Quantum-like states from classical systems

Abstract

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.

Figures (12)

• Figure 1: (a) This work studies maps between a suitably designed classical system (here Schrödinger's cat has been reinterpreted as a cockatoo by the artist) and a 'quantum-like' (QL) state space that mimics attributes of the state space of a quantum system. The map is mediated by a graph. The graph provides an abstract interpretation of the classical system allowing us to associate a state space with the classical system. (b) This perspective involves a mapping that takes properties of a classical system encoded with a suitable topology of phase relationships to a representation in a state space that has similar properties to a quantum state space.
• Figure 2: (a) Drawing of a small $d$-regular graph and the spectrum representative of a large $d$-regular graph, showing that the single emergent state is separated in the spectrum from the many other states that we refer to as 'random states'. (b) A QL bit is constructed by coupling together two $d$-regular subgraphs. The coupling edges, shown in red, are added randomly from each vertex in $G_{a1}$ to each vertex in $G_{a2}$ with probability 0.2. Realistically, the QL bit will likely not show the subgraphs separated in space, like we display here for clarity; instead the vertices can be positioned randomly. (c) Adjacency matrix of a QL bit showing the diagonal blocks hosting the adjacency matrices for each subgraph. These blocks are coupled by edges in the off-diagonal blocks labeled $c$ that hybridize the subgraphs.
• Figure 3: Examples of graph Cartesian products and corresponding spectra for (a) $C_5 \Box C_5$ (b) $C_5 \Box C_5 \Box C_5$ (c) $C_5 \Box C_5 \Box C_5 \Box C_5$. (d) Procedure for the physical construction of the product $C_5 \Box C_5$, see text. The vertices of each base graph ($C_5$) are labeled $0, 1, \dots, 4$. The vertices of the product graph $G \Box H$ are, accordingly, $(i,j)$, where $i$ labels a vertex in $G$ and $j$ labels a vertex in $H$.
• Figure 4: (a) Calculated spectra of $G_1 \Box G_2 \Box G_3$ where the base graphs are disordered $d$-regular random graphs. The plot shows the ensemble spectrum for a graph with diagonal ('frequency') disorder $\sigma = 2.0$. The emergent state is produced by the tensor product of emergent states of each graph. Other states are various products of sets of random states, denoted $\{r\}$, and emergent states, as indicated. (b) Spectrum of the product of two QL bits. Reprinted from G. D. Scholes and G. Amati, 2025, Quantumlike Product States Constructed from Classical Networks, Phys. Rev. Lett. 134:060202.
• Figure 5: Outline of how the Cartesian product of QL bits, where each $d$-regular subgraph comprises $n$ vertices, produces a much larger graph, where each subgraph comprises $n^2$ vertices and is $2d$-regular. This large graph can be contracted to an optimal graph, where each $d$-regular subgraph comprises $n$ vertices, that retains the qualitative features of the large graph.

Theorems & Definitions (8)

• Definition 1
• Definition 2
• Proposition 1
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7