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.
Highlights
Introduction and motivationsQuantum information concepts and perspectives have inspired surprising new insights into the understanding of the gravitational holography, e.g., [1,2,3,4,5,6,7,8]
Similar to the non-decreasing property of Uhlmann’s fidelity in (2.19), we summarize our observation as an universal conclusion that the purification complexity CIM derived from the quantum Fisher information metric is non-increasing under any trace-preserving quantum operations acting on the reference state and target state simultaneously, i.e., CIM ≥ CIM = CIM U σAU †, U ρAU † ≥ CIM (E, E) . (2.33)
We have shown the equivalence between purification complexity PFS and mixed-state complexity CIM based on the quantum Fisher information metric
Summary
Quantum information concepts and perspectives have inspired surprising new insights into the understanding of the gravitational holography, e.g., [1,2,3,4,5,6,7,8]. Due to the absence of a well-posed definition for the boundary dual of holographic complexity, some progresses have been made toward defining the computational complexity of states in quantum field theory in recent years, e.g., Nielsen’s geometric method [10, 14, 15], Fubini-Study method [16] and path-integral complexity proposal [17, 18]. In view of the difficulties in the minimization for purification complexity, we would like to generalize the Fubini-Study metric method for pure-state complexity to arbitrary quantum states ρA by defining the geodesic distance in the space of density matrix equipped with a special metric as the complexity measure for mixed states.
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