Purification complexity without purifications

Abstract

We generalize the Fubini-Study method for pure-state complexity to generic quantum states by taking Bures metric or quantum Fisher information metric (QFIM) 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 purification. We also find the purification complexity is non-increasing under any trace-preserving quantum operations. We also study the mixed Gaussian states as an example to explicitly illustrate our conclusions for purification complexity.

Quantum extremal islands made easy. Part III. Complexity on the brane

“…From the viewpoint of circuit complexity, a natural definition of mixed-state complexity is the so-called purification complexity, which can be understood as the complexity of the optimal purification. That is, one identifies the purification of the mixed state in question with the smallest complexity, e.g., see [53][54][55][56] for recent studies. Let us add, however, that there are other possible definitions of circuit complexity of mixed states which do not require the concept of purifications [53,54,57].…”

“…However, we should recall that the quantum term C QFT (R ∪ B) also includes the UV divergent terms associated with the cut-off surface near the asymptotic AdS boundary. These are less interesting for our purposes and so we point out that they can be eliminated by considering the mutual complexity, e.g., [24,[53][54][55]…”

Subsystem complexity after a global quantum quench

“…In order to understand the relation between entanglement and complexity, it is useful to study the optimal circuits and the corresponding circuit complexity when both the reference and the target states are mixed states [75][76][77][78][79]. The approach to the complexity of mixed states based on the purification complexity [75,76,79] is general, but evaluating JHEP05(2021)022 this quantity for large systems is technically complicated.…”

Islands and complexity of eternal black hole and radiation subsystems for a doubly holographic model

“…For details see[73][74][75][76][77][78][79] and references therein 3. It is worth noting that in their paper, the authors argued that for a gravitating bath, one does not get a Page curve and instead end up with a constant entanglement curve.…”