High-Dimensional Single-Photon Quantum Gates: Concepts and Experiments (1702.07299v1)

Abstract: Transformations on quantum states form a basic building block of every quantum information system. From photonic polarization to two-level atoms, complete sets of quantum gates for a variety of qubit systems are well known. For multi-level quantum systems beyond qubits, the situation is more challenging. The orbital angular momentum modes of photons comprise one such high-dimensional system for which generation and measurement techniques are well-studied. However, arbitrary transformations for such quantum states are not known. Here we experimentally demonstrate a four-dimensional generalization of the Pauli X-gate and all of its integer powers on single photons carrying orbital angular momentum. Together with the well-known Z-gate, this forms the first complete set of high-dimensional quantum gates implemented experimentally. The concept of the X-gate is based on independent access to quantum states with different parities and can thus be easily generalized to other photonic degrees-of-freedom, as well as to other quantum systems such as ions and superconducting circuits.

High-Dimensional Single-Photon Quantum Gates: Concepts and Experiments

This paper explores the implementation of high-dimensional quantum gates using single photons characterized by their orbital angular momentum (OAM). The authors focus on constructing a complete set of transformations within a four-dimensional quantum system, leveraging the higher-dimensional analogs of the Pauli gates. The paper is anchored in addressing challenges posed by manipulating quantum states beyond traditional qubit systems and stands out by providing empirical evidence of such gates on OAM modes.

Key Contributions

1. Four-Dimensional X-Gate Design: This work extends traditional two-dimensional Pauli transformations to a four-dimensional system. The highlight is the realization of a four-dimensional generalization of the Pauli X-gate, which acts as a cyclic ladder operator in this expanded Hilbert space. This fundamental transformation is not merely a conceptual leap but is demonstrated through rigorous experimental methodology.
2. Integration with the Z-Gate: A complete set of gates requires both X and Z gates, and the paper integrates the new X-gate with the existing four-dimensional Z-gate, achieved through optical elements like Dove prisms, facilitating mode-dependent phases.
3. Experimental Implementation: The implementation employs intricate interferometric setups, including a Mach-Zehnder interferometer with parity sorting capabilities. It demonstrates the X-gate, its integer powers, including X² and X³ gates (with X³ serving as X†), in transforming single photon states within the OAM basis.

Numerical Results and Validation

The experimental results notably achieve high transformation efficiencies across various gates—X, X², and X†—with average transformation probabilities exceeding 87%. Additionally, the preservation of the coherence of quantum superpositions during transformations validates the effectiveness of these gates in real-world quantum protocols.

Implications and Future Directions

Practical Implications: The development of these high-dimensional transformations carries significant potential for quantum communication using single photons. The ability to encode large amounts of information in a small state space could enhance protocols such as quantum key distribution, dense coding, and secret sharing within quantum networks.

Theoretical Implications: The X-gate's reliance on parity separation presents a versatile approach that can be generalized across different quantum systems, including frequency modes or trapped ions. This adaptability could broaden applications and inspire new experimental methodologies in quantum research.

Future Developments: The realization of high-dimensional quantum gates lays groundwork for complex quantum algorithms, especially those leveraging the advantages of high-dimensional state spaces in error correction or entanglement distribution. Moreover, extending this approach to entangle multiple particles or integrate higher-dimensional controlled-NOT operations remains a promising avenue.

The paper conclusively establishes the feasibility of high-dimensional gate operations on single photons, opening pathways to advanced quantum computing frameworks capable of harnessing the enhanced capabilities afforded by multi-level quantum systems.