Correlations Linked to Entanglement Via Universal Property

Correlations represent a fundamental aspect of physics, driving advances across science and technology, and originate from the inability to describe a system as independent parts. Elizabeth Agudelo of TU Wien, Atominstitut & Vienna Center for Quantum Science and Technology, Laura Ares and Jan Sperling from Paderborn University, Institute for Photonic Quantum Systems (PhoQS), Theoretical Quantum Science, working in collaboration with colleagues at TU Wien, Atominstitut & Vienna Center for Quantum Science and Technology and Paderborn University, Institute for Photonic Quantum Systems (PhoQS), Theoretical Quantum Science, now demonstrate a generalised understanding of these correlations through arbitrary products. Their research establishes a universal link between these general products and the more familiar tensor products, effectively connecting broader classes of non-product states to entanglement. This work constructs a framework for analysing correlations using an extended resource theory, applicable even to systems beyond two components, and offers potential insights into diverse areas such as fermionic states, multi-photon factorisation, and even the intriguing relationship between prime numbers and single-party entanglement.

Scientists have expanded our understanding of how connections form between quantum systems. This work reveals a surprising link between all such connections and the more familiar phenomenon of quantum entanglement, potentially unlocking new ways to harness quantum mechanics for advances in computing and even mathematics. Researchers have developed a new way to view quantum correlations, extending beyond traditional entanglement to encompass a wider range of interconnectedness between quantum systems.

This introduces a generalised notion of correlations based on arbitrary products of quantum states, revealing a link between these broader correlations and entanglement. Remarkably, the research establishes a universal property connecting any such product to the standard tensor product, allowing researchers to relate previously distinct non-product states to entanglement.

The team constructed a set of “free operations”, equivalent to allowed moves in a game, for these arbitrary products. By extending the well-established framework of local operations and classical communication from entanglement theory, they created a complete resource theory of correlations for general products. This generalisation extends beyond two components, offering a universally applicable connection to multipartite entanglement, where multiple parties share quantum information.

The implications of this work are far-reaching, potentially impacting diverse fields from quantum computing to our understanding of fundamental mathematical concepts. Applications highlighted within the study demonstrate the usefulness of this approach, including a novel way to factorize fermionic states, essential for describing electrons, and multi-photon states into their single-photon components.

Perhaps most intriguingly, the research suggests a potential connection between prime numbers and a form of single-party entanglement, opening up unexpected avenues for exploration. Entanglement is now being viewed through a broader lens, potentially unifying disparate areas of physics and mathematics, though characterising it remains a complex challenge, requiring joint analysis of subsystems and often proving computationally difficult.

Specifically, a product state arises when a composite quantum state can be expressed as a combination of independent states of its parts, while entangled states defy this simple description. Determining if a state is separable, capable of being built from these independent parts, is notoriously hard, even with relaxed error tolerances. Entanglement is studied within quantum resource theories, which assess its operational usefulness in quantum protocols and define what operations are considered “free”.

The diversity of ways quantum non-separability can manifest itself prompts the question of whether similar frameworks can be applied to other forms of correlations. In this paper, researchers investigate the problem of “quantum factorization” for various types of products, demonstrating that all non-factorizable states can be linked to entanglement through a unique linear map.

This result extends to multiple factors and mixed states, allowing for classical correlations within ensembles of product states. The established universal property allows generalisation of the local-operation-and-classical-communication paradigm to arbitrary products, defining the corresponding resource-free operations. At the heart of this work lies the concept of a general product, a bilinear map that combines quantum states to produce a new product state.

If a state can be written as the result of this product, it is considered factorizable; otherwise, it is termed entangled with respect to that product. This introduces a broader definition of entanglement, applicable to all bilinear products, with potential for multilinear generalizations. This map, denoted L◦, allows any non-factorizable state to be directly related to entanglement via the tensor product, demonstrating that all resources of quantum correlations can be represented, up to this linear map, as a consequence of tensor entanglement.

This finding has implications for both the theoretical and experimental verification of entanglement, suggesting existing techniques can be adapted for broader types of correlations. Beyond this core connection, the study extends the local-operation-and-classical-communication (LOCC) paradigm, central to standard entanglement theory, to encompass arbitrary products of states, defining resource-free operations applicable to any product under study.

