KLN theorem and infinite statistics

Lei Chen, Xiao-Ming Chen, Yalong Yan

Abstract One-decoy-state quantum key distribution with infinite length pulses can even approach the asymptotic case, i.e., infinite decoy states and infinite length pulses. Due to the limitation of experimental time, we must consider the statistical fluctuation caused by the finite number of pulses. The central limit theorem analysis method is an effective method, which can easily give the upper and lower bounds of experimental observations. Combining the central limit theorem method with the theorem that the ratio of two independent normal random variables can be approximated as a normal random variable under certain conditions, we propose a more efficient method to estimate the secure key rate. Compared with using central limit theorem method directly, our method has fewer bounds, smaller estimation deviation and higher estimation accuracy. The simulation results show that our method can significantly improve the secure key rate and transmission distance when considering statistical fluctuation and keeping the pulse number and security parameters unchanged, especially in the case of small pulse number (e.g. 107).

A.K. Mishra, G. Rajasekaran

Naoto Tsuji, Masahito Ueda

We derive the fluctuation theorem for quantum-state statistics that can be obtained when we initially measure the total energy of a quantum system at thermal equilibrium, let the system evolve unitarily, and record the quantum-state data reconstructed at the end of the process. The obtained theorem shows that the quantum-state statistics for the forward and backward processes is related to the equilibrium free-energy difference through an infinite series of independent relations, which gives the quantum work fluctuation theorem as a special case, and reproduces the out-of-time-order fluctuation-dissipation theorem near thermal equilibrium. The quantum-state statistics exhibits a system-size scaling behavior that differs between integrable and non-integrable (quantum chaotic) systems as demonstrated numerically for one-dimensional quantum lattice models.

Robert H. Romer

Laura Ruetsche

This chapter catalogs circumstances under which the reassuring uniqueness results presented in Chapter 2 break down. The Stone-von Neumann theorem presupposes a continuity condition; with that presupposition suspended, it becomes possible to equip the quantum theory of a single mechanical system with exact position eigenstates. The Stone-von Neumann theorem presupposes the classical theory to be quantized to be set in Euclidean space; when systems such as a bead on a circle are quantized, the theorem breaks down and unitarily inequivalent representations of the physics ensue. Finally, the Stone-von Neumann and Jordan-Wigner theorems presuppose finitely many degrees of freedom. They therefore fail to apply to quantum field theories (because a field, assigning a value to each point of space[time] has infinitely many degrees of freedom), or theories at the thermodynamic (infinite volume) limit of quantum statistical mechanics. The chapter presents simple examples of infinite systems and their unitarily inequivalent quantizations: the free Klein-Gordon field and the infinite spin chain.