High Fidelity Quantum State Transformation Achieves Optimisation under Locality Constraints

Quantum state transformation, the process of manipulating the properties of quantum systems, presents a significant challenge when restricted by limitations on how different parts of the system can interact, a condition known as locality. Sasan Sarbishegi and Maryam Sadat Mirkamali, both from the Department of Physics at Sharif University of Technology, and their colleagues, now present a new framework for optimising these transformations, whether they occur with certainty or as a probability. Their method constructs efficient local channels, essentially pathways for manipulating quantum information, that convert an initial quantum state into a desired target state with remarkably high accuracy. This achievement substantially improves the process of quantum distillation, a crucial technique for enhancing weakly defined quantum states, and establishes a versatile tool with broad implications for future quantum information processing technologies.

To achieve this goal, local quantum channels are parametrized on a complex Stiefel manifold and optimized using gradient-based methods. The research demonstrates that this approach significantly enhances entanglement distillation for weakly entangled states via two complementary strategies: optimized local state transformation and probabilistic local transformation. These results establish the method as a powerful and versatile tool for a broad class of quantum information processing tasks. Transforming one quantum state into another under various operational constraints is a fundamental problem in quantum control.

Entanglement Purification via Repeated Distillation

Entanglement is a vital resource for quantum technologies, but it is fragile and degrades due to noise during transmission or storage. Entanglement distillation takes multiple weakly entangled pairs and purifies them into fewer, highly entangled pairs. Existing protocols struggle when initial entanglement is very weak, limiting their use in noisy real-world scenarios. Researchers have generalized entanglement distillation to work even with very noisy states, making it effective for any two-qubit state. The team employed a two-pronged approach, both involving optimization techniques.

First, they pre-processed the initial state to resemble a specific class of partially entangled states, known as R-states, before applying a standard entanglement distillation protocol. This pre-processing step effectively boosts the entanglement. Second, they directly optimized the local quantum operations, constrained by the rules of local operations and classical communication, to maximize the fidelity of the resulting entangled state. The optimization process utilized a mathematical space, the Stiefel manifold, to ensure physically realistic operations. Key to understanding this work are concepts like qubits, the basic unit of quantum information, and entanglement, where linked qubits instantly influence each other.

The fully entangled fraction (FEF) measures the strength of entanglement in a mixed quantum state, with a higher FEF indicating stronger entanglement. Fidelity measures how close a quantum state is to a target state, and a Bell state represents a maximally entangled pair of qubits. The pre-processing step allows a standard protocol to be used with any two-qubit state, overcoming previous limitations. The direct optimization approach achieves fidelities as high as theoretically possible, given the constraints. This method outperforms the pre-processing approach and is not limited to entanglement distillation, with potential applications in distributed state discrimination and quantum network routing.

The researchers have also made their code publicly available to promote reproducibility and further research. In essence, this research addresses the problem of maintaining a clear signal when sending information using entangled particles. The team provides two solutions: cleaning up the signal first by pre-processing the particles, or optimizing the measurement process to extract the clearest possible signal. Both methods improve the reliability of entangled signals, even with weak initial entanglement, which is crucial for building practical quantum technologies.

Optimizing Local Quantum Channels for Entanglement Distillation

Scientists have created a numerical framework for transforming quantum states while respecting locality constraints, achieving high fidelity between initial and target states. This work focuses on optimizing local quantum channels, which are defined on a complex Stiefel manifold and refined using gradient-based methods, to convert an input state into a desired output state. Experiments demonstrate the effectiveness of this approach for enhancing entanglement distillation, particularly for weakly entangled states, through optimized state transformation and probabilistic local transformation strategies. The team developed a method to explore the space of local channels, defining a cost function that measures the distance between the transformed state and the target state, then minimizing this function to identify the optimal local channel.

A complementary approach involved probabilistic local transformation, interpreting channels as generalized quantum measurements with post-selection rules, and optimizing the fidelity between the post-selected state and the target state by varying channel parameters. Applying these methods to entanglement distillation, researchers overcame limitations of existing protocols, such as DEJMPS and BBPSSW, which struggle when the fully entangled fraction (FEF) of an input state falls below 0.5. Results demonstrate the ability to map states with FEF values of 0.5 or less to R-states, ideal for the extreme photon loss (EPL) protocol, enabling effective distillation even from weakly entangled inputs. Furthermore, scientists directly optimized a single-step probabilistic local transformation protocol, achieving increased fidelity with respect to a Bell state, specifically the |Ψ+⟩⟨Ψ+| state. Measurements confirm that this tailored approach maximizes fidelity by customizing operators to the known input state, delivering a significant advancement in quantum information processing.

Optimised Local Transformations for Quantum States

Scientists have developed a new numerical framework capable of transforming quantum states, both predictably and with a degree of randomness, while adhering to constraints imposed by locality. This method constructs optimised local channels, effectively reshaping an initial quantum state into a desired target state with high fidelity, and utilises a complex mathematical space called a Stiefel manifold alongside gradient-based optimisation techniques to achieve this transformation. The team successfully demonstrated the effectiveness of this approach by enhancing the process of entanglement distillation, employing both optimised and probabilistic local transformations to improve outcomes with weakly entangled states. The research establishes a versatile tool applicable to a wide range of quantum information processing tasks, as evidenced by successful tests on specific examples.

In one demonstration, the method accurately transformed arbitrary quantum states to achieve a maximum expected value for a defined observable, closely matching the theoretical limit across numerous trials. Furthermore, the framework effectively increased the fully entangled fraction of a given quantum state, a crucial measure of entanglement quality, surpassing the performance of existing techniques like one-sided amplitude damping channels in certain scenarios. The authors acknowledge that the current framework relies on iterative optimisation, which may require substantial computational resources for highly complex states or transformations, and that the performance gains are most pronounced for specific classes of initial states. Future work will likely focus on extending the method to handle more complex systems and exploring its application to other quantum information tasks, potentially including quantum error correction and quantum communication protocols.