[Paper Review] Coherent-state optical qudit cluster state generation and teleportation via homodyne detection

Jaewan Kim, Juhui Lee+4 more|arXiv (Cornell University)|Dec 29, 2010

Quantum Information and Cryptography3 citations

TL;DR

This paper proposes a deterministic scheme to generate optical qudit cluster states using coherent states with large amplitude, where each coherent state acts as a qudit in an even superposition of pseudo-number states. By applying a weak cross-Kerr nonlinearity-based generalized Controlled-Z gate, maximally entangled qudit pairs are created, enabling a scalable, all-optical qudit teleportation protocol via homodyne detection, demonstrating a practical path toward large-scale quantum computation with continuous-variable systems.

ABSTRACT

Defining a computational basis of pseudo-number states, we interpret a coherent state of large amplitude, $|α|\gg\frac{d}{2π}$, as a qudit --- a $d$-level quantum system --- in a state that is an even superposition of $d$ pseudo-number states. A pair of such coherent-state qudits can be prepared in maximally entangled state by generalized Controlled-$Z$ operation that is based on cross-Kerr nonlinearity, which can be weak for large $d$. Hence, a coherent-state optical qudit cluster state can be prepared by repetitive application of the generalized Controlled-$Z$ operation to a set of coherent states. We thus propose an optical qudit teleportation as a simple demonstration of cluster state quantum computation.

Motivation & Objective

• To develop a deterministic, scalable method for generating optical qudit cluster states using coherent states with large amplitude.
• To overcome the limitations of probabilistic linear optical gates in quantum computing by leveraging qudits instead of qubits.
• To enable practical large-scale quantum computation using continuous-variable optical systems with feasible nonlinearities.
• To demonstrate qudit cluster state quantum computation via a teleportation protocol based on homodyne measurement.

Proposed method

• Define a computational basis of pseudo-number states from coherent states with |α| ≫ d/(2π), treating them as d-level qudits in even superposition.
• Use a generalized Hadamard transformation to map between computational basis (pseudo-number states) and conjugated basis (pseudo-phase states), which correspond to coherent states at different phases.
• Implement a generalized Controlled-Z gate using cross-Kerr nonlinearity with Hamiltonian H = −ℏχn̂₁n̂₂, enabling entanglement between coherent-state qudits.
• Perform qudit teleportation by entangling a qudit with a coherent state via Controlled-Z, then measuring the qudit in the conjugated (pseudo-phase) basis using double-arm homodyne detection.
• Reconstruct the teleported state on the target qudit using feedforward operations based on measurement outcomes, with the recovery unitary expressed as X⁻ˢRZ⁻ᵏ.
• Utilize the fact that the measurement outcome in the conjugated basis directly determines the required correction operations, enabling deterministic teleportation.

Experimental results

Research questions

• RQ1 Can coherent states with large amplitude be used as effective qudits in a d-level quantum system?
• RQ2 Is it possible to generate a cluster state of optical qudits using deterministic, weak nonlinear interactions?
• RQ3 Can qudit teleportation be achieved via homodyne detection without requiring strong nonlinearities?
• RQ4 How does the cross-Kerr nonlinearity strength affect the feasibility of qudit cluster state generation for large d?
• RQ5 Can the measurement outcome in the conjugated basis fully determine the correction operations for state teleportation?

Key findings

• Coherent states with |α| ≫ d/(2π) can be interpreted as qudits in an even superposition of d pseudo-number states, enabling a continuous-variable qudit encoding.
• The generalized Controlled-Z gate based on cross-Kerr nonlinearity can be implemented with weak nonlinearity, making it feasible for large d (e.g., d ~ 10⁵) when |α|² ~ 10¹⁰.
• A qudit teleportation protocol is demonstrated using homodyne detection, where measurement outcomes in the conjugated basis fully determine the required correction operations.
• The teleportation outcome is equivalent to applying the unitary operation X⁻ˢRZ⁻ᵏ on the target qudit, with R being the basis reversal operator.
• The scheme enables deterministic, scalable cluster state generation and quantum computation using only single-qudit measurements and feedforward correction, avoiding the need for probabilistic gates.
• The method is robust under realistic nonlinearity constraints, as the required χt is on the order of 10⁻⁴, compatible with current experimental limits.