Spin networks of quantum channels

TL;DR

This work extends spin networks from unitary SU(2) holonomies to open-system dynamics by promoting edges to CPTP maps described by Kraus operators $\hat{K}_i$, while preserving gauge invariance. Through vectorization, the authors express CPTP maps as $\hat{\Lambda}=\sum_i\hat{K}_i^*\otimes\hat{K}_i$ acting on $\mathrm{vec}(\hat{\rho})$, enabling a generalized spin-network formalism with functions $\Psi[ A ]=\psi(\hat{\Lambda}_{e})$ and a Liouville-space projection ensuring gauge invariance. They define a consistent kinematical Hilbert space, $\mathcal{H}_{\rm kin}=L^2\left(\prod_e CPTP_{r_e},\prod_e d\Lambda_{r_e}\right)$, and illustrate the construction with a Wilson loop $W[\hat{\Lambda}]=\mathrm{tr}(\hat{\Lambda})$ and a dipole spin network, highlighting how unitary limits recover the standard spin-network framework. The formalism accommodates environmental effects and provides a path toward studying decoherence and potential emergent sectors in quantum gravity, with potential applications to cosmology and black-hole thermodynamics.

Abstract

Spin networks in Loop Quantum Gravity are traditionally described by unitary holonomies corresponding to noiseless transformations. In this work, we extend this framework to incorporate general quantum channels that model effects of environment, which can become significant at the Planck scale. Specifically, we demonstrate that the transformation properties of Kraus operators, which define completely positive trace-preserving (CPTP) maps, are consistent with the gauge invariance of spin networks. This enables the introduction of generalized spin network states that can be expressed in terms of the Kraus operators. Furthermore, the associated notion of an inner product is proposed, allowing for introduction of the Hilbert space. We illustrate these constructions with examples involving a Wilson loop and a dipole network.

Figures (5)

• Figure 1: Pictorial representation of the extention introduced in this work.
• Figure 2: A diagram of mappings used in our formulation.
• Figure 3: Results of Monte Carlo integration of Wilson loop norm of $\hat{\Lambda}_r$ for $r=1$ corresponding to unitary evolution and for $r=2$ for rank 2 environment, which encapsulates, e.g., the bit flip model considered earlier. Both results are considered for a fundamental representation, i.e., 2-dimensional system corresponding to the fundamental representation. Obtained values converge within a feasible number of Monte Carlo trials. For the unitary case results coincide with the theoretical calculations. Horizontal dashed lines are drawn for values 2 and $\sqrt{2}$.
• Figure 4: Relation between square of the norm of Wilson loop and rank of the $\hat{\Lambda}_r$ operator. Computed values follow a $2^{1/r}$ trend. As the degrees of freedom of environment increase, the probability gets more diluted. Results obtained for 10000 samples for each point.
• Figure 5: A dipole spin network state in the generalized description.