Xin Wang

Determining the ultimate limits of quantum communication, such as the quantum capacity of a channel and the distillable entanglement of a shared state, remains a central challenge in quantum information theory, primarily due to the phenomenon of superadditivity. This work develops Riemannian optimization methods to establish significantly tighter, computable two-sided bounds on these fundamental quantities. For upper bounds, our method systematically searches for state and channel extensions that minimize known information-theoretic bounds. We achieve this by parameterizing the space of all possible extensions as a Stiefel manifold, enabling a universal search that overcomes the limitations of ad-hoc constructions. Combined with an improved upper bound on the one-way distillable entanglement based on a refined continuity bound on quantum conditional entropy, our approach yields new state-of-the-art upper bounds on the quantum capacity of the qubit depolarizing channel for large values of the depolarizing parameter, strictly improving the previously best-known bounds. For lower bounds, we introduce Riemannian optimization methods to compute multi-shot coherent information. We establish lower bounds on the one-way distillable entanglement by parameterizing quantum instruments on the unitary manifold, and on the quantum capacity by parameterizing code states with a product of unitary manifolds. Numerical results for noisy entangled states and different channels demonstrate that our methods successfully unlock superadditive gains, improving previous results. Together, these findings establish Riemannian optimization as a principled and powerful tool for navigating the complex landscape of quantum communication limits. Furthermore, we prove that amortization does not enhance the channel coherent information, thereby closing a potential avenue for improving capacity lower bounds in general. This result can be of independent interest.

Quantum state purification protocols, which mitigate noise by converting multiple copies of noisy quantum states into fewer copies with a lower noise level, have applications in quantum communication and computation with imperfect devices. Here, we systematically study the task of state purification in distributed quantum systems, demanding that purification be achieved by local operations and classical communication (LOCC). We prove that, in the presence of depolarizing noise, no LOCC purification protocol starting from two copies can work blindly for all the states in three important sets: the set of all pure two-qubit states, the set of all two-qubit maximally entangled states, and the Bell basis. In stark contrast, we show that a targeted, single-state purification is always achievable in the presence of depolarizing noise, and we provide an explicit analytical LOCC protocol for every given two-qubit state. For arbitrary finite sets of pure states and arbitrary noise profiles, we develop an optimization-based algorithm that systematically designs LOCC purification protocols, and we demonstrate it through concrete examples. Overall, our results identify both fundamental limitations and practical noise reduction strategies for distributed quantum information processing.

The inherent irreversibility of quantum dynamics for open systems poses a significant barrier to the inversion of unknown quantum processes. To tackle this challenge, we propose the framework of virtual combs that exploit the unknown process iteratively with additional classical post-processing to simulate the process inverse. Our research establishes a path to achieving the exact inverse of unknown channels with certain conditions, accompanied by a no-go theorem that underscores the intrinsic limitations imposed by quantum mechanics on such tasks. Notably, we demonstrate that an n-slot virtual comb can exactly reverse a depolarizing channel with one unknown noise parameter out of n+1 potential candidates, and a 1-slot virtual comb can exactly reverse an arbitrary pair of quantum channels. We further explore the approximate inverse of an unknown channel within a given channel set. For any unknown depolarizing channels within a specified noise region, we unveil a worst-case error decay of O(n^−1) of reversing the channel via virtual combs. Moreover, we show that virtual combs with constant slots can be applied to universally reverse unitary operations and investigate the trade-off between the slot number and the sampling overhead.

The quantum capacity of a noisy quantum channel determines the maximal rate at which we can code reliably over asymptotically many uses of the channel, and it characterizes the channel’s ultimate ability to transmit quantum information coherently. In this paper, we derive single-letter upper bounds on the quantum and private capacities of quantum channels. The quantum capacity of a quantum channel is always no larger than the quantum capacity of its extended channels, since the extensions of the channel can be considered as assistance from the environment. By optimizing the parametrized extended channels with specific structures such as the flag structure, we obtain new upper bounds on the quantum capacity of the original quantum channel. Furthermore, we extend our approach to estimating the fundamental limits of private communication and one-way entanglement distillation. As notable applications, we establish improved upper bounds to the quantum and private capacities for fundamental quantum channels of great interest in quantum information, some of which are also the sources of noise in superconducting quantum computing. In particular, our upper bounds on the quantum capacities of the depolarizing channel and the generalized amplitude damping channel are strictly better than previously best-known bounds.

Random thermal states of many-body Hamiltonians underpin studies of thermalization, chaos, and quantum phase transitions, yet their generation remains costly when each Hamiltonian must be prepared individually. We introduce the thermal-drift channel, a measurement-based operation that implements a tunable nonunitary drift along a chosen Pauli term. Based on this channel, we present a measurement-controlled sampling algorithm that generates thermal states together with their Hamiltonian ``labels'' for general physical models. We prove that the total gate count of our algorithm scales cubically with system size, quadratically with inverse temperature, and as the inverse error tolerance to the two-thirds power, with logarithmic dependence on the allowed failure probability. We also show that the induced label distribution approaches a normal distribution reweighted by the thermal partition function, which makes an explicit trade-off between accuracy and effective range. Numerical simulations for a 2D Heisenberg model validate the predicted scaling and distribution. As an application, we compute unfolding-free level-spacing ratio statistics from sampled thermal states of a 2D transverse-field Ising model and observe a crossover toward the Wigner-Dyson prediction, demonstrating a practical and scalable route to chaos diagnostics and random matrix universality studies on near-term quantum hardware.

Estimating nonlinear properties of quantum states, particularly observable-weighted moments $\mathrm{Tr}(\mathcal{O}\rho^k)$, is a central task in quantum information science. In this work, we present a unified framework establishing the optimal sample complexity for this task and a resource-efficient protocol for its implementation. Theoretically, we prove that $\tilde{\Theta}(k)$ samples of an $m$-qubit state $\rho$ are sufficient and necessary to \textit{simultaneously} estimate the full hierarchy of moments $\mathrm{Tr}(\mathcal{O}\rho), \dots, \mathrm{Tr}(\mathcal{O}\rho^k)$. This reveals that estimating the entire hierarchy is asymptotically as efficient as estimating the single highest-order term. To realize this, we introduce a circuit architecture leveraging qubit reuse that requires only $2m+1$ physical qubits and $\mathcal{O}(k)$ depth. This approach achieves the near-optimal sample complexity of $\mathcal{O}(k \log k / \varepsilon^2)$ with significantly reduced hardware overhead. We demonstrate the framework's utility by bounding maximum eigenvalues, performing virtual cooling on the Heisenberg model, and experimentally measuring higher-order Rényi entropies on a superconducting processor.

Transformations of quantum channels, such as the transpose, complex conjugate, and adjoint, are fundamental to quantum information theory. Given access to an unknown channel, a central problem is whether these transformations can be implemented physically with quantum supermaps. While such supermaps are known for unitary operations, the situation for general quantum channels is fundamentally different. In this work, we establish a strict hierarchy of physical realizability for the transposition, complex conjugation, and adjoint transformation of an unknown quantum channel. We present a probabilistic protocol that exactly implements the transpose with a single query. In contrast, we prove no-go theorems showing that neither the complex conjugate nor the adjoint can be implemented by any completely positive supermap, even probabilistically. We then overcome this impossibility by designing a virtual protocol for the complex conjugate based on quasi-probability decomposition, and show its optimality in terms of the diamond norm. As a key application, we propose a protocol to estimate the expectation values resulting from the Petz recovery map of an unknown channel, achieving an improved query complexity compared to existing methods.