Entanglement Breaks Expected Rules with Holistic Systems

Entanglement, a cornerstone of quantum theory, exhibits markedly different behaviours when subjected to physical constraints like symmetries or superselection rules. Roberto D. Baldijão from the Perimeter Institute for Theoretical Physics and the International Centre for Theory of Quantum Technologies, Marco Erba of the International Centre for Theory of Quantum Technologies, and David Schmid from the Perimeter Institute for Theoretical Physics, working with colleagues including John H. Selby and Ana Belén Sainz from the International Centre for Theory of Quantum Technologies, the Okinawa Institute of Science and Technology Graduate University, and the Basic Research Community for Physics e.V., investigate entanglement within tomographically nonlocal theories. This collaborative research demonstrates that a lack of tomographic locality generates two distinct types of entanglement, termed ‘entanglement’ and ‘entanglement’, and reveals their operational consequences. Crucially, the study proves that this tomographically-nonlocal entanglement cannot facilitate Bell nonlocality, steering, or teleportation, yet remains adequate for dense coding and perfect data hiding, thereby clarifying previously unexplained entanglement phenomena arising from failures in tomographic locality, even within standard quantum theory when considering fermions or fundamental superselection rules.

For decades, physicists have sought to fully understand how entanglement behaves under unusual conditions. Now, a theoretical analysis reveals entanglement can take surprising forms when fundamental measurement limits apply, clarifying long-standing puzzles and its potential uses in quantum technologies. Scientists are increasingly focused on understanding the subtle nature of entanglement, a key feature of quantum theory, and how it changes when physical constraints are applied.

These constraints can produce unexpected phenomena including the local broadcasting of entangled states and breakdowns in entanglement monogamy. Such effects are particularly apparent in tomographically nonlocal theories, where composite systems exhibit holistic degrees of freedom inaccessible through local measurements. Recent research within the framework of generalised probabilistic theories (GPTs) explores entanglement in these theories, revealing that failures in tomographic locality give rise to two separate types of entanglement: tomographically-local entanglement and tomographically-nonlocal entanglement.

Analyses demonstrate that tomographically-nonlocal entanglement cannot be used for tasks like Bell nonlocality, steering, or teleportation, despite being sufficient for dense coding and perfectly secure data hiding. This framework clarifies puzzling aspects of entanglement arising when tomographic locality fails, a situation that can occur even within standard quantum theory when considering fermions or fundamental superselection rules.

The GPT framework offers a rigorous way to study physical theories by focusing on their operational features: preparation, transformation, measurement, and probability assignment. By abstracting away from specific mathematical structures like Hilbert spaces, GPTs allow comparison of different theories and exploration of alternatives to quantum theory.

Many prior works assume tomographic locality, simplifying the framework but potentially obscuring important features. Unlike theories with tomographic locality, those without it possess composite systems with holistic degrees of freedom, where dual processes cannot be distinguished. There is ongoing debate regarding whether the fundamental theory governing our world is tomographically local, particularly concerning the role of superselection rules.

If these rules are fundamental, understanding the resulting failure of tomographic locality becomes essential. Real quantum theory and fermionic quantum theory, both restricted versions of standard quantum theory, exhibit tomographic nonlocality. Since these theories display distinctive features, even when compared to unrestricted quantum theory, understanding the operational consequences of such constraints is important.

Researchers have investigated how tomographic nonlocality affects the forms of entanglement that can exist within a theory. Entanglement behaves differently in tomographically-nonlocal theories than in unrestricted quantum theory, displaying counter-intuitive mathematical and operational characteristics. Some theories, including real quantum theory and fermionic quantum theory, contain three maximally entangled states that are non-monogamous and locally broadcastable. Also, certain tomographically-nonlocal theories exhibit entanglement even when every system possesses a state space identical to that of a classical system.

State and effect space construction underpins generalised probabilistic theory

Generalised probabilistic theories (GPTs) provided the framework for this work, allowing investigation of entanglement beyond standard quantum mechanics. Initially, the research focused on defining the state and effect spaces for individual systems, A and B, and their composite system AB. Each system’s state space, SA, is a subset of a real, finite-dimensional vector space A, with states represented as vectors within this space.

Correspondingly, the effect space, EA, comprises vectors in the dual of A, assigning probabilities to states and adhering to conditions ensuring valid probability assignments. Crucially, both SA and EA are defined as convex sets containing a unique unit effect, uA, which normalizes states and defines measurements as collections of effects summing to this unit.

