Complexity of Mixed States in QFT and Holography

TL;DR

The paper develops purification complexity for Gaussian mixed states in a free scalar QFT, defining the minimal circuit complexity over all purifications and focusing on essential ancillae and mode-by-mode purifications. It provides analytic results for a single oscillator, elucidates diagonal vs physical basis dependencies, and extends to many-mode Gaussian states, including thermal states and vacuum subregions in 2D, with careful treatment of degeneracies and minimal ancilla counts. By comparing to holographic subregion proposals (CV, CA, CV2.0), the work reveals qualitative similarities in volume-law divergences and mutual complexity structures, while highlighting basis-dependent differences in subleading terms and sign patterns. The results demonstrate that purification complexity encodes information beyond entanglement entropy and offers a bridge between QFT and holography, guiding understanding of mixed-state complexity in field theories. The study lays groundwork for future exploration of alternate cost functions, non-Gaussian purifications, and interacting theories.

Abstract

We study the complexity of Gaussian mixed states in a free scalar field theory using the 'purification complexity'. The latter is defined as the lowest value of the circuit complexity, optimized over all possible purifications of a given mixed state. We argue that the optimal purifications only contain the essential number of ancillary degrees of freedom necessary in order to purify the mixed state. We also introduce the concept of 'mode-by-mode purifications' where each mode in the mixed state is purified separately and examine the extent to which such purifications are optimal. We explore the purification complexity for thermal states of a free scalar QFT in any number of dimensions, and for subregions of the vacuum state in two dimensions. We compare our results to those found using the various holographic proposals for the complexity of subregions. We find a number of qualitative similarities between the two in terms of the structure of divergences and the presence of a volume law. We also examine the 'mutual complexity' in the various cases studied in this paper.

Figures (27)

• Figure 1: Circuit with the ancillary degrees of freedom. The mixed state $\hat{\rho}_\mathcal{A}$ that we want to prepare is obtained at the final step after tracing out the ancillae.
• Figure 2: Possible values for the pure state complexity $\mathcal{C}_1^{\textrm{\tiny phys}}(|\psi_{12}\rangle)$ in the physical basis as a function of $\theta$, for all possible sign combinations according to eq. \ref{['complexitypositionabcd']} for fixed values of $\bar{r}$ and $\alpha$. The complexity of the mixed state purified by $|\psi_{12}\rangle$ is obtained by minimizing over the uppermost envelope of each of these plots.
• Figure 3: The difference between the complexity obtained for $\theta_c$ at the intersection of cases (a) and (c) and the exact purification complexity of one-mode Gaussian states in the physical basis $\mathcal{C}_{1,c}^{\textrm{\tiny phys}}$ ρ̂_1$-\mathcal{C}_1^{\textrm{\tiny phys}}$ ρ̂_1$$ as a function of $\bar{r}$ for some fixed values of $\alpha$. We see that the complexity obtained at the intersection between cases (a) and (c) with $\mathcal{C}_{1,c}^{\textrm{\tiny phys}}$ ρ̂_1$$ in eq. \ref{['pos_ac']} ceases to be optimal for some region of the parameter $\bar{r}$ for large enough values of $\alpha$.
• Figure 4: Purification complexity of one-mode Gaussian states in the physical basis $\mathcal{C}_{1}^{\textrm{\tiny phys}}$ ρ̂_1$$ as a function of $\alpha$ for some fixed values of $\bar{r}$. The fact that the curves with $\bar{r}=-6$ and $\bar{r}=-10$ coincide after a certain value of $\alpha$ is due to the fact that this minimization is obtained at the minimum of case (c) which is $\bar{r}$ independent.
• Figure 5: Illustration of the different ways to purify a multi-mode Gaussian state $\hat{\rho}_A$. We refer to the purifications of the form $\Psi_{11^c} \otimes \Psi_{22^c} \otimes \cdots \otimes \Psi_{NN^c}$ as mode-by-mode purifications.