bibkey: wu2021imaginarity title: Operational Resource Theory of Imaginarity authors: Kang-Da Wu, Tulja Varun Kondra, Swapan Rana, Carlo Maria Scandolo, Guo-Yong Xiang, Chuan-Feng Li, Guang-Can Guo, Alexander Streltsov year: 2021 url: https://arxiv.org/abs/2007.14847v2 doi: 10.1103/PhysRevLett.126.090401 claim: Relative to a fixed basis, the robustness of imaginarity equals half the trace norm of the difference between a state and its transpose; real operations admit real Kraus operators. strata_touched: [] license: citation-only triage: anchor

Fixed real structure and imaginarity

Wu et al., arXiv:2007.14847v2, dated 2 March 2021, define real states relative to a specified basis and real operations through Kraus operators with real entries. The setup on page 2 follows the resource theory of Hickey and Gour, Quantifying the imaginarity of quantum mechanics, Journal of Physics A 51, 414009 (2018).

Theorem 2, equation (5), gives

$$I_R(\rho )=\frac{1}{2}\Vert \rho -{\rho }^{\mathsf{T}}{\Vert }_1,F_I(\rho )=\frac{1+I_R(\rho )}{2}.$$

Here $I_R$ is the robustness defined in equation (4), and $F_I$ is the maximal fidelity with a maximally imaginary qubit under real operations, defined in equation (3). For a Hermitian density matrix, ${\rho }^{\mathsf{T}}=\overline{\rho }$. The measure and identity are existing results. They depend on the chosen real structure and are not lower bounds over all changes of basis.

Theorem 1, equation (2), describes stochastic pure-state conversion under real operations using $|⟨{\psi }^{\ast }|\psi ⟩|$. Neither theorem asserts that an entire parameter-dependent preparation can be made real by one fixed coordinate change. Nor do they determine the minimum dimension of an exact programmable channel with prescribed Fisher-calibration and curvature data.

The quantum-volume comparison concerns one fixed conjugation preserved by every state of a preparation curve, while allowing an arbitrary CPTP processor. Conjugating and averaging that processor is justified separately by the target channel’s conjugation covariance. A minimum dimension gap additionally requires the channel-specific sharp-state rigidity and continuation obstruction. A quantitative use of $I_R$ requires a lower bound derived from those constraints; the trace-norm identity alone does not provide one.

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