The main purpose of thispaper is to show that composite quantum-like (QL) systems can closely mimic the separable states of quantum systems, and that suitable physical systems exhibiting these states exist. It is shown that QL graphs can closely emulate states of composite quantum systems, such as coupled two-level systems that display separable linear combinations of states. Examples of classical systems are suggested that show these states. These include multipole moments of waves or networks of phase oscillators. The work indicates that composite QL states can be manifest in complex network structures relevant to quantum biology or engineered into circuits, or even possibly soft matter.
Journal reference
Scholes GD.2026 Existence, structure and properties of quantum-like states.Proc. R. Soc. A 482:20260115
The main purpose of thispaper is to show that composite quantum-like (QL) systems can closely mimic the separable states of quantum systems, and that suitable physical systems exhibiting these states exist. It is shown that QL graphs can closely emulate states of composite quantum systems, such as coupled two-level systems that display separable linear combinations of states. Examples of classical systems are suggested that show these states. These include multipole moments of waves or networks of phase oscillators. The work indicates that composite QL states can be manifest in complex network structures relevant to quantum biology or engineered into circuits, or even possibly soft matter.
This work studies how a suitably-designed classical system generates with a quantum-like (QL) state space mediated by a graph. The graph plays a special dual role by directing the topology of the classical network and defining a state space that comprises superpositions of states in a tensor product basis. The basis for constructing QL graphs and their properties is reviewed and extended. An optimization of the graph product is developed to produce a more compact graph with the essential properties required to produce states that mimic many of the properties of quantum states. This provides a concrete visualization of the correlation structure in a quantum state space. The question of whether and, if so, how, entanglement can be exhibited by these QL systems is discussed critically and contrasted to the concept of `classical entanglement' in optics.
Recent work has exposed the idea that interesting quantum-like (QL) probability laws, including interference effects, can be manifest in classical systems. Here, we propose a model for QL states and QL bits. We suggest a way that huge, complex systems can host robust states that can process information in a QL fashion. Axioms that such states should satisfy are proposed. Specifically, it is shown that building blocks suited for QL states are networks, possibly very complex, that we defined based on k -regular random graphs. These networks can dynamically encode a lot of information that is distilled into the emergent states we can use for QL processing. Although the emergent states are classical, they have properties analogous to quantum states. Concrete examples of how QL functions are possible are given. The possibility of a ‘QL advantage’ for computing-type operations and the potential relevance for new kinds of function in the brain are discussed and left as open questions.
Recent publications have introduced the concept of quantum-like (QL) bits, along with their associated QL-states and QL-gate operations, emerging from the dynamics of complex synchronized networks. The present work builds on this framework to present a simulation of quantum dynamics in spin systems. We focus on the transverse field Ising model, a widely studied Hamiltonian for interacting spin systems. We show how circuits, built using the QL analog of quantum gates, can be used to prepare an arbitrary initial wavepacket and simulate the time evolution of the resulting state for extracting physical observables such as magnetization. Beyond the statevector dynamics, we introduce a set of metrics that characterize how QL-circuit operations transform the underlying graph connections. These metrics quantify the microscopic coupling, phase coherence, and effective inter-block coupling generated by the circuit. These graph-level quantities provide constraints and design targets for potential physical implementations, indicating where interconnection resources, phase control, and effective coupling strength must be engineered in a hardware realization. Overall, this study demonstrates how classical resources can be harnessed to simulate quantum systems and we discuss the advantages and limitations.
Abstract In recent work we have developed a theory for classical systems that mimic the state space of quantum systems. We call the states ‘quantum-like’ (QL) because they show many properties similar to those of a quantum system. Here we study the QL states of a large classical system comprising a network of coupled phase oscillators. In particular, we seek to understand how the nonlinearity of the classical system manifests in the linear representation of the state space. To that end, we study how a classical network is viewed from the perspective of the time evolution of its emergent state. We investigate how mixedness of the QL states diminish or increase as the underlying classical system synchronizes or de-synchronizes respectively. In the limit of a very strongly phase-locked classical network—that is, where couplings between phase oscillators are very large—the state space evolves according to unitary dynamics. When the system is set up so that the classical phase oscillators synchronize with time, we find that the purity measure of the QL states correspondingly increases. Whereas, in cases where the coupling between oscillators is too small to allow the network to synchronize, the classical variables act like a kind of environment that promotes dephasing of initially-prepared superpositions.
Recent publications have introduced the concept of quantum-like (QL) bits, along with their associated QL states and QL gate operations, which emerge from the dynamics of complex, synchronized networks. The present work extends these ideas to multi-level QL resources, referred to as QL dits, as higher-dimensional analogs of QL bits. We employ systems of k-regular graphs to construct QL-dits for arbitrary dimensions, where the emergent eigenspectrum of their adjacency matrices defines the QL-state space. The tensor product structure of multi-QL dit systems is realized through the Cartesian product of graphs. Furthermore, we examine the potential computational advantages of employing d-nary QL systems over two-level QL bit systems, particularly in terms of classical resource efficiency. Overall, this study generalizes the paradigm of using synchronized network dynamics for QL information processing to include higher-dimensional QL resources.
Quantum mechanics of composite systems, gives rise to certain special states called entangled states. A physical system, that is in an entangled state displays an intricate correlation between its subsystems. There are also some composite quantum states (classically correlated states or separable states) that are not entangled. It is generally claimed, often without a rigorous proof to support, that these intricate correlations of an entangled state cannot occur in a classical system. This expository article, provides an elementary proof that entangled states cannot arise in the setting of classical mechanics. In addition, a detailed description of the origin of entanglement in quantum systems is included. The mathematical concepts that are necessary for this purpose are presented. The absence of entanglement in the classical setting is due to the fact that every pure classical state of a composite system is a product state, that is, a tensor product of two pure states of the subsystems. In contrast, there are pure composite quantum states that cannot be expressed in the form of a product state or even by a convex sum of product states. Roughly speaking, this is because classical states are positive valued functions on the phase-space while quantum states are positive linear operators. The structure of the tensor product between two commutative spaces of scalar valued functions is drastically different from that of the tensor product between two non-commutative spaces of linear operators. In other words, entanglement is a non-commutative phenomenon.