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arXiv:0904.0923

Scrutinizing single-qubit quantum channels: Theory and experiment with trapped ions

Thilo Hannemann, Christof Wunderlich, Martin Plesch, Mário Ziman, Vladimír Bužek

Published Apr 6, 2009• 3 citations

Abstract

We report experimental implementation of various types of qubit channels using an individual trapped ion. We analyzed experimental data and we performed tomographic reconstruction of quantum channels based on these data. Specifically, we studied phase damping channels, where the damping acts either in the xy-plane of the Bloch sphere or in an arbitrary plane that includes the origin of the Bloch sphere. We also experimentally realized and consequently analyzed quantum channels that in addition to phase damping affect also a polarization rotation. We used three reconstruction schemes for estimation of quantum channels from experimental data: (1) a linear inverse method, (2) a maximum likelihood estimation, and (3) a constrained maximum likelihood estimation. We took into account realistic experimental conditions where imperfect test-state preparations and biased measurements are incorporated into the estimation schemes. As a result we found that imperfections present in the process of preparation of test states and as well as in measurements of the considered ion trap system do not limit the control of the implementation of the desired channel. Even imperfect preparation of test state and subsequent measurements still provide sufficient resources for the complete quantum-channel tomography.

DOI: 10.48550/arxiv.0904.0923

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This thesis deals with the implementation of quantum channels using linear optics. We begin with overviews of some important concepts in both quantum information and quantum optics. First, we discuss the quantum bit and describe the evolution of the states via quantum channels. We then discuss both quantum state and process tomography, methods for how to determine which states and operations we are experimentally implementing in the lab. Second, we discuss topics in quantum optics such the generation of single photons, polarization entanglement, and the construction of an entangling gate. \n \nThe first experiment is the implementation of a quantum damping channel, which intentionally can add a specific type and amount of decohering noise to a photonic qubit. Specifically, we realized a class of quantum channels which contains both the amplitude-damping channel and the bit-flip channel, and did so with a single, static, optical setup. Many quantum channels, and some gates, can only be implemented probabilistically when using linear optics and postselection. Our main result is that the optical setup achieves the optimal success probability for each channel. Using a novel ancilla-assisted tomography, we characterize each case of the channel, and find process fideilities of

0.98 \\pm 0.01

for the amplitude-damping channel and

0.976 \\pm 0.009

for the bit-flip. \n \nThe second experiment is an implementation of a protocol for quantum computing on encrypted data. The protocol provides the means for a client with very limited quantum power to use a server's quantum computer while maintaining privacy over the data. We perform a quantum process tomography for each gate in a universal set, showing that only when the proper decryption key is used on the output states, which is hidden from the server, then the action of the quantum gate is recovered. Otherwise, the gate acts as the completely depolarizing channel.

Abstract Quantum tomography is currently the mainly employed method to characterize a quantum system and therefore plays a fundamental role when trying to characterize the action of a particular channel. Nonetheless, quantum tomography works on the premise of a full characterization and description of the devices preparing the quantum state and realizing the measurements. Such an assumption was recently relaxed in Dall’Arno et al (arXiv: 1805.01159 ) and Dall’Arno et al (2017 Proc. R. Soc. A 473 20160721), where a theoretical framework for the device-independent inference of quantum channels was developed and experimentally implemented with superconducting qubits. Here, based on such a framework, we present a complete experimental test on a photonic setup of two semi-device-independent protocols that can be employed for the validation of the tomographic reconstruction or the characterization of a given quantum channel, not relying on many assumptions on the adopted device. Our implementation paves the way to the development of new experimental methods not relying on the assumptions typically taken for granted in all the previous protocols.

We introduce and demonstrate experimentally: (1) a framework called "gate set tomography" (GST) for self-consistently characterizing an entire set of quantum logic gates on a black-box quantum device; (2) an explicit closed-form protocol for linear-inversion gate set tomography (LGST), whose reliability is independent of pathologies such as local maxima of the likelihood; and (3) a simple protocol for objectively scoring the accuracy of a tomographic estimate without reference to target gates, based on how well it predicts a set of testing experiments. We use gate set tomography to characterize a set of Clifford-generating gates on a single trapped-ion qubit, and compare the performance of (i) standard process tomography; (ii) linear gate set tomography; and (iii) maximum likelihood gate set tomography.