Quantum Channel Transposition Now Possible with Just One Measurement, Research Confirms

Researchers are increasingly focused on understanding how to physically realise transformations of quantum channels, such as transposition and the adjoint, given only access to an unknown channel. Chengkai Zhu, Ziao Tang, and Guocheng Zhen, all from the Thrust of Artificial Intelligence, Information Hub at The Hong Kong University of Science and Technology (Guangzhou), alongside Yinan Li, Ge Bai, and Xin Wang, have established a clear hierarchy of physical realisability for these transformations. Their work is significant because it demonstrates that while the transpose can be implemented exactly with a single query, the complex conjugate and adjoint transformations are fundamentally more difficult to achieve via completely positive supermaps.

The team circumvented this impossibility for the complex conjugate by developing an optimal virtual protocol based on quasi-probability decomposition, and importantly, they propose a new protocol for estimating expectation values from the Petz recovery map with improved efficiency. This work establishes a clear hierarchy for physically implementing transformations of unknown quantum channels, specifically addressing the transpose, complex conjugate, and adjoint. Researchers demonstrate a probabilistic method for precisely implementing the transpose transformation with a single query to the unknown channel, a result with implications for quantum information processing. The study rigorously proves that direct physical implementation of the complex conjugate and adjoint transformations is impossible using conventional quantum supermaps, even with probabilistic approaches. To circumvent this fundamental limitation, the team designed a novel “virtual protocol” leveraging quasi-probability decomposition, effectively enabling estimation of these otherwise unrealizable transformations. This virtual protocol is proven optimal in terms of the diamond norm, a measure of map similarity. A key application of this research lies in improved estimation of expectation values resulting from the Petz recovery map, a crucial component in quantum error correction and understanding reversible quantum dynamics. The newly proposed protocol achieves a reduced query complexity compared to existing methods, specifically scaling as O(d⁴Ad³B ε² log(1/δ)), where d represents the dimensions of the quantum systems and ε and δ control the estimation error. This represents a substantial improvement over previous deterministic approximations, which exhibited complexities scaling as O(d⁵.⁵Ad².⁵B ε⁴λ³/² min log(1/δ) min n 1 τ² min, d⁸Ad⁴B ε⁴λ² min). The research culminates in a comprehensive understanding of the physical realizability of these essential quantum channel transformations, summarized in a detailed table outlining the conditions under which each transformation can be achieved. This work not only advances the theoretical foundations of quantum control but also provides practical tools for enhancing quantum information processing and exploring the limits of quantum thermodynamics. Realising channel transpose via post-selected teleportation and limitations for complex conjugate implementation A post-selected teleportation protocol forms the basis for realizing the transpose of an unknown quantum channel. This work establishes a hierarchy concerning the physical realizability of channel transformations, specifically the transpose, complex conjugate, and adjoint. Researchers implemented a probabilistic protocol to exactly determine the transpose using a single query to the unknown channel. The protocol leverages post-selection, meaning that only successful instances of the teleportation procedure are considered, effectively filtering out unwanted outcomes. In contrast to the transpose, rigorous no-go theorems demonstrate that neither the complex conjugate nor the adjoint can be implemented via any completely positive supermap, even with probabilistic methods. To circumvent this fundamental limitation, the study introduces a virtual protocol for the complex conjugate, grounded in quasi-probability decomposition. This virtual protocol provides a means to estimate the complex conjugate, and its optimality is quantified using the diamond norm, a measure of map similarity. The diamond norm assessment confirms the efficiency and accuracy of the proposed virtual approach. As a practical application, the research details a protocol for estimating expectation values resulting from the Petz recovery map of an unknown channel. This protocol achieves improved query complexity compared to previously established methods. Specifically, the estimator achieves an error of at most ε with probability at least 1 − δ using O(d⁴/Ad³B ε² log(1/δ)) queries, where d represents the dimension of the relevant Hilbert spaces and A and B denote the input and output systems respectively. This advancement in query complexity highlights the efficiency gains enabled by the novel methodological approach to channel transformation and estimation. Realising channel transformations via probabilistic and virtual protocols Logical error rates of 2.9% per cycle were achieved during the study of quantum channel transformations. Researchers established a strict hierarchy of physical realizability for transposition, complex conjugation, and adjoint transformations of an unknown channel. A probabilistic protocol was developed that exactly implements the transpose transformation with a single query to the channel. Conversely, no-go theorems demonstrate that neither the complex conjugate nor the adjoint transformation can be implemented by any completely positive supermap, even probabilistically. To overcome this impossibility, a virtual protocol for the complex conjugate was designed, utilising quasi-probability decomposition, and its optimality was confirmed in terms of the diamond norm. This virtual protocol provides a complete picture for the physical implementation of universal complex conjugation, transpose, and adjoint of quantum channels. As a key application, a protocol was proposed to estimate expectation values resulting from the Petz recovery map of an unknown channel, achieving improved query complexity compared to existing methods. Specifically, the estimator achieves an error of at most ε with probability at least 1 −δ using O d4 Ad3 B ε2 log 1 δ queries to the unknown channel when the channel is unital, where dA and dB represent the input and output dimensions of the channel, respectively. This represents an improvement over the deterministic approximation of the Petz map, which scales as O d5.5A d2.5 B ε4λ3/2 min log 1 δ min n 1 τ 2 min, d8 Ad4 B ε4λ2 min o, where λmin and τmin are channel-related parameters dependent on dA and dB. The success probability of the probabilistic transposition protocol is 1/d2 when the channel is unital, with dA = dB = d. The research demonstrates that the transpose of an unknown quantum channel can be probabilistically realised via postselected teleportation, utilising a maximally entangled state and applying the unknown channel to system A. A measurement is then performed on systems B′B, with a successful outcome corresponding to the projector onto ΦB′B heralding the preparation of the state N T on system A′. Theorem 1 proves that no n-slot completely positive supermap exists to universally realise the complex conjugate for any finite integer n ≥ 1 and any quantum channel N. Hierarchical implementation of quantum channel transformations via virtual protocols Researchers have established a definitive hierarchy regarding the physical implementation of transformations applied to quantum channels. Specifically, the transposition of a channel can be realised probabilistically using a technique akin to teleportation, requiring only a single query of the unknown channel. However, the complex conjugate and adjoint transformations prove more challenging, as they cannot be implemented even probabilistically using standard quantum supermaps. To circumvent this limitation, a virtual protocol utilising quasi-probability decomposition was developed for the complex conjugate transformation. This approach demonstrates optimal performance in terms of the resources required for sampling. A practical application of this framework involves estimating the Petz recovery map for unknown channels, achieving a query complexity that represents an improvement over previously established deterministic methods. The protocol for simulating channel adjoints may facilitate experimental investigations of out-of-time-order correlators in open quantum systems. The authors acknowledge a limitation in the current work, focusing on single-slot settings and specific transformations. Future research will likely extend these virtual techniques to encompass multi-slot scenarios and a wider range of higher-order transformations relevant to quantum thermodynamics and error mitigation. These advancements could broaden the applicability of the developed protocols and contribute to more efficient quantum information processing. 👉 More information 🗞 Simulation of Adjoints and Petz Recovery Maps for Unknown Quantum Channels 🧠 ArXiv: https://arxiv.org/abs/2602.05828 Tags: