Löwdin Method Boosts Quantum Coherence, Outperforms Rivals

Quantifying the resources available in quantum systems requires careful mathematical treatment, and a long-standing challenge involves choosing the best way to transform sets of quantum states into a more manageable form. Gökhan Torun from Izmir University of Economics, along with colleagues, now demonstrates that a technique called Löwdin symmetric orthogonalization consistently outperforms standard methods in characterizing these resources, particularly coherence and superposition. The team reveals that this approach avoids ambiguities arising from the order in which states are processed, generating more stable and physically meaningful representations of quantum information. Furthermore, they introduce a new set of probabilistic weights, building upon Löwdin orthogonalization, which allows for a consistent and basis-independent measurement of resource content, offering a significant advance in the field of quantum resource theory.

Researchers utilize Löwdin’s Symmetric Orthogonalization (LSO) to construct an orthogonal basis from a non-orthogonal one, and conversely, to obtain a non-orthogonal basis from an orthogonal set, avoiding ambiguity when analysing quantum coherence. Unlike the Gram-Schmidt Orthogonalization (GSO) procedure, which is sensitive to the order of input states, LSO applies a symmetric transformation that treats all vectors equally and minimizes deviation from the original basis. This approach generates basis sets with enhanced stability and physical relevance, facilitating the analysis of superpositions in non-orthogonal quantum states. Building on LSO, the research introduces Löwdin weights, probabilities that quantify the contribution of each original basis state to the resulting orthogonalized states, characterizing the transformation and its impact on quantum coherence.

Löwdin Orthogonalization Preserves Quantum State Symmetry

Researchers have developed a new method for transforming sets of vectors into orthogonal forms, demonstrating significant advantages over traditional approaches like the Gram-Schmidt process. This method, Löwdin symmetric orthogonalization (LSO), preserves the inherent symmetry and key characteristics of the original vectors, a crucial feature often lost in standard transformations. The team’s work focuses on accurately representing quantum states and resources, where maintaining the integrity of superposition and coherence is paramount. The conventional Gram-Schmidt process, while widely used, is sensitive to the order in which vectors are processed, leading to potentially arbitrary and unreliable results when dealing with physically meaningful systems.

In contrast, LSO applies a global transformation that treats all vectors equally, minimizing distortions and ensuring a more stable and physically relevant representation. This is achieved through a mathematical process involving the overlap matrix, which characterizes the relationships between the original vectors and ensures the new orthogonal basis closely resembles the original. Importantly, LSO introduces “Löwdin weights,” which assign probabilities to the non-orthogonal basis, providing a consistent measure of resource content and enabling accurate quantification of coherence and state delocalization. The researchers demonstrated that LSO excels at preserving the symmetry of quantum states, a critical factor in accurately modeling and predicting their behavior.

This preservation of symmetry is a significant improvement over the Gram-Schmidt process, which can distort these crucial characteristics. The team’s findings have implications for quantum resource theory, offering a more reliable framework for analyzing and quantifying coherence and superposition. By accurately representing the underlying quantum states, LSO promises to enhance our ability to harness and control these delicate phenomena, potentially leading to advancements in quantum computing and communication. The method’s ability to consistently quantify resource content, through the use of Löwdin weights, provides a powerful new tool for researchers in this rapidly evolving field.

Löwdin Weights Quantify Coherence and Delocalization

This work demonstrates that Löwdin symmetric orthogonalization (LSO) offers advantages over both the standard Gram-Schmidt procedure and its canonical variant when analysing superpositions of orthogonal and non-orthogonal states. Unlike methods dependent on the order of input states, LSO applies a symmetric transformation that preserves the original basis’s geometry, establishing a clear relationship between non-orthogonal and orthonormal representations. A key development is the introduction of Löwdin weights, which define a probability distribution over the orthonormalized basis and allow for unambiguous, basis-independent quantification of coherence. Information-theoretic measures derived from these weights, such as Shannon entropy and participation ratios, characterise state delocalization and the interplay between superposition asymmetry and basis non-orthogonality.

The researchers confirm the framework’s effectiveness through examples in two and three-dimensional systems, demonstrating its ability to quantify coherence across a range of superposition states. While the study focuses on discrete systems, the authors suggest future research could extend these techniques to higher-dimensional or continuous-variable systems, explore applications in quantum information protocols, and investigate the distillation of superposition states using Löwdin weights. Further work could also explore alternative descriptions of superposition in non-orthogonal bases, potentially deepening the understanding of coherence in complex quantum systems.