Layer-by-layer disentangling two-dimensional topological quantum codes

While local unitary transformations are used for identifying quantum states that are in the same topological class, nonlocal unitary transformations are also important for studying the transition between different topological classes. In particular, it is an important task to find suitable nonlocal transformations that systematically sweep different topological classes. Here, regarding the role of dimension in the topological classes, we introduce partially local unitary transformations, namely Greenberger-Horne-Zeilinger (GHZ) disentanglers, which reduce the dimension of the initial topological model by a layer-by-layer disentangling mechanism. We justify the importance of such a mechanism for characterizing topological phases by considering two important examples. To this end, we apply GHZ disentanglers to two-dimensional (2D) topological quantum codes and show that they are converted to many copies of Kitaev's ladders. It implies that the above disentanglers cause a transition from an intrinsic topological phase to a symmetry-protected topological phase. Then, we show that while Kitaev's ladders are building blocks of both color code and toric code, there are different patterns of entangling ladders in 2D color code and toric code. It shows that different topological features of these topological codes are reflected in different patterns of entangling ladders. In this regard, we propose that the layer-by-layer disentangling mechanism can be used as a systematic method for classification of topological orders based on finding different patterns of long-range entanglement in topological lattice models.