Establishing a GPT requires more than defining these spaces. Tomography demands the ability to distinguish any two states or effects using appropriate measurements. Since the work does not assume the non-restriction hypothesis, it explicitly notes when this property is relevant. Instead of relying on complex Hilbert spaces, the research utilizes a vector space analogous to the space of Hermitian operators, allowing for a more abstract and general description of quantum systems.

For instance, a quantum system with d levels is described using d × d density matrices, with the full state space consisting of positive semidefinite operators with trace less than or equal to one. Extending this to composite systems, the study details how the state and effect spaces of subsystems A and B relate to those of the combined system AB. By considering the composite system AB, researchers aimed to explore entanglement forms arising from failures of tomographic locality, a condition where composite systems possess inaccessible degrees of freedom. Once established, this framework enabled the analysis of two distinct types of entanglement and their operational consequences for tasks like Bell nonlocality and dense coding.

Distinguishing Tomographic Entanglement Forms and their Functional Capabilities

The research distinguished two forms of entanglement: tomographically-local entanglement and tomographically-nonlocal entanglement, clarifying how entanglement manifests within theories lacking complete local tomography. Tomographically-nonlocal entanglement proves ineffective for Bell scenarios, steering, and teleportation, tasks readily achievable with standard, tomographically-local entanglement found in unrestricted quantum theory.

However, despite this limitation, tomographically-nonlocal entanglement suffices for both dense coding and perfect data hiding across a wide range of generalised probabilistic theories. Projectors were introduced to isolate holistic and non-holistic degrees of freedom within composite systems, decomposing states and effects to pinpoint entanglement accessible through local tomography and isolating the source of tomographic nonlocality.

By employing these projectors, researchers systematically identified the components of a state’s entanglement linked to local tomography versus those arising from holistic behaviour. Real quantum theory exhibits tomographically-nonlocal entanglement enabling dense coding and data hiding even when encoding and decoding are restricted to local operations.

The study reveals that certain entangled states in real quantum theory can be locally broadcast, while others can be infinitely shared, a previously puzzling observation now explained by this refined understanding of entanglement types. These behaviours are exclusive to states possessing a tomographically-nonlocal component, unlike those exhibiting tomographically-local entanglement.

The framework clarifies perplexing features of entanglement observed when tomographic locality fails, as occurs in systems governed by superselection rules or symmetries. Beyond simply identifying these two entanglement forms, the research establishes their operational consequences, revealing a clear separation between entanglement useful for creating non-classical correlations and that sufficient for secure communication. Inside real quantum theory, this distinction explains why some entangled states exhibit broadcasting and infinite sharing, while others do not.

Constrained quantum systems reveal entanglement’s diverse functional properties

Scientists have long grappled with the subtle nature of entanglement, a phenomenon where two particles become linked regardless of distance. Recent work demonstrates that imposing constraints on physical systems, symmetries or fundamental rules, can dramatically alter this entanglement, creating forms previously unseen. For years, the focus remained on entanglement as a resource for quantum technologies, but this research shifts attention to understanding entanglement’s behaviour under unusual conditions, revealing a richer, more complex picture.

Researchers have pinpointed operational differences. Certain types of entanglement prove useless for tasks like quantum teleportation, yet remain effective for secure data hiding. This distinction isn’t merely academic; it addresses a longstanding puzzle concerning systems governed by rules differing from standard quantum theory, such as those involving fermions.

These systems now appear to offer a natural setting for observing these unique entanglement forms. Defining these distinctions requires careful mathematical treatment, and the framework employed, generalised probabilistic theories, introduces its own level of abstraction. The immediate impact lies in refining our theoretical understanding, rather than immediately translating to practical devices.

The inability of some entangled states to support teleportation highlights a limitation in assuming all entanglement is equally valuable. The work relies on simplified models; scaling up to more complex systems will undoubtedly present challenges. The field may see increased exploration of systems with inherent constraints, moving beyond the idealized conditions of most quantum experiments.

This work suggests that focusing on the type of entanglement, not just its presence, is vital. Future research might extend this framework to explore entanglement in systems with more complex symmetries, or even to investigate its role in condensed matter physics. By clarifying the origins of unusual entanglement, this work opens avenues for a more complete theory of quantum correlations.