Global asymmetry of many-qubit correlations: A lattice-gauge-theory approach

We introduce a bridge between the familiar gauge field theory approaches used in many areas of modern physics such as quantum field theory and the stochastic local operations and classical communication protocols familiar in quantum information. Although the mathematical methods are the same, the meaning of the gauge group is different. The measure we introduce, “twist,” is constructed as a Wilson loop from a correlation-induced holonomy. The measure can be understood as the global asymmetry of the bipartite correlations in a loop of three or more qubits; if the holonomy is trivial (the identity matrix), the bipartite correlations can be globally untwisted using general local qubit operations, the gauge group of our theory, which turns out to be the group of Lorentz transformations familiar from special relativity. If it is not possible to globally untwist the bipartite correlations in a state using local operations, the twistedness is given by a nontrivial element of the Lorentz group, the correlation-induced holonomy. We provide several analytical examples of twisted and untwisted states for three qubits, the most elementary nontrivial loop one can imagine.

Figure 1

Graphical representation of twist $\xi \left(abc\right)$ for a state of three qubits. The ribbons between the qubits represent the twist—the degree of asymmetry in the correlations in each two-qubit link. The twist in each link is not invariant by itself, however. One can untwist each link by careful choice of local gauge transformation (representing a local operation) $U\in {\text{SO}}^{+}(1,3)$. (b) Local transformation on $b$ that results in the link $\Lambda (b,a)$ being the identity $\Lambda (b,a)\to U_b\Lambda (b,a)$. Similarly, (c) corresponds to the local transformation on qubit $c$ that untwists $\Lambda (c,b)U_b^{-1}$; i.e., $\Lambda (c,b)U_b^{-1}\to U_c\Lambda (c,b)U_b^{-1}$. One can still perform a local transformation on qubit $a$; however, the total twist in the link does not change. Even though locally we can untwist each link, we cannot gauge away the twist globally.