Classification of informative subsets in quantum encrypted cloning on qudits

TL;DR

This paper classifies which subsets of a quantum encrypted cloning system can leak information, revealing how confidentiality depends on the system's dimension and subset composition.

Contribution

It extends the classification of information leakage in quantum encrypted cloning from qubits to arbitrary finite dimensions, using a GCD condition.

Findings

Leakage depends on solutions to a system of congruences.

Complete uninformative state occurs only when the congruence system has only trivial solutions.

Dimension-dependent boundary of confidentiality is established for qudit systems.

Abstract

Encrypted cloning offers a means of introducing redundancy into quantum storage while respecting the no-cloning theorem: an unknown state is encoded into multiple signal-noise pairs, and only authorized subsets can recover the original information. However, the leakage properties of unauthorized subsets particularly for higher-dimensional systems (qudits) have remained unexplored. In this work, we systematically classify the informative subsets of the storage register in the qudit encrypted-cloning protocol. We focus on unauthorized subsets of size $n$ that contain exactly one qudit from each signal-noise pair. We show that the presence or absence of information leakage is determined by the solution set of a system of congruences whose coefficients depend on the dimension $d$ and on the numbers of signal and noise qudits in the subset. The reduced state is completely uninformative if and only if the congruence system admits only the trivial solution; otherwise, it retains a residual dependence on the input state through specific generalized Pauli operators. Low-dimensional examples ($n=1,2,3$) are worked out explicitly, and the complete classification is expressed in terms of a greatest-common-divisor condition. Our results extend the parity-based classification known for qubits ($d=2$) to arbitrary finite dimensions, revealing a dimension-dependent boundary of confidentiality in encrypted cloning.