Existence, structure, and properties of quantum-like states

Abstract

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.

Figures (4)

• Figure 1: (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 2: (a) Schematic representation of the graph formed by the Cartesian product of two QL bit graphs. (b) The same graph, but with edges added to indicate higher order correlations.
• Figure 3: 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.
• Figure 4: Comparison the graph product structure, tensor product construction of the basis of states, and the corresponding Hasse diagram for the set structure on the product states.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Proposition 1
• Theorem 1
• Lemma 1
• Definition 5
• Lemma 2
• Definition 6