Month: January 2026

Topological Quantum Color Code Model on Infinite Lattice

Our work, topological quantum color code model on infinite lattices, co-authored with Shiyu Cao and Sheng Tan, has been posted on arXiv1. In this study, we carry out a rigorous Doplicher–Roberts–Haag (DHR) analysis for the quantum color code model—a framework central to understanding the structure of topological order in infinite lattice.

It is known that the topological phases of the color code model are equivalent to those of the double-layer toric code model. In this work, we present a rigorous proof of this equivalence within the framework of quasi-local C*-algebras.

We classify the model’s irreducible anyon superselection sectors and construct explicit string operators that generate anyonic excitations from the ground state. We further examine the fusion and braiding properties of these excitations and show that the resulting category is equivalent to \mathsf{Rep}(D(\mathbb{Z}_2 \times \mathbb{Z}_2)). We also prove the Haag duality for the gound state of color code model.

  1. Since I have recently been moving and settling down in Changsha and have been somewhat nomadic, this post comes out a little bit late. ↩︎

Temporal Kirkwood-Dirac Quasiprobability Distribution and Unification of Temporal State Formalisms through Temporal Bloch Tomography

Recently, we post the paper “temporal kirkwood-dirac quasiprobability distribution and unification of temporal state formalisms through temporal bloch tomography” on arXiv. This is joint work with Kavan Modi and Dagomir Kaszlikowski, in which we extend the Kirkwood–Dirac (KD) quasiprobability distribution to the temporal setting and develop a unified framework for temporal states.

The traditional (spatial) KD quasiprobability distribution is defined for density operators, which serve as a phase-space–like representation of a quantum state (see this post for a brief introduction). In this work, we generalize the KD formalism to temporal quantum processes, which consist of an initial state together with a sequence of quantum evolutions, \mathfrak{P} = (\rho_{t_0}, \mathcal{E}_{t_1 \leftarrow t_0}, \ldots, \mathcal{E}_{t_n \leftarrow t_{n-1}}), where \rho_{t_0} is the initial state and each \mathcal{E}_{t_j \leftarrow t_{j-1}} denotes a completely positive trace-preserving (CPTP) map describing the evolution from t_{j-1} to t_j. The system state at an intermediate time t_k is then obtained recursively as \rho_{t_k} = \mathcal{E}_{t_k \leftarrow t_{k-1}} \circ \cdots \circ \mathcal{E}_{t_1 \leftarrow t_0}(\rho_{t_0}).

Within this framework, we define the multi-time doubled temporal KD quasiprobability distribution by

\overleftrightarrow{Q}_{KD}(a_n,\ldots,a_0; b_n,\ldots,b_0)

= \mathrm{Tr}[ \Pi_{a_n|A_n}^{t_n} \mathcal{E}_{t_n \leftarrow t_{n-1}}(\cdots \Pi_{a_1|A_1}^{t_1}( \mathcal{E}_{t_1 \leftarrow t_0}( \Pi_{a_0|A_0}^{t_0} \rho_{t_0} \Pi_{b_0|B_0}^{t_0})) \Pi_{b_1|B_1}^{t_1} \cdots) \Pi_{b_n|B_n}^{t_n}]

By taking the left and right marginals of this doubled distribution, we recover the corresponding temporal KD quasiprobability distributions \overleftarrow{Q}_{KD}(a_n,\ldots,a_0) and \overrightarrow{Q}_{KD}(b_n,\ldots,b_0). These two distributions are related by complex conjugation, reflecting the intrinsic temporal structure encoded in the doubled formalism. The temporal Margenau-Hill (MH) quasiprobability distribution is defined as real part of the temporal KD distribution.

The temporal KD quasiprobability may take values outside the interval [0,1]. This phenomenon reflects the intrinsically quantum nature of multi-time processes, in direct analogy with the negativity and nonclassical features of the spatial KD quasiprobability distribution. Using the Heisenberg picture, we provide a general and unified explanation for the origin of this quantumness.

A crucial application of temporal KD quasiprobability distribution is that it provides a general framework for temporal state.

There exist several formalisms of temporal states. In our paper, we show that using the temporal KD quasiprobability distribution, we can compute correlators of Pauli operators at different time steps. Then, by applying the Bloch representation of the state, we can derive several distinct temporal state formalisms. This provides a unified picture of temporal states.

For example, with doubled temporal KD distribution, we obtain the doubled KD temporal state:

\overleftrightarrow{\Upsilon}= \frac{1}{d^{n+1}}\sum_{\mu_0,\ldots,\mu_n \atop \nu_0,\ldots,\nu_n=0}^{d^2-1}\overleftrightarrow{T}^{\mu_n,\ldots,\mu_0;\,\nu_n,\ldots,\nu_0}\left(\bigotimes_{i=0}^n \sigma_{\mu_i}\right)\otimes\left(\bigotimes_{j=0}^n \sigma_{\nu_j}\right)

This is the most information-complete temporal state, and we show that it coincides with the doubled density operator. The temporal states obtained from the left or right temporal KD distribution correspond to the left and right reduced states of the doubled KD temporal state. Similarly, the left and right MH distributions coincide and give the left/right MH temporal states. For the two-time case, the left and right MH temporal states are identical to the pseudo-density operator; however, for general multi-time processes, they are no longer the same. In general, the MH temporal states can be understood as the Hermitianization of the corresponding KD temporal state.