Quantum State Recovery via Direct Sum Formalism Without Measurement Outcomes
Taiga Suzuki, Masayuki Ohzeki
TL;DR
The paper tackles recovering a quantum state after measurement without relying on the measurement outcome. It introduces a direct-sum Hilbert space framework to transfer the state's amplitude into the orthogonal complement (a quasi-copy) and perform a measurement there, enabling probabilistic recovery of the original state with no dependence on the observed outcome. A four-step protocol is constructed using Kraus operators on the direct-sum space, yielding a recovery probability $P[\text{rev}] = \cos^2\phi$ that is independent of the measurement result. The authors show that information gained by the initial measurement is erased upon successful reversal, and that the information-gain vs reversibility trade-off matches the known QRM bound, highlighting a universal aspect of quantum measurement reversibility. The work provides operational and informational insights and suggests potential simplifications for experiments exploiting direct-sum structures in quantum information processing.
Abstract
This study proposes a new approach to quantum state recovery following measurement. Specifically, we introduce a special operation that transfers the probability amplitude of the quantum state into its orthogonal complement. This operation is followed by a measurement performed on this orthogonal subspace, enabling the undisturbed original quantum state to be regained. Remarkably, this recovery is achieved without dependence of the post-measurement operation on the measurement outcome, thus allowing the recovery without historical dependence. This constitutes a highly nontrivial phenomenon. From the operational perspective, as the no-cloning theorem forbids perfect and probabilistic cloning of arbitrary quantum states, and traditional post-measurement reversal methods typically rely on operations contingent on the measurement outcomes, it questions fundamental assumptions regarding the necessity of historic dependence. From an informational perspective, since this recovery method erases the information about the measurement outcome, it's intriguing that the information can be erased without accessing the measurement outcome. These results imply the operational and informational non-triviality formulated in a direct-sum Hilbert space framework.