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

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 multilevel 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 generalized to other photonic degrees of freedom and potentially also to other quantum systems.

Figure 1

The conceptual diagrams for the three types of quantum logic gates. The input states for each case is ($-2$, $-1$, 0, 1). (a) A spiral phase plate (SPP) adds $+1$ to the mode, leading to ($-1$, 0, 1, 2). Afterwards, the first parity sorter separates even and odd modes, while the second one combines them again—which forms a large interferometer. Within the interferometer, the even modes have an odd number of reflections, which leads to the correct output modes ($-1$, 0, 1, $-2$). (b) For the $X^2$ gate, the input modes are directly separated into even and odd modes. After a reflection in each arm, the even modes are increased by 2. The two arms are recombined at the PS2, and all of the modes are reflected for changing the signs of the modes. That leads to (0, 1, $-2$, $-1$). (c) In the $X^{†}$ transformation, the different parity modes are separated again, and the even part gets reflected twice, while the odd modes are reflected once. After recombination, the $\ell$ of the modes is decreased by one, which leads to (1, $-2$, $-1$, 0). Note that, in the experiment, it can be adjusted whether even or odd modes are reflected at the parity sorter. In this conceptual diagram, for simplicity we have chosen PS1 to reflect even modes and PS2 to reflect odd modes.

Figure 2

Experimental setup for the four-dimensional $X$ gate (additional experimental details are partially transparent). A 405 nm cw laser pumps a type-II ppKTP crystal (not shown), creating photon pairs entangled in OAM. The idler photon is used for heralding the signal photon in a particular OAM mode. After passing through a $\ell =+1$ spiral phase plate, the signal photon is input into a parity-sorting interferometer, which separates the odd and even OAM components of the photon. After traversing a series of mirrors, the odd and even components are coherently recombined in a Mach-Zehnder interferometer through the use of a polarizing beam splitter (PBS), a half-wave plate (HWP), and an effective polarizer (in the form of a SLM which acts only on horizontally polarized light). A spatial light modulator (SLM) and single mode fiber are used to perform projective measurements of OAM modes and their superpositions.

Figure 3

Data showing the operation of the (a) identity, (b) $X$ gate, (c) $X^2$ gate, and (d) $X^{†}$ gate on the four-dimensional set of input states $\{|-2⟩,|-1⟩,|0⟩,|1⟩\}$. Each row shows the measured normalized coincidence rate in every output mode for a given input mode. The $X$ gate implements a clockwise cyclic transformation $(-2\to -1\to 0\to 1\to -2)$, the $X^2$ gate swaps the odd and even modes $(-1\leftrightarrow 1,-2\leftrightarrow 0)$, and the $X^{†}$ gate performs a counterclockwise cyclic transformation $(1\leftarrow -2\leftarrow -1\leftarrow 0\leftarrow 1)$. The average transformation efficiency for the $X$, $X^2$, and $X^{†}$ gates are 87.3%, 90.4%, and 88.4%, respectively.