New Algorithm Efficiently Calculates Key Quantum Information Measure

Quantifying the correlation between quantum systems remains a central challenge in quantum information theory, with measures like mutual information providing crucial insights into entanglement and its role in quantum computation and communication. Researchers now present a novel numerical method for calculating a particularly challenging quantity, the doubly minimized Petz Rényi mutual information (PRMI), a refinement of standard mutual information that accounts for the specific structure of quantum states. Laura Burri, from the Institute for Theoretical Physics at ETH Zurich, and colleagues demonstrate that an alternating minimization algorithm converges efficiently to the PRMI for any order, extending previous work limited to classical-classical states. Their findings, detailed in the paper “Alternating minimization for computing doubly minimized Petz Renyi mutual information”, offer a practical approach to characterise quantum correlations in complex systems and advance the development of quantum technologies.

Quantum information theory increasingly relies on quantifying correlations within quantum systems, moving beyond traditional entanglement measures, and mutual information serves as a fundamental tool for this purpose. Recent research focuses on generalizations of this concept, specifically Rényi mutual information, which offers a parameterised family of correlation measures with varying sensitivities to different aspects of the quantum state. The doubly minimized Petz Rényi mutual information (PRMI) emerges as a particularly intriguing variant. Defined through a minimization procedure involving the Petz divergence and product states, the PRMI provides a nuanced approach to understanding quantum correlations and their implications for information processing.

The Petz divergence, a measure of distinguishability between quantum states, plays a central role in defining the PRMI, quantifying how well two states can be distinguished by a measurement and providing a way to assess their dissimilarity. Minimizing this divergence over all possible product states yields the PRMI, effectively identifying the ‘closest’ product state to the given bipartite state. A bipartite state describes a quantum system composed of two subsystems.

Despite its theoretical importance, calculating the PRMI remains a significant hurdle, as no closed-form expression exists for the PRMI of order α not equal to 1, even for classical-classical states, motivating the development of numerical methods for approximating the PRMI. Researchers are exploring various approaches, including optimization techniques and approximations based on the limit of repeated state copies, and the PRMI possesses an operational interpretation in the context of binary quantum state discrimination for α within the range of (1/2, 1).

Researchers are increasingly reliant on numerical methods to compute the Petz Rényi mutual information, as a closed-form expression remains elusive. This work establishes the asymptotic convergence of alternating minimization, a common iterative technique, when applied to compute the doubly minimized Petz Rényi mutual information for any order. Alternating minimization involves repeatedly optimising the solution with respect to different variables.

The core of the analysis centres on demonstrating the rate of convergence, specifically how quickly the algorithm approaches the true value of the Petz Rényi mutual information. Researchers prove linear convergence of the objective function values, meaning the error decreases proportionally with each iteration for certain variables within the algorithm. However, for other variables, the convergence is sublinear, implying a slower, yet still guaranteed, approach to the optimal value. Understanding these differing rates of convergence is vital for optimising the algorithm’s performance and determining appropriate stopping criteria.

The mathematical framework employed relies heavily on concepts from functional analysis, particularly the properties of operators and their behaviour within Hilbert spaces. Hilbert spaces provide a mathematical structure for representing quantum states and operators, allowing researchers to rigorously analyse their properties and interactions. Establishing conditions for convergence requires careful consideration of the properties of these operators, ensuring that the optimisation process remains stable and well-defined. The proof leverages monotone operator theory to demonstrate the convergence of the algorithm.

This work provides a solid theoretical foundation for the practical computation of the Petz Rényi mutual information. By rigorously establishing the convergence of alternating minimization, researchers can confidently deploy this algorithm in various quantum information processing tasks, including quantum coding, quantum hypothesis testing, and quantum cryptography.

The findings are particularly valuable given the practical need for efficient computation of the α-mutual information, spanning diverse areas within quantum information, including quantum state discrimination, quantum data compression, and the characterisation of quantum entanglement. Future work should focus on practical implementation and performance evaluation of the algorithm on realistic quantum systems. Investigating the algorithm’s robustness to noise and errors inherent in experimental setups is crucial, and furthermore, exploring the scalability of the method to larger quantum systems remains a key challenge. Comparative studies against alternative algorithms for computing α-mutual information would provide valuable insights into its relative strengths and weaknesses, and extending the analytical framework to encompass other relevant quantum information measures also presents a promising avenue for future research.