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arXiv:1607.02667 · Physical Review A

Improved classical and quantum random access codes

Ola Liabøtrø

Published May 8, 2017• 18 citations
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Abstract

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.

DOI
10.1103/physreva.95.052315

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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≤d2m\ensuremath−1d\ensuremath−1n\ensuremath{\le}\frac{{d}^{2m}\ensuremath{-}1}{d\ensuremath{-}1}n\ensuremath≤d\ensuremath−1d2m\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=4m\ensuremath−1,m,p)(n={4}^{m}\ensuremath{-}1,m,p)(n=4m\ensuremath−1,m,p) QRACs, p=121+[(3+1)m\ensuremath−1]\ensuremath−1p=\frac{1}{2}{1+{[{(\sqrt{3}+1)}^{m}\ensuremath{-}1]}^{\ensuremath{-}1}}p=21​1+[(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 pq\ensuremath≥pc{p}_{q}\ensuremath{\ge}{p}_{c}pq​\ensuremath≥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 ((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.
Optimal Average Success Probabilities of Binary (n,n−1)(n,n-1)(n,n−1) and (n,n−2)(n,n-2)(n,n−2) Quantum Random Access Codes
A binary (n,m)(n,m)(n,m) quantum random access code (QRAC) compresses an nnn-bit classical string into an mmm-qubit quantum state, from which a decoder attempts to recover a randomly selected target bit. Of particular interest is the optimal average probability of success, Pn,mQ,avg,optP^{Q,\mathrm{avg},\mathrm{opt}}_{n,m}Pn,mQ,avg,opt​, which was numerically conjectured to satisfy the bound Pn,mQ,avg,opt≤12+12mnP^{Q,\mathrm{avg},\mathrm{opt}}_{n,m}\leq \frac{1}{2}+\frac{1}{2}\sqrt{\frac{m}{n}}Pn,mQ,avg,opt​≤21​+21​nm​​. Recent constructions of (n,n−1)(n,n-1)(n,n−1) QRACs by Suzuki and (n,n−2)(n,n-2)(n,n−2) QRACs by Akibue et al. meet this bound exactly, raising the question of their strict optimality. In this work, we settle this question by proving the conjectured upper bound for m∈{n−1,n−2}m\in\{n-1,n-2\}m∈{n−1,n−2}, thereby precisely determining Pn,n−1Q,avg,optP^{Q,\mathrm{avg},\mathrm{opt}}_{n,n-1}Pn,n−1Q,avg,opt​ and Pn,n−2Q,avg,optP^{Q,\mathrm{avg},\mathrm{opt}}_{n,n-2}Pn,n−2Q,avg,opt​. The proof utilizes a translation recently studied by Lin and de Wolf from local to global reconstruction via pretty good measurement, along with dimensional and positive-semidefinite constraints on an induced channel.
Random access coding is an information task that has been extensively studied and found many applications in quantum information. In this scenario, Alice receives an n -bit string x , and wishes to encode x into a quantum state , such that Bob, when receiving the state , can choose any bit and recover the input bit x i with high probability. Here we study two variants: parity-oblivious random access codes (RACs), where we impose the cryptographic property that Bob cannot infer any information about the parity of any subset of bits of the input apart from the single bits x i ; and even-parity-oblivious RACs, where Bob cannot infer any information about the parity of any even-size subset of bits of the input. In this paper, we provide the optimal bounds for parity-oblivious quantum RACs and show that they are asymptotically better than the optimal classical ones. Our results provide a large non-contextuality inequality violation and resolve the main open problem in a work of Spekkens et al (2009 Phys. Rev. Lett. 102 010401 ). Second, we provide the optimal bounds for even-parity-oblivious RACs by proving their equivalence to a non-local game and by providing tight bounds for the success probability of the non-local game via semidefinite programming. In the case of even-parity-oblivious RACs, the cryptographic property holds also in the device independent model.
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.
Analytical construction of (n,n−1)(n, n-1)(n,n−1) quantum random access codes saturating the conjectured bound
Quantum Random Access Codes (QRACs) embody the fundamental trade-off between the compressibility of information into limited quantum resources and the accessibility of that information, serving as a cornerstone of quantum communication and computation. In particular, the (n,n−1)(n, n-1)(n,n−1)-QRACs, which encode nnn bits of classical information into n−1n-1n−1 qubits, provides an ideal theoretical model for verifying quantum advantage in high-dimensional spaces; however, the analytical derivation of optimal codes for general nnn has remained an open problem. In this paper, we establish an analytical construction method for (n,n−1)(n, n-1)(n,n−1)-QRACs by using an explicit operator formalism. We prove that this construction strictly achieves the numerically conjectured upper bound of the average success probability, P=1/2+(n−1)/n/2\mathcal{P} = 1/2 + \sqrt{(n-1)/n}/2P=1/2+(n−1)/n​/2, for all nnn. Furthermore, we present a systematic algorithm to decompose the derived optimal POVM into standard quantum gates. Since the resulting decoding circuit consists solely of interactions between adjacent qubits, it can be implemented with a circuit depth of O(n)O(n)O(n) even under linear connectivity constraints. Additionally, we analyze the high-dimensional limit and demonstrate that while the non-commutativity of measurements is suppressed, an information-theoretic gap of O(log⁡n)O(\log n)O(logn) from the Holevo bound inevitably arises for symmetric encoding. This study not only provides a scalable implementation method for high-dimensional quantum information processing but also offers new insights into the mathematical structure at the quantum-classical boundary.