Compression and replication of quantum information

Abstract

This dissertation explores how quantum data can be transmitted and distributed to many users, focusing on two dual tasks of quantum information processing: compression and replication. The first task concerns on compressing identical copies of an unknown quantum state or gate, in a way that the original copies can be approximately recovered. For various types of quantum states, reliable protocols achieving the minimum of the total memory cost are designed. The total memory cost turns out to depend only on the number of input copies and the number of free parameters of the state. The compression of unitary gates is also studied.

The second task concerns on how quantum states and gates can be optimally cloned, especially when the numbers of input and output copies are large. This part of the dissertation begins with super-replication of quantum states, an intriguing phenomenon where quantum states can produce a large number of almost perfect replicas for a small probability. Ultimate limits on how fast quantum information can be replicated are derived for both deterministic and probabilistic quantum devices.

This dissertation further explores the way to certify the advantage of using genuine quantum resources and to see whether this advantage persists in the asymptotic scenario of many copies. A relevant question asked here is whether a task can be optimally realized by first measuring the input state or, equivalently, whether incoherent operations can attain the optimality. Two distinct phenomena are observed for the two tasks here: In the limit of large output copies, replication of most quantum states can be optimally accomplished by measure-and-prepare devices. Compression, on the other hand, cannot. From the fundamental aspect, this dissertation aims at providing a new angle to look at the resource theory of asymmetry and shedding light on a quantum Shannon theory of asymmetry.