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arXiv:2607.01564

An Information-Theoretic Principle for Optimal Quantum Encoding: Tight Frames and Equiangular Ensembles

Farhad Farokhi, Shuixin Xiao

quant-phcs.ITSubmitted Jul 2, 2026• Announced Jul 3, 2026• 0 citations• Revised Oct 8, 2026 (v2)

v1 · Jul 2, 2026 v2 · Oct 8, 2026

Abstract

Optimal encoding of classical data for quantum-assisted statistical inference is investigated from an information-theoretic perspective. We prove that the accuracy of any quantum-computing inference procedure is upper bounded by the maximal quantum leakage from the classical data through its quantum encoding, establishing leakage as a universal, task-agnostic quality measure for encoders. The optimal encoding strategy, i.e., an encoding strategy that maximizes the maximal quantum leakage, is proved to be attained by pure states. When there are enough qubits, basis encoding is proved to be universally optimal. However, when the dimension of the system is small, phase encoding is optimal. For the latter, the optimal encoding is not unique. That is, any tight frame, any ensemble whose average state is the maximally mixed state, is in fact optimal. Within tight frames, equiangular tight frames (ETFs) are distinguished as the uniquely symmetric optimal encodings, i.e., they saturate the Welch lower bound on pairwise overlaps. Prominent special cases are the qubit trine, the regular simplex, and symmetric informationally complete positive operator-valued measures (SIC-POVMs), for which the ETF structure and explicit codeword constructions are provided. Numerical examples are presented to validate the theoretical predictions.

Comments: Fixed a few mistakes and typos

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Optimal encoding of classical data for quantum-assisted statistical inference is investigated from an information-theoretic perspective. We prove that the accuracy of any quantum-computing inference procedure is upper bounded by the maximal quantum leakage from the classical data through its quantum encoding, establishing leakage as a universal, task-agnostic quality measure for encoders. The optimal encoding strategy, i.e., an encoding strategy that maximizes the maximal quantum leakage, is proved to be attained by pure states. When there are enough qubits, basis encoding is proved to be universally optimal. However, when the dimension of the system is small, phase encoding is optimal. For the latter, the optimal encoding is not unique. That is, any tight frame, any ensemble whose average state is the maximally mixed state, is in fact optimal. Within tight frames, equiangular tight frames (ETFs) are distinguished as the uniquely symmetric optimal encodings, i.e., they saturate the Welch lower bound on pairwise overlaps. Prominent special cases are the qubit trine, the regular simplex, and symmetric informationally complete positive operator-valued measures (SIC-POVMs), for which the ETF structure and explicit codeword constructions are provided. Numerical examples are presented to validate the theoretical predictions.

Optimal encoding of classical data for statistical inference using quantum computing is investigated. A universal encoder is sought that is optimal for a wide array of statistical inference tasks. Accuracy of any statistical inference is shown to be upper bounded by a term that is proportional to maximal quantum leakage from the classical data, i.e., the input to the inference model, through its quantum encoding. This demonstrates that the maximal quantum leakage is a universal measure of the quality of the encoding strategy for statistical inference as it only depends on the quantum encoding of the data and not the inference task itself. The optimal universal encoding strategy, i.e., the encoding strategy that maximizes the maximal quantum leakage, is proved to be attained by pure states. When there are enough qubits, basis encoding is proved to be universally optimal. An iterative method for numerically computing the optimal universal encoding strategy is presented.

Maximal quantum leakage (MQL) is a worst-case information leakage measure that quantifies an adversary's inference advantage gained from accessing quantum encoding of classical data with arbitrary measurements. While MQL admits an exact characterization for a given ensemble of quantum states, its robustness to implementation imperfections has not been systematically studied. In this paper, we analyze the sensitivity of maximal quantum leakage under perturbations of the quantum encoding. We establish a continuity bound in terms of the trace distance between ideal and perturbed quantum states, and show, via an example, that this bound is attainable. We further derive fidelity-based and relative-entropy-based sufficient conditions for bounding the variation of maximal quantum leakage, and illustrate numerically that these conditions can be loose.

Despite being pursued with a lot of dedication in frame theory and quantum information theory, maximal equiangular tight frames have only been confirmed to exist for lowest dimensions. This is especially puzzling since maximal sets of mutually unbiased bases, which are also known to be optimal packings but have a slightly larger number of vectors, are known to exist in complex Hilbert spaces of any prime power dimension. This talk presents a method by which we can construct a set of

M^2+1

vectors in

M

complex dimensions that form a tight complex orthoplectic Grassmannian frame that is in addition related to a weighted complex projective 2-design with optimality properties for quantum state tomography.

Any given density matrix can be represented as an infinite number of ensembles of pure states. This leads to the natural question of how to uniquely select one out of the many, apparently equally suitable, possibilities. Following Jaynes' information-theoretic perspective, this can be framed as an inference problem. We propose the Maximum Geometric Quantum Entropy Principle to exploit the notions of Quantum Information Dimension and Geometric Quantum Entropy. These allow us to quantify the entropy of fully arbitrary ensembles and select the one that maximizes it. After formulating the principle mathematically, we give the analytical solution to the maximization problem in a number of cases and discuss the physical mechanism behind the emergence of such maximum entropy ensembles.

Any given density matrix can be represented as an infinite number of ensembles of pure states. This leads to the natural question of how to uniquely select one out of the many, apparently equally-suitable, possibilities. Following Jaynes' information-theoretic perspective, this can be framed as an inference problem. We propose the Maximum Geometric Quantum Entropy Principle to exploit the notions of Quantum Information Dimension and Geometric Quantum Entropy. These allow us to quantify the entropy of fully arbitrary ensembles and select the one that maximizes it. After formulating the principle mathematically, we give the analytical solution to the maximization problem in a number of cases and discuss the physical mechanism behind the emergence of such maximum entropy ensembles.

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,

\mathcal{P} = 1/2 + \sqrt{(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(\log n)

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.