Khabbazi Oskouei Samad, Mancini Stefano, Wilde Mark M
Department of Mathematics, Islamic Azad University, Varamin-Pishva Branch, 33817-7489 Iran.
School of Science and Technology, University of Camerino, Via M. delle Carceri 9, 62032 Camerino, Italy.
Proc Math Phys Eng Sci. 2019 Jan;475(2221):20180612. doi: 10.1098/rspa.2018.0612. Epub 2019 Jan 9.
In this paper, we prove a quantum union bound that is relevant when performing a sequence of binary-outcome quantum measurements on a quantum state. The quantum union bound proved here involves a tunable parameter that can be optimized, and this tunable parameter plays a similar role to a parameter involved in the Hayashi-Nagaoka inequality (Hayashi & Nagaoka 2003 , 1753-1768. (doi:10.1109/TIT.2003.813556)), used often in quantum information theory when analysing the error probability of a square-root measurement. An advantage of the proof delivered here is that it is elementary, relying only on basic properties of projectors, Pythagoras' theorem, and the Cauchy-Schwarz inequality. As a non-trivial application of our quantum union bound, we prove that a sequential decoding strategy for classical communication over a quantum channel achieves a lower bound on the channel's second-order coding rate. This demonstrates the advantage of our quantum union bound in the non-asymptotic regime, in which a communication channel is called a finite number of times. We expect that the bound will find a range of applications in quantum communication theory, quantum algorithms and quantum complexity theory.
在本文中，我们证明了一个量子并集界，它在对量子态执行一系列二值结果量子测量时是相关的。这里证明的量子并集界涉及一个可优化的可调参数，并且这个可调参数起到了与Hayashi - Nagaoka不等式（Hayashi & Nagaoka 2003，1753 - 1768。（doi:10.1109/TIT.2003.813556））中涉及的一个参数类似的作用，该不等式在量子信息理论中分析平方根测量的误差概率时经常被使用。这里给出的证明的一个优点是它很基础，仅依赖于投影算子的基本性质、毕达哥拉斯定理和柯西 - 施瓦茨不等式。作为我们量子并集界的一个重要应用，我们证明了量子信道上经典通信的顺序解码策略实现了该信道二阶编码率的一个下界。这展示了我们的量子并集界在非渐近区域的优势，在该区域中通信信道被调用有限次数。我们期望这个界将在量子通信理论、量子算法和量子复杂性理论中找到一系列应用。
