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arXiv:0810.2937

Quantum Random Access Codes with Shared Randomness

Andris Ambainis, Debbie Leung, Laura Mancinska, Maris Ozols

Published Oct 16, 2008• 48 citations
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Abstract

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 .

DOI
10.48550/arxiv.0810.2937

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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.

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Quantum Random Access Codes encoding every possible number of bits into m qubits

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A (Quantum) Random Access Code ((Q)RAC) is a scheme that encodes $n$ bits into $m$ (qu)bits such that any of the $n$ bits can be recovered with a worst case probability $p>\frac{1}{2}$. Such a code is denoted by the triple $(n,m,p)$. It is known that $n<4^m$ for all QRACs and $n<2^m$ for classical RACs. These bounds are also known to be tight, as explicit constructions exist for $n=4^m-1$ and $n=2^m-1$ for quantum and classical codes respectively. We generalize (Q)RACs to a scheme encoding $n$ $d$-levels into $m$ (qu)-$d$-levels such that any $d$-level can be recovered with the probability for every wrong outcome value being less than $\frac{1}{d}$. We construct explicit solutions for all $n\leq \frac{d^{2m}-1}{d-1}$. For $d=2$, the constructions coincide with those previously known. We show that the (Q)RACs are $d$-parity-oblivious, generalizing ordinary parity-obliviousness. We further investigate optimization of the success probabilities. For $d=2$, 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 $(n=4^m-1,m,p)$ QRACs, $p=\frac{1+\frac{1}{(\sqrt{3}+1)^m-1}}{2}$ is possible and show that it is an upper bound for the measure operators that we use. When we compare $(n,m,p_q)$ QRACs with classical $(n,2m,p_c)$ RACs, we can always find $p_q\geq p_c$, but the classical code gives information about every input bit simultaneously, while the QRAC only gives information about a subset. For several different $(n,2,p)$ QRACs, we see the same trade-off, as the best $p$ 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.

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A random access code (RAC) encodes an $L$-bit string into a $k$-bit message, $L>k$, so that any requested bit can be recovered with high probability; a quantum RAC (QRAC) uses $k$ qubits instead. We give a geometric characterization of optimal classical $(L,k)$-RACs under average and worst-case decoding criteria. The average criterion is reduced to choosing $2^k$ representatives in $\{0,1\}^L$, while the worst-case criterion is reduced to a minimax problem over $2^k$ points in $[0,1]^L$ with a distance-like objective. This framework proves optimality for several parameter families, with many optimal constructions arising from standard infinite families of binary linear codes. It also yields two explicit classical--quantum separations. First, for every $L>1$, we construct a $(L,1)$-QRAC whose average decoding success probability strictly exceeds the optimal classical value. Second, for the family $(2^k-1,k)$, we prove worst-case optimality of a classical RAC and construct a QRAC with strictly larger worst-case success probability. For the family $(L,L-1)$, the framework identifies a classical RAC that is average-case optimal and, under a stated conjecture, also worst-case optimal. The same viewpoint further recovers explicit $(L,L-1)$-QRACs attaining a previously conjectured upper-bound value.

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Entanglement-assisted random access codes

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An $(n,m,p)$ random access code (RAC) makes it possible to encode $n$ bits in an $m$-bit message in such a way that a receiver of the message can guess any of the original $n$ bits with probability $p$ greater than $\frac{1}{2}$. In quantum RACs (QRACs), one transmits $n$ qubits. The full set of primitive entanglement-assisted random access codes (EARACs) is introduced, in which parties are allowed to share a two-qubit singlet. It is shown that via a concatenation of these, one can build for any $n$ an $(n,1,p)$ EARAC. QRACs for $n>3$ exist only if parties also share classical randomness. We show that EARACs outperform the best of known QRACs not only in the success probabilities but also in the amount of communication needed in the preparatory stage of the protocol. Upper bounds on the performance of EARACs are given and shown to limit also QRACs.

