Complex Tensors Defy Expected Simplification in Large Systems

A thorough investigation into large random tensors reveals the surprising non-factorisation of moments, challenging understandings derived from matrix theory. Sylvain Carrozza of the University of Burgundy Europe and colleagues present an explicit example of a non-factorising trace-invariant and identify conditions under which factorisation does occur in the large N limit. Three key theorems relating combinatorial structures, termed ‘tree-like dominant pairings’, to factorisation are proven, demonstrating this behaviour in several previously studied trace-invariants. The work advances the theory of multipartite quantum entanglement by enabling the explicit calculation of Rényi entanglement entropy in uniform random quantum states, dependent on large N factorisation.

Non-factorising trace-invariants and large N factorisation of Rényi entanglement entropy

Trace-invariants, mathematical properties of tensors remaining constant regardless of rotation, exhibited non-factorization in complex Gaussian tensors, a surprising result given established matrix theory where such simplification is common. In matrix theory, the moments of a random matrix typically factorize in the large N limit, meaning they can be expressed as a product of simpler terms related to individual matrix elements or their correlations. This factorization significantly simplifies calculations and allows for analytical progress in many areas of physics, including quantum chaos and statistical mechanics. However, this behaviour does not generally hold for real Gaussian tensors of size N. Building on previous work that constructively demonstrated this non-factorization, the researchers now present a specific non-factorizing trace-invariant, lacking explicit examples in earlier studies. This particular trace-invariant provides a concrete instance where the expected factorization fails, solidifying the distinction between matrix and tensor behaviour. The work identifies conditions under which factorization does occur, establishing three theorems linked to ‘tree-like dominant pairings’, combinatorial structures that streamline calculations within tensors.

These theorems prove that previously studied trace-invariants factorize at large N, enabling explicit computation of Rényi entanglement entropy, a measure of quantum entanglement, in random quantum states. Rényi entanglement entropy is a generalisation of the more commonly known von Neumann entropy, providing a more flexible tool for quantifying entanglement. The ability to calculate this entropy efficiently is crucial for understanding the properties of complex quantum systems. Without factorization, analysing multipartite quantum entanglement becomes sharply more difficult, as complex calculations are significantly hindered. The computational cost of calculating entanglement entropy grows rapidly with the number of entangled particles, making efficient methods essential. The findings reveal that factorization occurs under specific conditions, proving three theorems linked to ‘tree-like dominant pairings’ which simplify tensor calculations. Currently, these results apply only to the large N limit and do not yet indicate how easily these principles translate to practical, real-world quantum systems with finite dimensions, prompting further investigation into the limitations of this approach and its applicability to systems with fewer degrees of freedom. The large N limit, while mathematically tractable, represents an idealization; real quantum systems always have finite dimensionality, and the behaviour observed in the limit may not accurately reflect the behaviour of these systems.

Dominant pairings enable efficient calculation within complex tensor networks

Understanding complex systems through the lens of tensors, multidimensional data structures important for modelling everything from quantum entanglement to material properties, is an increasingly important focus for scientists. Tensors represent a natural extension of matrices, allowing for the representation of higher-order correlations between variables. They are particularly useful in describing systems with many interacting components, such as quantum many-body systems or complex networks. While matrices, simpler two-dimensional arrays, readily simplify calculations through factorization, breaking down complex problems into manageable parts, tensors often do not. The team now establishes conditions where factorization is possible, specifically identifying ‘tree-like dominant pairings’ within these tensors that streamline calculations. These pairings represent a specific way of grouping tensor indices that leads to a simplification of the overall expression.

Recent findings demonstrated that not all tensor arrangements lend themselves to simplification, creating doubt about universal factorization. The initial expectation, based on analogies with matrix theory, was that factorization would be a general property of large random tensors. However, these results showed that this is not the case, and that non-factorizing tensors are common. However, identifying specific ‘tree-like dominant pairings’ within these complex structures unlocks a pathway to manageable calculations, even when complete factorization isn’t possible. These structures are vital for analysing multipartite quantum entanglement, a key area of quantum information theory, and future work will likely expand these techniques to unlock even more complex quantum systems. Multipartite entanglement, involving more than two entangled particles, is particularly challenging to analyse, and the development of efficient methods for calculating entanglement entropy is crucial for progress in this field.

Factorization can simplify calculations involving complex, high-dimensional tensors, multi-dimensional data arrays. Unlike matrices, tensors do not always allow for this streamlining, but specific arrangements, termed ‘tree-like dominant pairings’, are now identified where it is indeed possible. The team demonstrated, by establishing three theorems, that these pairings predict when expectation values of certain tensor properties, known as trace-invariants, will simplify at large scales, with the variance of the complex Gaussian tensor components represented as 1{N D. The trace-invariant represents a specific combination of tensor elements that remains unchanged under certain transformations. The theorems establish a connection between the structure of these pairings and the factorisation properties of the corresponding trace-invariants. This advances the understanding of multipartite quantum entanglement, enabling more efficient computation of Rényi entanglement entropy, a measure of quantum connectedness, within complex quantum systems. Specifically, the ability to efficiently compute Rényi entanglement entropy allows researchers to characterise the entanglement structure of random quantum states and to understand how entanglement affects the properties of these states. The identification of these dominant pairings provides a powerful tool for tackling the computational challenges associated with analysing large tensor networks and opens up new avenues for research in quantum information theory and beyond.

The researchers demonstrated that moments of complex random tensors do not consistently simplify with increasing size, unlike their matrix counterparts. This finding matters because it impacts the efficient calculation of properties crucial for analysing multipartite quantum entanglement, where systems involve more than two entangled particles. By identifying ‘tree-like dominant pairings’ within these tensors, they established conditions under which certain tensor properties, termed trace-invariants, do simplify at larger scales. The authors suggest this work will likely expand techniques for unlocking more complex quantum systems and improving the computation of Rényi entanglement entropy.