MIT OpenCourseWare: Quantum States of Light and Single-Photon Qubits in Interferometers

In this MIT OpenCourseWare lecture, single-photon qubits are realized using dual-rail encoding across two optical modes. The instructor demonstrates how a phase shifter and a beam splitter implement any single-qubit operation on the two-mode state, and how the same building blocks, arranged as an interferometer, realize rotations on the qubit. The talk then introduces a nonlinear medium to create interactions between two qubits, enabling two-qubit gates and the generation of entangled states. The session links these optical elements to core quantum information concepts such as measurement, entanglement as a resource, and the path toward universal photonic quantum computation with linear optics plus nonlinear interactions.

Overview of single-qubit photonic encoding

The lecture begins by framing photons as qubits in a dual-rail, two-mode representation. A single photon can occupy mode A or mode B, corresponding to the logical states |1,0> and |0,1>. The instructor emphasizes that the most practical qubit realization uses the presence of the photon in one of two spatial modes rather than relying on zero or one photons in a single mode. This dual-rail approach establishes a two-level system whose state is completely described by the amplitudes in the two modes, allowing a clean mapping to qubit language and facilitating geometric interpretations on the Bloch sphere.

Phase shifts and beam splitters form the two essential optical elements. A phase shifter adds a relative phase between the two modes, while a beamsplitter mixes the two modes, distributing the photon between them. Together they generate arbitrary single-qubit rotations when applied in sequence. The beam splitter is described by a unitary transformation acting on the two mode operators, mixing A and B with coefficients determined by the beam-splitter angle. The phase shifter contributes a relative phase between the modes, providing the missing axis rotation necessary for full single-qubit control.

Matrix picture and Bloch-sphere picture

The discussion introduces a matrix representation of the beam splitter and phase shifter, showing that the combination of these two operations yields a rotation in the qubit space. In particular, a two-beam-splitter interferometer reduces to a rotation around the X-axis on the Bloch sphere, when the constituent rotations are reinterpreted as sequential turns about the Y and Z axes. This geometric viewpoint gives intuition for how simple optical devices can implement any single-qubit gate, making linear optics a powerful platform for quantum information processing even before introducing interactions between photons.

Two-qubit interactions and nonlinear optics

To realize universal quantum computation, interactions between qubits are required. The lecturer introduces a nonlinear medium, specifically a cross-phase modulation (XPM) element, whose phase shift on one mode depends on the photon number in another mode. When placed inside an interferometer, this nonlinear interaction effectively makes the beamsplitter angle depend on a second photonic qubit. The result is a nonlinear Mach-Zehnder-like interferometer whose action is equivalent to a two-qubit gate. The nonlinear Hamiltonian adds a phase shift that is conditional on the presence of photons in the interacting modes, enabling entangling operations and the construction of two-qubit gates from linear optics with an intensity-dependent control element.

Entanglement, measurements, and the dual-rail subspace

The speaker emphasizes the dual-rail photon state space, called the dual-rail photon state space, where exactly one photon resides in either of the two modes. This restriction defines the single-qubit subspace, simplifying the description of multi-qubit devices and making measurements with finite efficiencies straightforward. Entanglement arises naturally when photons pass through the interferometric network: even with linear optics, starting from a product state with one photon in each of two qubits and guiding them through a nonlinear interaction can produce entangled outputs. The lecture highlights how entanglement is a resource central to advanced quantum phenomena such as teleportation and quantum metrology.

Outlook and connections to broader quantum information topics

The discussion closes by connecting the optical implementation to broader topics in quantum information science, including the Deutsch-Jozsa algorithm as a notional example of how the same optical toolkit can realize quantum algorithms. The lecturer foreshadows topics for next week, such as quantifying entanglement, Einstein-Podolsky-Rosen correlations, Bell inequalities, and the prospect of entangling not only photons but other systems like atoms. The overarching message is that a small set of optical elements, when used within a well-structured framework, can realize universal quantum gates and enable the exploration of fundamental quantum phenomena using light as a highly controllable platform.