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Improved classical and quantum random access codes

Ola Liabøtrø

A (quantum) random access code ((Q)RAC) is a scheme that encodes $n$ bits into $m$ (qu)bits such that any of the $n$ bits can be recovered with a worst case probability $p>\frac{1}{2}$. We generalize (Q)RACs to a scheme encoding $n$ $d$-levels into $m$ (quantum) $d$-levels such that any $d$-level can be recovered with the probability for every wrong outcome value being less than $\frac{1}{d}$. We construct explicit solutions for all $n\ensuremath{\le}\frac{{d}^{2m}\ensuremath{-}1}{d\ensuremath{-}1}$. For $d=2$, the constructions coincide with those previously known. We show that the (Q)RACs are $d$-parity oblivious, generalizing ordinary parity obliviousness. We further investigate optimization of the success probabilities. For $d=2$, 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 $(n={4}^{m}\ensuremath{-}1,m,p)$ QRACs, $p=\frac{1}{2}{1+{[{(\sqrt{3}+1)}^{m}\ensuremath{-}1]}^{\ensuremath{-}1}}$ is possible, and show that it is an upper bound for the measure operators that we use. We then compare $(n,m,{p}_{q})$ QRACs with classical $(n,2m,{p}_{c})$ RACs. We can always find ${p}_{q}\ensuremath{\ge}{p}_{c}$, but the classical code gives information about every input bit simultaneously, while the QRAC only gives information about a subset. For several different $(n,2,p)$ QRACs, we see the same trade-off, as the best $p$ 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.

arXiv:1607.02667•Physical Review A•May 8, 2017•18 citations
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 .
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.
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.
A random access code (RAC) encodes an LLL-bit string into a kkk-bit message, L>kL>kL>k, so that any requested bit can be recovered with high probability; a quantum RAC (QRAC) uses kkk qubits instead. We give a geometric characterization of optimal classical (L,k)(L,k)(L,k)-RACs under average and worst-case decoding criteria. The average criterion is reduced to choosing 2k2^k2k representatives in {0,1}L\{0,1\}^L{0,1}L, while the worst-case criterion is reduced to a minimax problem over 2k2^k2k points in [0,1]L[0,1]^L[0,1]L with a distance-like objective. This framework proves optimality for several parameter families, with many optimal constructions arising from standard infinite families of binary linear codes. It also yields two explicit classical--quantum separations. First, for every L>1L>1L>1, we construct a (L,1)(L,1)(L,1)-QRAC whose average decoding success probability strictly exceeds the optimal classical value. Second, for the family (2k−1,k)(2^k-1,k)(2k−1,k), we prove worst-case optimality of a classical RAC and construct a QRAC with strictly larger worst-case success probability. For the family (L,L−1)(L,L-1)(L,L−1), the framework identifies a classical RAC that is average-case optimal and, under a stated conjecture, also worst-case optimal. The same viewpoint further recovers explicit (L,L−1)(L,L-1)(L,L−1)-QRACs attaining a previously conjectured upper-bound value.
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.
An (n,m,p)(n,m,p)(n,m,p) random access code (RAC) makes it possible to encode nnn bits in an mmm-bit message in such a way that a receiver of the message can guess any of the original nnn bits with probability ppp greater than 12\frac{1}{2}21​. In quantum RACs (QRACs), one transmits nnn qubits. The full set of primitive entanglement-assisted random access codes (EARACs) is introduced, in which parties are allowed to share a two-qubit singlet. It is shown that via a concatenation of these, one can build for any nnn an (n,1,p)(n,1,p)(n,1,p) EARAC. QRACs for n>3n>3n>3 exist only if parties also share classical randomness. We show that EARACs outperform the best of known QRACs not only in the success probabilities but also in the amount of communication needed in the preparatory stage of the protocol. Upper bounds on the performance of EARACs are given and shown to limit also QRACs.
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​. We generalize (Q)RACs to a scheme encoding nnn ddd-levels into mmm (quantum) 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 $n\ensuremath{\le}\frac{{d}^{2m}\ensuremath{-}1}{d\ensuremath{-}1}$. 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 $(n={4}^{m}\ensuremath{-}1,m,p)$ QRACs, $p=\frac{1}{2}{1+{[{(\sqrt{3}+1)}^{m}\ensuremath{-}1]}^{\ensuremath{-}1}}$ is possible, and show that it is an upper bound for the measure operators that we use. We then 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 ${p}_{q}\ensuremath{\ge}{p}_{c}$, 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.