Quantum random access codes (QRACs) are key tools for a variety of protocols in quantum information theory. This paper gives an upper bound on the guessing success probability in the classical case of random access codes using mutually unbiased bases as measurement bases in a 3-dimensional Hilbert space and gives an encoding strategy capable of exceeding the classical bound. This encoding strategy holds for both 3-1 and 4-1 QRACs. This result is useful in areas such as random number expansion.
Quantum random access codes (QRACs) are key tools for a variety of protocols in quantum information theory. This paper gives an upper bound on the guessing success probability in the classical case of random access codes using mutually unbiased bases as measurement bases in a 3-dimensional Hilbert space and gives an encoding strategy capable of exceeding the classical bound. This encoding strategy holds for both 3-1 and 4-1 QRACs. This result is useful in areas such as random number expansion.
Quantum random access codes (QRACs) ask how well N classical bits can be encoded into M qubits while allowing any single bit to be recovered. Although the Nayak bound remains the standard general upper bound on the decoding probability, numerical evidence suggests a stronger upper bound in the small-qubit regime.
In this work, we formulate the optimal decoding probability in terms of decoding measurements, reformulating QRAC design as a spectral problem for noncommuting measurements. Using this formulation, we give an elementary proof of the Nayak bound by simplifying the Chernoff-bound argument. Moreover, we refine the argument to obtain upper bounds that improve over Nayak's bound in the entire finite-size regime.
The equality conditions of our bounds justify defining mutually unbiased projector-valued measurements (MUPVMs), a generalization of mutually unbiased bases. We show that decoding measurement of any two-qubit QRAC attaining the conjectured bound must form MUPVMs. We also show that any MUPVM, assisted by one ancillary qubit, yields a QRAC with optimal N-scaling decoding probability. Finally, we propose a new MUPVM-based construction for the (M+2,M)-QRAC family attaining the conjectured bound.
Quantum random access codes (QRACs) provide a basic tool for demonstrating the advantages of quantum resources and protocols, which have a wide range of applications in quantum information processing tasks. However, the investigation and application of high-dimensional (d) multi-input $(n)\phantom{\rule{4pt}{0ex}}{n}^{(d)}\ensuremath{\rightarrow}1$ QRACs are still lacking. Here, we present a general method to find the maximum success probability of ${n}^{(d)}\ensuremath{\rightarrow}1$ QRACs. In particular, we give the analytical solution for maximum success probability of ${3}^{(d)}\ensuremath{\rightarrow}1$ QRACs when measurement bases are mutually unbiased bases (MUBs). Based on the analytical solution, we show the relationship between MUBs and ${n}^{(d)}\ensuremath{\rightarrow}1$ QRACs. First, we provide a systematic method of searching for the operational inequivalence of MUBs (OI-MUBs) when the dimension d is a prime power. Second, we theoretically prove that, surprisingly, the commonly used Galois MUBs are not the optimal measurement bases to obtain the maximum success probability of ${n}^{(d)}\ensuremath{\rightarrow}1$ QRACs, which indicates a breakthrough according to the traditional conjecture regarding the optimal measurement bases. Furthermore, based on high-fidelity high-dimensional quantum states of orbital angular momentum, we experimentally achieve two-input and three-input QRACs up to dimension 11. We experimentally confirm the OI-MUBs when d=5. Our results open alternative avenues for investigating the foundational properties of quantum mechanics and quantum network coding.
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>21. Such a code is denoted by the triple (n,m,p). It is known that n<4m for all QRACs and n<2m for classical RACs. These bounds are also known to be tight, as explicit constructions exist for n=4m−1 and n=2m−1 for quantum and classical codes respectively. We generalize (Q)RACs to a scheme encoding nd-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 d1. We construct explicit solutions for all n≤d−1d2m−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=4m−1,m,p) QRACs, 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) QRACs with classical (n,2m,pc) RACs, we can always find pq≥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) 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.
I introduce a new notion, that extends the mutually unbiased bases (MUB) conditons to more than two bases. These, I call the nUB conditions, and the corresponding bases n-fold unbiased. They naturally appear while optimizing generic n-to-one quantum random access code (QRAC) strategies. While their existence in general dimensions is an open question, they nevertheless give close-to-tight upper bounds on QRAC success probabilities, and raise fundamental questions about the geometry of quantum states.
Analytical construction of (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)-QRACs, which encode n bits of classical information into n−1 qubits, provides an ideal theoretical model for verifying quantum advantage in high-dimensional spaces; however, the analytical derivation of optimal codes for general n has remained an open problem. In this paper, we establish an analytical construction method for (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, for all n. 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) 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(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.
Mutually unbiased bases (MUBs) constitute the canonical example of incompatible quantum measurements. One standard application of MUBs is the task known as quantum random access code (QRAC), in which classical information is encoded in a quantum system, and later part of it is recovered by performing a quantum measurement. We analyze a specific class of QRACs, known as the ${2}^{d}\ensuremath{\rightarrow}1$ QRAC, in which two classical dits are encoded in a d-dimensional quantum system. It is known that among rank-1 projective measurements MUBs give the best performance. We show (for every d) that this cannot be improved by employing nonprojective measurements. Moreover, we show that the optimal performance can only be achieved by measurements which are rank-1 projective and mutually unbiased. In other words, the ${2}^{d}\ensuremath{\rightarrow}1$ QRAC is a self-test for a pair of MUBs in the prepare-and-measure scenario. To make the self-testing statement robust we propose measures which characterize how well a pair of (not necessarily projective) measurements satisfies the MUB conditions and show how to estimate these measures from the observed performance. Similarly, we derive explicit bounds on operational quantities like the incompatibility robustness or the amount of uncertainty generated by the uncharacterized measurements. For low dimensions the robustness of our bounds is comparable to that of currently available technology, which makes them relevant for existing experiments. Last, our results provide essential support for a recently proposed method for solving the long-standing existence problem of MUBs.