bibkey: wu2024distributedimaginarity
title: Resource theory of imaginarity in distributed scenarios
authors: Kang-Da Wu, Tulja Varun Kondra, Carlo Maria Scandolo, Swapan Rana, Guo-Yong Xiang, Chuan-Feng Li, Guang-Can Guo, Alexander Streltsov
year: 2024
url: https://arxiv.org/abs/2301.04782v1
doi: 10.1038/s42005-024-01649-y
claim: An explicitly real joint Kraus construction implements a given complex operation when supplied with a maximally imaginary qubit; the construction preserves that supplied qubit.
strata_touched: []
license: citation-only
triage: anchor

Imbit-assisted real implementation

The journal locator is Communications Physics 7, 171, Eq. (11), the real Kraus construction immediately preceding it and the completeness argument immediately following it. The corresponding arXiv locator is arXiv:2301.04782v1, Resource Theory of Imaginarity: New Distributed Scenarios, Appendix A, equations (22)–(23).

For complex Kraus matrices $K_j$, let $|\hat{\pm }⟩=(|0⟩\pm i|1⟩)/\sqrt{2}$ and $P_{\pm }=|\hat{\pm }⟩⟨\hat{\pm }|$. The source constructs

$$L_j=K_j\otimes P_{+}+\overline{K_j}\otimes P_{-}.$$

These matrices are real, because the two terms exchange under entrywise conjugation. Their completeness operator is $P\otimes P_{+}+P^{\mathsf{T}}\otimes P_{-}$, where $P={\sum }_j{K}_j^{†}{K}_j\le I$. With the supplied program $P_{+}$, the joint operation produces ${\sum }_j{K}_j\rho K_j^{†}\otimes P_{+}$. This is a real operation on the joint complex algebra. It does not physically conjugate an unknown input.

The use in the five-mode source problem is a restricted controlled real rotation, together with a direct calculation for a mixed qubit extracted from an actual source copy. A further source-specific circuit retains one imaginary-sign qubit while measuring relative signs of the actual extracted qubits using controlled real rotations and real Hadamards. Its mixed-source orbit weights and conditional response budgets are proved in the owning five-mode continuation, rather than supplied by this literature theorem. The literature theorem does not provide that source, select its unknown sign, distill a perfect imbit from its noisy states, or establish a simultaneous unknown-label processor. Those are separate source-specific obligations. Its exact finite real controls and joint interaction remain declared services, rather than native operations supplied by a classical description.

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