The researchers proved preservation of ◦-factorization under factor-wise transformations, meaning these transformations do not create a quantum resource. Furthermore, the research addresses mixed states by introducing classically correlated states through the closure of the convex hull of pure product states, taking the form σA|B, representing a weighted sum of pure product states with a joint probability distribution P. Allowing for signed measures P, including negative quasiprobabilities, enables the construction of an affine resource theory as long as σA|B remains a valid quantum state.

Any state not within this closure is defined as quantum ◦-correlated, and crucially, these classical correlations also relate to tensor-separable states through the linear map L◦.

Generalised Products and Universal Links to Multipartite Entanglement

Researchers formulated a generalised concept of correlations using arbitrary products, moving beyond the standard tensor product typically employed in quantum mechanics. This approach establishes a universal connection between these general products and tensor products, effectively linking broader, non-product states to the more familiar notion of entanglement.

To define the boundaries of these correlations, the work extends the local-operation-and-classical-communication (LOCC) paradigm, a cornerstone of entanglement theory, to encompass general product types, creating a resource theory of correlations applicable to any product scheme. This extends beyond just two factors, providing a universally relatable link to multipartite entanglement.

Specifically, the team demonstrated that any state expressible as a product, denoted ‘◦’, can be rewritten as a tensor product through a linear operator L◦, holding true for all ◦-factorizable states. This universality allows existing techniques from tensor entanglement theory to be adapted for analysing ◦-entanglement, simply by modifying them with the map L◦.

Subsequently, the study considered transformations preserving ◦-factorization, revealing that factor-wise transformations, analogous to local operations in entanglement theory, do not generate quantum resources. Statistical ensembles were then used to extend the concept to classically ◦-correlated states, defined as a convex hull of pure states and distinguished from quantum ◦-correlated states by their separability.

Beyond bipartite systems, the research extended the concept to multi-factor correlations, demonstrating that a linear map can relate multipartite entanglement to ◦-entanglement with multiple factors. To illustrate the framework’s versatility, the work presented examples including fermionic states described by the exterior product, and multi-photon states factorized into single photons.

Furthermore, a single-system, continuous-variable example was constructed using infinite-dimensional states, revealing that specific coefficient values determine the presence of tripartite ◦-entanglement. This approach, by linking seemingly disparate concepts, offers a new perspective on understanding quantum correlations and their underlying resourcefulness.

Generalised correlations expand resource theory beyond entanglement

Scientists have long sought to broaden our understanding of correlation, moving beyond the limitations of traditional entanglement theory. This recent work achieves precisely that, formulating a generalised notion of correlation using arbitrary products rather than solely tensor products. While entanglement remains a cornerstone of quantum information science, restricting correlation to this framework has proven a persistent challenge.

The difficulty stems from entanglement’s inherent requirement for shared quantum states, a condition not always present or easily achievable in complex systems. Establishing a universal connection between these broader correlations and standard entanglement offers a powerful new lens through which to view quantum phenomena. By extending the established framework of local operations and classical communication, researchers have created a resource theory applicable to these general products.

This isn’t merely a mathematical exercise; it provides a pathway to understanding systems where entanglement is difficult to define or maintain, such as those involving many interacting particles. The practical implications extend beyond fundamental physics, with applications including factoring fermionic and multi-photon states, potentially streamlining quantum computations and optical technologies.

Perhaps most surprisingly, the work suggests a tantalising link between prime numbers and a form of single-party entanglement, a connection that, if fully explored, could reshape our understanding of number theory. Translating these theoretical advances into tangible technologies remains a significant hurdle. The field stands poised for further exploration.

Beyond this group’s immediate efforts to refine the resource theory, we can anticipate investigations into the specific types of correlations generated in diverse physical systems. Once these correlations are better characterised, the focus will likely shift towards developing methods to control and manipulate them, opening doors to new quantum devices and algorithms. This work offers a more inclusive framework, promising a richer and more complete picture of quantum interconnectedness.