Complexity for quantum field theory states and applications to thermofield double states

This paper studies the complexity between states in quantum field theory by introducing a Finsler structure based on ladder operators (the generalization of creation and annihilation operators). Two simple models are shown as examples to clarify the differences between complexity and other conceptions such as complexity of formation and entanglement entropy. When it is applied into thermofield double (TFD) states in $d$-dimensional conformal field theory, results show that the complexity density between them and corresponding vacuum states are finite and proportional to $T^{d-1}$, where $T$ is the temperature of TFD state. Especially, a proof is given to show that fidelity susceptibility of a TFD state is equivalent to the complexity between it and corresponding vacuum state, which gives an explanation why they may share the same object in holographic duality. Some enlightenments to holographic conjectures of complexity are also discussed.

Figure 1

The schematic explanations about the triangle inequality. ${\varphi }_{1R}$ is one quantum circuit of minimal gates to realize $|{\psi }_1⟩\to |R⟩$, ${\varphi }_{R2}$ is one quantum circuit of minimal gates to realize $|R⟩\to |{\psi }_2⟩$, and ${\varphi }_{12}$ is one quantum circuit of minimal gates to realize $|{\psi }_1⟩\to |{\psi }_2⟩$. As the combination ${\varphi }_{R2}\circ {\varphi }_{1R}$ is a possible quantum circuit to realize $|{\psi }_1⟩\to |{\psi }_2⟩$ with the gates number $\mathcal{C}(|{\psi }_2⟩,|R⟩)+\mathcal{C}(|R⟩,|{\psi }_1⟩)$ which should be larger than or equal to the gates number of ${\varphi }_{12}$, we see that $\mathcal{C}(|{\psi }_2⟩,|R⟩)+\mathcal{C}(|R⟩,|{\psi }_1⟩)\ge \mathcal{C}(|{\psi }_2⟩,|{\psi }_1⟩)$.

Figure 2

The schematic example that the value of $\mathcal{C}(\widehat{U})$ depends on the choice of $\mathcal{U}$. The operators set $\mathcal{U}$ is a subset of a larger operators set ${\mathcal{U}}^{\prime }$. The curve length is just given by Euclidean metric. For the operators set $\mathcal{U}$, the shortest curve from $\widehat{I}$ to $\widehat{U}$ is given by ${\widehat{c}}_1$. However, if one extend the operators set $\mathcal{U}$ to ${\mathcal{U}}^{\prime }$, the shortest curve from $\widehat{I}$ to $\widehat{U}$ becomes ${\widehat{c}}_2$. This shows that the complexity of $\widehat{U}$ depends on the choice of operators set.

Figure 3

Curves ${\varphi }_{IM},{\varphi }_{MF},{\varphi }_1,{\varphi }_2,{\varphi }_3,\dots$ stand for the possible quantum circuits which can covert the initial state $|I⟩$ into the finial state $|F⟩$. The medial state $|M⟩$ is the necessary state that all the physically realizable quantum circuit will bring the initial state $|I⟩$ into the medial state $|M⟩$ before it reaches the finial state $|F⟩$. The black dashed curve $\tilde{\varphi }$ stands for a quantum circuit which can connect states $|I⟩$ and $|F⟩$ without passing through the medial state $|M⟩$. But this curve is forbidden by some physical rules.

Figure 4

The schematic explanation about why the complexity from $|A⟩$ to a TFD state is finite. The left and right sides are MERA approximations for states $|A⟩$ and $|\mathrm{TFD}⟩$ in tensor network representation. The some green rectangles at the middle of two tensor networks stand for the quantum circuit. The horizonal direction stands for the ($d-1$)-dimensional spatial directions and the vertical direction is the length scale (inverse of momentum). For convenience, the spatial direction and momentum are shown in 1-dimensional case and only one copy of a double state is shown in the figure. One can image that the $i$th layer is the state $|{\psi }_{k_i}⟩≔\sum _{n=0}^{\infty }{e}^{-\pi n{\omega }_{{\overrightarrow{k}}_i}/a}|n_i{⟩}_L|n_i{⟩}_R$.