Physics Reports 544 (2014) 241–322 Contents lists available at ScienceDirect Physics Reports journal homepage: www.elsevier.com/locate/physrep Quantum cloning machines and the applications Heng Fan a,b,∗, Yi-Nan Wang c, Li Jing c, Jie-Dong Yue a, Han-Duo Shi c, Yong-Liang Zhang c, Liang-Zhu Mu c a Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China b Collaborative Innovation Center of Quantum Matter, Beijing 100190, China c School of Physics, Peking University, Beijing 100871, China article info a b s t r a c t Article history: No-cloning theorem is fundamental for quantum mechanics and for quantum information Accepted 30 June 2014 science that states an unknown quantum state cannot be cloned perfectly. However, we can Available online 10 July 2014 try to clone a quantum state approximately with the optimal fidelity, or instead, we can try editor: J. Eichler to clone it perfectly with the largest probability. Thus various quantum cloning machines have been designed for different quantum information protocols. Specifically, quantum cloning machines can be designed to analyze the security of quantum key distribution protocols such as BB84 protocol, six-state protocol, B92 protocol and their generalizations. Some well-known quantum cloning machines include universal quantum cloning machine, phase-covariant cloning machine, the asymmetric quantum cloning machine and the probabilistic quantum cloning machine. In the past years, much progress has been made in studying quantum cloning machines and their applications and implementations, both theoretically and experimentally. In this review, we will give a complete description of those important developments about quantum cloning and some related topics. On the other hand, this review is self-consistent, and in particular, we try to present some detailed formulations so that further study can be taken based on those results. ' 2014 Elsevier B.V. All rights reserved. Contents 1. Introduction.......................................................................................................................................................................................... 243 1.1. Quantum information, qubit and quantum entanglement................................................................................................... 244 1.2. Quantum gates......................................................................................................................................................................... 246 2. No-cloning theorem............................................................................................................................................................................. 247 2.1. A simple proof of no-cloning theorem ................................................................................................................................... 247 2.2. No-broadcasting theorem ....................................................................................................................................................... 248 2.3. No-broadcasting for correlations............................................................................................................................................ 249 2.4. A unified no-cloning theorem from information theoretical point of view ........................................................................ 250 2.5. No-cloning and no-signaling................................................................................................................................................... 253 2.6. No-cloning for unitary operators............................................................................................................................................ 254 2.7. Other developments and related topics................................................................................................................................. 254 3. Universal quantum cloning machines ................................................................................................................................................ 255 3.1. Symmetric UQCM for qubit..................................................................................................................................................... 255 ∗ Corresponding author at: Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China. E-mail address: [email protected] (H. Fan). http://dx.doi.org/10.1016/j.physrep.2014.06.004 0370-1573/' 2014 Elsevier B.V. All rights reserved. 242 H. Fan et al. / Physics Reports 544 (2014) 241–322 3.2. Symmetric UQCM for qudit..................................................................................................................................................... 257 3.3. Asymmetric quantum cloning ................................................................................................................................................ 259 3.4. A unified UQCM ....................................................................................................................................................................... 261 3.5. Singlet monogamy and optimal cloning ................................................................................................................................ 263 3.6. Mixed-state quantum cloning ................................................................................................................................................ 265 3.7. Universal NOT gate .................................................................................................................................................................. 266 3.8. Minimal input set, six-state cryptography and other results ............................................................................................... 266 3.9. Other developments and related topics................................................................................................................................. 267 4. Probabilistic quantum cloning ............................................................................................................................................................ 269 4.1. Probabilistic quantum cloning machine................................................................................................................................. 270 4.2. A novel quantum cloning machine......................................................................................................................................... 270 4.3. Probabilistic quantum anti-cloning and NOT gate ................................................................................................................ 271 4.4. Other developments and related topics................................................................................................................................. 271 5. Phase-covariant and state-dependent quantum cloning .................................................................................................................. 272 5.1. Quantum key distribution protocols ...................................................................................................................................... 272 5.2. General state-dependent quantum cloning........................................................................................................................... 273 5.3. Quantum cloning of two non-orthogonal states ................................................................................................................... 275 5.4. Phase-covariant quantum cloning: economic quantum cloning for equatorial qubits ...................................................... 276 5.5. One to many phase-covariant quantum cloning machine for equatorial qubits................................................................. 277 5.6. Phase quantum cloning: comparison between economic and non-economic.................................................................... 279 5.7. Phase-covariant quantum cloning for qudits......................................................................................................................... 280 5.8. Symmetry condition and minimal sets in determining quantum cloning machines.......................................................... 281 5.9. Quantum cloning machines of arbitrary set of MUBs ........................................................................................................... 282 5.10. Quantum cloning in mean king problem as a quantum key distribution protocol............................................................. 286 5.11. Other developments and related topics................................................................................................................................. 287 6. Local cloning of entangled states, entanglement in quantum cloning............................................................................................. 289 6.1. Local cloning of Bell states ...................................................................................................................................................... 289 6.2. Local cloning and local discrimination................................................................................................................................... 290 6.3. Entanglement of quantum cloning........................................................................................................................................