Abstract
We consider an environment for an open quantum system described by a ‘quantum network geometry with flavor’ (QNGF) in which the nodes are coupled quantum oscillators. The geometrical nature of QNGF is reflected in the spectral properties of the Laplacian matrix of the network which display a finite spectral dimension, determining also the frequencies of the normal modes of QNGFs. We show that an a priori unknown spectral dimension can be indirectly estimated by coupling an auxiliary open quantum system to the network and probing the normal mode frequencies in the low frequency regime. We find that the network parameters do not affect the estimate; in this sense it is a property of the network geometry, rather than the values of, e.g., oscillator bare frequencies or the constant coupling strength. Numerical evidence suggests that the estimate is also robust both to small changes in the high frequency cutoff and noisy or missing normal mode frequencies. We propose to couple the auxiliary system to a subset of network nodes with random coupling strengths to reveal and resolve a sufficiently large subset of normal mode frequencies.
Highlights
Networks [1, 2] describe discrete topologies that can capture the architecture of complex systems, from the brain to societies
In this paper we investigate the Quantum Network Geometry with Flavor” (QNGF) which is the environment of a quantum open system and is formed by a network of quantum harmonic oscillators coupled according to the topology of the Network Geometry with Flavor (NGF)
Finite spectral dimension is characteristic of network geometries
Summary
Networks [1, 2] describe discrete topologies that can capture the architecture of complex systems, from the brain to societies. For any dimension d and any flavor s the NGFs capture the main characteristics of complex networks including modularity, smallworld properties and heterogeneous degree distribution These topologies display a finite spectral dimension which is the signature of their geometrical nature. The experimental implementation of the theoretical framework and the probing schemes, have been recently proposed using a multi-mode optical set-up [24] Building on these results, in this paper we investigate the QNGF which is the environment of a quantum open system (the probe) and is formed by a network of quantum harmonic oscillators coupled according to the topology of the NGF. Using the theory of open quantum systems combined with data science we are able to probe the value of the spectral dimension of the QNGF.
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