We introduce the concept of quantum supermap, describing the most general transformation that maps an input quantum operation into an output quantum operation. Since quantum operations include as special cases quantum states, effects, and measurements, quantum supermaps describe all possible transformations between elementary quantum objects (quantum systems as well as quantum devices). After giving the axiomatic definition of supermap, we prove a realization theorem, which shows that any supermap can be physically implemented as a simple quantum circuit. Applications to quantum programming, cloning, discrimination, estimation, information-disturbance trade-off, and tomography of channels are outlined.
Quantum supermaps are a higher-order generalization of quantum maps, taking quantum maps to quantum maps. It is known that any completely positive, trace non-increasing (CPTNI) map can be performed as part of a quantum measurement. By providing an explicit counterexample we show that, instead, not every quantum supermap sending a quantum channel to a CPTNI map can be realized in a measurement on quantum channels. We find that the supermaps that can be implemented in this way are exactly those transforming quantum channels into CPTNI maps even when tensored with the identity supermap. We link t…
A complex conjugation of unitary quantum map is a second-order map (supermap) that maps a unitary operator $U$ to its complex conjugate $U^*$. First, we present a deterministic quantum protocol that universally implements the complex conjugation supermap when we are given a blackbox quantum circuit, guaranteed to implement some unitary operation, whose only known description is its dimension. We then discuss the complex conjugation supermap in the context of entanglement theory and derive a conjugation-based expression of the $G$-concurrence. Finally, we present a physical process involving id…
Mathematical Institute, Slovak Academy of Sciences（斯洛伐克科学院数学研究所）；Université Paris-Saclay, CNRS, ENS Paris-Saclay, Inria, CentraleSupélec, Laboratoire Méthodes Formelles（巴黎萨克雷大学、法国国家科学研究中心、巴黎高等师范学院、法国国家信息与自动化研究所、中央理工-高等电力学院、形式方法实验室）
University of Oxford（牛津大学）；HKU-Oxford Joint Laboratory for Quantum Information and Computation（香港大学-牛津大学量子信息与计算联合实验室）；University College London（伦敦大学学院）；Université Paris-Saclay, CNRS, ENS Paris-Saclay, Inria, CentraleSupélec, Laboratoire Méthodes Formelles（巴黎萨克雷大学、法国国家科学研究中心、巴黎高等师范学院、法国国家研究与数学与软件研究所、中央理工电院、形式方法实验室）；The University of Hong Kong（香港大学）；Perimeter Institute for Theoretical Physics（理论物理珀IMETER研究所）
Simulation of Adjoints and Petz Recovery Maps for Unknown Quantum Channels
The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；Wuhan University（武汉大学）；Hubei Center for Applied Mathematics（湖北应用数学中心）；Hubei Key Laboratory of Computational Science（湖北省计算科学重点实验室）；Wuhan Institute of Quantum Technology（武汉量子技术研究院）
This paper addresses the problem of designing universal quantum circuits to transform $k$ uses of a $d$-dimensional unitary input-operation into a unitary output-operation in a probabilistic heralded manner. Three classes of protocols are considered, parallel circuits, where the input-operations can be simultaneously, adaptive circuits, where sequential uses of the input-operations are allowed, and general protocols, where the use of the input-operations may be performed without a definite causal order. For these three classes, we develop a systematic semidefinite programming approach that fin…
Critical to the construction of large scale quantum networks, i.e. a quantum internet, is the development of fast algorithms for managing entanglement present in the network. One fundamental building block for a quantum internet is the distribution of Bell pairs between distant nodes in the network. Here we focus on the problem of transforming multipartite entangled states into the tensor product of bipartite Bell pairs between specific nodes using only a certain class of local operations and classical communication. In particular we study the problem of deciding whether a given graph state, a…
When transforming pairs of independent quantum operations according to the fundamental rules of quantum theory, an intriguing phenomenon emerges: some such higher-order operations may act on the input operations in an indefinite causal order. Recently, the formalism of process matrices has been developed to investigate these noncausal properties of higher-order operations. This formalism predicts, in principle, statistics that ensure indefinite causal order even in a device-independent scenario, where the involved operations are not characterised. Nevertheless, all physical implementations of …
We study the problem of transforming a set of pure bipartite states into another using deterministic LOCC (local operations and classical communication). Necessary conditions for the existence of such a transformation are obtained using LOCC constraints on state transformation, entanglement, and distinguishability. These conditions are shown to be independent but not sufficient. We discuss their satisfiability and classify all possible input-output pairs of sets accordingly. We also prove that strict inclusions hold between LOCC, separable, and positive partial transpose operations for set tra…
Stabilizer states form an important class of states in quantum information, and are of central importance in quantum error correction. Here, we provide an algorithm for deciding whether one stabilizer (target) state can be obtained from another stabilizer (source) state by single-qubit Clifford operations (LC), single-qubit Pauli measurements (LPM), and classical communication (CC) between sites holding the individual qubits. What's more, we provide a recipe to obtain the sequence of LC+LPM+CC operations which prepare the desired target state from the source state, and show how these operation…
We propose a passive all optical device capable of transforming the orbital angular momentum (OAM) state of light conditioned over the polarization states. The efficiency of this device is ensured due to its linear optical nature. As applications of this device, we show CNOT and SWAP operations between polarization and OAM qubits, non-interferometric OAM mode sorter and generalized Pauli X operation on a four-dimensional subspace of OAM.
Transforming Traditional Neural Networks into Neuromorphic Quantum-Cognitive Models: A Tutorial with Applications
Artificial Intelligence and Cyber Futures Institute, Charles Sturt University（查尔斯斯特大学人工智能与网络未来学院）；Cyber Futures Institute, Charles Sturt University（查尔斯斯特大学网络未来学院）
