Purification Complexity without Purifications

TL;DR

This paper extends the Fubini-Study approach to quantify the complexity of mixed quantum states using the Bures metric, showing that mixed-state complexity equals purification complexity without explicit purification, and is non-increasing under quantum operations.

Contribution

It introduces a generalized complexity measure for mixed states based on the Bures metric, linking it to purification complexity without explicit purification procedures.

Findings

Mixed-state complexity equals purification complexity for purified states.

Purification complexity is non-increasing under trace-preserving quantum operations.

Explicit example with Gaussian states illustrates the theoretical results.

Abstract

We generalize the Fubini-Study method for pure-state complexity to generic quantum 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 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.