Circuit Complexity of Mixed States
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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.
Cite this version of the work
Shan-Ming Ruan (2021). Circuit Complexity of Mixed States. UWSpace. http://hdl.handle.net/10012/17148