Semidefinite Programming Algorithm Accurately Computes Smooth Max-Mutual of Bipartite States in Any Dimension

Determining the shared information within quantum systems presents a significant challenge in quantum information theory, and researchers continually seek more accurate and efficient methods for its calculation. Christopher Popp, Tobias C. Sutter, and Beatrix C. Hiesmayr, all from the University of Vienna, now present a new algorithm based on semidefinite programming that computes a key measure of this shared information, known as the smooth max-mutual information, for quantum states of any dimension. The team’s approach delivers accurate results when certain conditions regarding the state’s properties are met, and importantly, provides a reliable upper bound even when those conditions are not fully satisfied. This advancement extends the power of semidefinite programming techniques, improving the ability to compute and estimate crucial measures relevant to diverse quantum processing tasks.

Smooth Max-Mutual Information and Quantum Key Rates

This paper details a new method for calculating the smooth max-mutual information between two quantum systems, a crucial quantity in quantum information theory. This measure is essential for understanding quantum key distribution, entanglement-assisted communication, and cryptography, but its calculation is often computationally challenging. The researchers aimed to develop an efficient and accurate method for determining this important value. The team leveraged semidefinite programming (SDP), a powerful optimization technique well-suited for handling the positive semidefinite matrices common in quantum information theory, to address this challenge.

They present a novel iterative algorithm based on SDP, repeatedly solving SDPs until a solution is reached. A significant contribution is the explicit characterization of a previously unexplored SDP that forms the core of this iterative procedure, with both primal and dual formulations established alongside a proof of strong duality. The algorithm converges to an optimal solution under certain conditions, specifically when the marginal states of the quantum system are full rank. The smoothness in smooth max-mutual information reflects the robustness of the measure against small changes in the quantum state, controlled by a parameter that determines the degree of smoothing. This algorithm improves the accuracy of key rate estimations in quantum key distribution and provides a tool for assessing the cryptographic capacities of quantum states.

Smooth Max-Mutual Information via Semidefinite Programming

Scientists developed an innovative iterative algorithm grounded in semidefinite programming (SDP) to compute the smooth max-mutual information of bipartite quantum states, applicable across any dimension. This method addresses a longstanding challenge in quantum information theory by providing a means to quantify correlations even when dealing with imperfect or noisy quantum states. The algorithm’s accuracy hinges on satisfying a specific rank condition for the marginal states within a defined smoothing environment, and it provides a reliable upper bound when this condition is not met. This builds on previous work employing iterative SDPs to calculate smooth min-entropy, but requires a distinct approach due to the differing smoothing structures of the two quantities. This algorithm, based on semidefinite programming (SDP), provides a robust method for assessing how well quantum states can be used for applications like secret-key distillation. The core of the algorithm involves an iterative process that refines the solution by alternately optimizing different variables within the SDP framework. This “seesaw” method converges to an accurate solution, providing an upper bound on the smooth max-mutual information, particularly for states where the marginal states remain full rank throughout the process. The algorithm’s accuracy hinges on the ability to characterize the underlying SDP with both primal and dual formulations, and the researchers have rigorously proven that these formulations exhibit strong duality, ensuring reliable results. Experiments demonstrate that the algorithm effectively computes the smooth max-mutual information, even in scenarios where previous SDP-based techniques failed.

Smooth Max-Mutual Information via Semidefinite Programming

This work presents a new iterative algorithm, based on semidefinite programming, for computing the smooth max-mutual information of bipartite quantum states. The algorithm accurately determines this quantity when the marginal states of the system meet a specific rank condition, and provides a useful bound otherwise. A key achievement is the characterization of a novel semidefinite program, for which both primal and dual formulations were established alongside a proof of strong duality, solidifying its theoretical basis within quantum information theory. The results extend existing techniques for evaluating operational quantities, complementing methods used to calculate measures like hypothesis testing mutual information and smooth min/max-entropy. Importantly, because smooth max-mutual information can be used alongside hypothesis testing mutual information to estimate the one-shot distillable key of a quantum state, this algorithm offers a tool for assessing the cryptographic potential of states that satisfy the necessary rank conditions. The authors acknowledge that the algorithm may not find optimal solutions when the rank condition is not met, and suggest future research could focus on relaxing this assumption to broaden the method’s applicability.