The power of quantum memories : faithful quantum data compression and quantum benchmarks

Abstract

Quantum memories are a powerful and expensive resource. This dissertation studies how to efficiently use this resource and how to demonstrate its advantages. Specifically, it studies how to compress information into quantum memories of the smallest possible size, and how to benchmark the performances of devices with an internal quantum memory.

Data compression reduces the cost of data transmission and the quantum memory requirement for data storage. We show that for states with translational symmetry, great savings of quantum memory could be achieved: our compression protocol uses a memory size exponentially smaller than the original system's size. For the case of independent and identically prepared states, the memory could be mostly classical, while a non-zero portion of quantum memory is indispensable for faithful compression. We also consider the general case of correlated multipartite states described by tensor network states. The compression takes advantage of an analogy between tensor networks and flow networks, relating the memory size with the minimum cut of a flow network associated to the family of tensor network states under consideration. To implement the compression, we design an algorithm that uses the minimum memory size dictated by the structure of the tensor network, reaching the optimal memory size for exact compression. We provide sufficient conditions for the algorithm to run in polynomial time, and show that the conditions are satisfied for matrix product states.

To measure the advantage of a quantum memory in a given task, we use quantum benchmarks. A quantum benchmark evaluates the performance of a device and compares it to the threshold that upper bounds the performance of devices without quantum memories. However, standard benchmarking techniques are sometimes hard to implement in practical scenarios, as they sometimes require infinitely many input states and infinitely many measurement setups. We solve the problem by proposing an experimental setup that yields the performance of the device over all possible inputs with only one fixed input and one fixed measurement. We apply our results to tasks involving continuous-variable quantum systems such as amplification, attenuation and purification of noisy coherent states, providing the first rigorous test for these tasks. The test is based on a unified framework for quantum benchmarks. With this framework, one is able to optimize the tests for deterministic and probabilistic devices, and construct a test whose benchmark does not depend on the success probability of the device. We further study the case when the task is group covariant, showing that our unified framework is able to exploit the symmetry and find the optimal tests explicitly.