Circuit Complexity of Mixed States

Abstract

Quantum information has produced fresh insights into foundational questions about the AdS/CFT correspondence. One fascinating concept, which has captured increasing attention, is quantum circuit complexity. As a natural generalization for the complexity of pure states, we investigate the circuit complexity of mixed states in this thesis.

First of all, we explore the so-called purification complexity which is defined as the lowest value of the circuit complexity, optimized over all possible purifications of a given mixed state. We focus on studying the complexity of Gaussian mixed states in a free scalar field theory using the ‘purification complexity’. 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. In order to compare with the results from using the various holographic proposals for the complexity of subregions, we explore the purification complexity for thermal states of a free scalar QFT, and for subregions of the vacuum state in two dimensions. 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 thesis.

In addition, we propose to generalize the Fubini-Study method for pure-state complexity to generic quantum states including mixed states by taking Bures metric or quantum Fisher information metric on the space of density matrices as the complexity measure. Due to Uhlmann’s theorem, we show that the mixed-state complexity exactly equals the purification complexity measured by the Fubini-Study metric for purified states but without explicitly applying any purifications. We also find that the purification complexity is nonincreasing under any trace-preserving quantum operations. As an illustration, we study the mixed Gaussian states as an example to explicitly show our conclusions for purification complexity.