq p≥p. The measurement on ρxρ_xρx​ that is used to recover, say, x1x_1x1​ may destroy all the information about the other bits; this is in fact what happens in the well-known QRAC that encodes n=2n=2n=2 bits into m=1m=1m=1 qubits. Does this generalize to large nnn, i.e., could there exist QRACs that are so "obfuscated" that one cannot get much more than one bit out of them? Here we show that this is not the case: for every QRAC there exists a measurement that (with high probability) recovers the full nnn-bit string xxx up to small Hamming distance, even for the worst-case xxx.
A (Quantum) Random Access Code ((Q)RAC) is a scheme that encodes nnn bits into mmm (qu)bits such that any of the nnn bits can be recovered with a worst case probability p>12p>\frac{1}{2}p>21​. Such a code is denoted by the triple (n,m,p)(n,m,p)(n,m,p). It is known that n<4mn<4^mn<4m for all QRACs and n<2mn<2^mn<2m for classical RACs. These bounds are also known to be tight, as explicit constructions exist for n=4m−1n=4^m-1n=4m−1 and n=2m−1n=2^m-1n=2m−1 for quantum and classical codes respectively. We generalize (Q)RACs to a scheme encoding nnn ddd-levels into mmm (qu)-ddd-levels such that any ddd-level can be recovered with the probability for every wrong outcome value being less than 1d\frac{1}{d}d1​. We construct explicit solutions for all n≤d2m−1d−1n\leq \frac{d^{2m}-1}{d-1}n≤d−1d2m−1​. For d=2d=2d=2, the constructions coincide with those previously known. We show that the (Q)RACs are ddd-parity-oblivious, generalizing ordinary parity-obliviousness. We further investigate optimization of the success probabilities. For d=2d=2d=2, we use the measure operators of the previously best known solutions, but improve the encoding states to give a higher success probability. We conjecture that for maximal (n=4m−1,m,p)(n=4^m-1,m,p)(n=4m−1,m,p) QRACs, p=1+1(3+1)m−12p=\frac{1+\frac{1}{(\sqrt{3}+1)^m-1}}{2}p=21+(3​+1)m−11​​ is possible and show that it is an upper bound for the measure operators that we use. When we compare (n,m,pq)(n,m,p_q)(n,m,pq​) QRACs with classical (n,2m,pc)(n,2m,p_c)(n,2m,pc​) RACs, we can always find pq≥pcp_q\geq p_cpq​≥pc​, but the classical code gives information about every input bit simultaneously, while the QRAC only gives information about a subset. For several different (n,2,p)(n,2,p)(n,2,p) QRACs, we see the same trade-off, as the best ppp values are obtained when the number of bits that can be obtained simultaneously is as small as possible. The trade-off is connected to parity-obliviousness, since high certainty information about several bits can be used to calculate probabilities for parities of subsets.
An n↦pm random access code (RAC) is an encoding of n bits into m bits such that any initial bit can be recovered with probability at least p , while in a quantum RAC (QRAC), the n bits are encoded into m qubits. Since its proposal, the idea of RACs was generalized in many different ways, e.g. allowing the use of shared entanglement (called entanglement-assisted random access code, or simply EARAC) or recovering multiple bits instead of one. In this paper we generalize the idea of RACs to recovering the value of a given Boolean function f on any subset of fixed size of the initial bits, which we call f -random access codes. We study and give protocols for f -random access codes with classical ( f -RAC) and quantum ( f -QRAC) encoding, together with many different resources, e.g. private or shared randomness, shared entanglement ( f -EARAC) and Popescu-Rohrlich boxes ( f -PRRAC). The success probability of our protocols is characterized by the noise stability of the Boolean function f . Moreover, we give an upper bound on the success probability of any f -QRAC with shared randomness that matches its success probability up to a multiplicative constant (and f -RACs by extension), meaning that quantum protocols can only achieve a limited advantage over their classical counterparts.