[Paper Review] Generation of photonic non-Gaussian states by measuring multimode Gaussian states

Daiqin Su, Casey R. Myers+1 more|arXiv (Cornell University)|Feb 6, 2019

Quantum Information and Cryptography8 citations

TL;DR

This paper presents a general analytic framework for generating photonic non-Gaussian states by postselecting on photon-number measurements of multimode Gaussian states. It derives exact expressions for the output Wigner function and success probability, enabling optimal design of optical circuits to achieve high-fidelity, high-probability generation of target non-Gaussian states such as cat states, weak cubic phase states, and bosonic code states.

ABSTRACT

We present a detailed analytic framework for studying multimode non-Gaussian states that are conditionally generated when few modes of a multimode Gaussian state are subject to photon-number-resolving detectors. From the output state Wigner function, we deduce that the state factorizes into a Gaussian gate applied to a finite Fock-superposition non-Gaussian state. The framework provides an approach to find the optimal strategy to generate a given target non-Gaussian state. We explore examples, such as the generation of cat states, weak cubic phase states, and bosonic code states, and achieve improvements of success probability over other schemes. Our framework also applies to the case in which the measured Gaussian state is mixed which is very important for the analysis of experimental imperfections such as photon loss. The framework has potential far-reaching implications to the generation of bosonic error-correcting codes and for the implementation of non-Gaussian gates using resource states, among other applications requiring non-Gaussianity

Motivation & Objective

• To develop a systematic analytic framework for probabilistic generation of non-Gaussian photonic states from multimode Gaussian states via photon-number-resolving detection.
• To unify and generalize existing heralding schemes for non-Gaussian state preparation under a single formalism.
• To enable optimization of optical circuits for maximum success probability and fidelity in generating specific target non-Gaussian states.
• To account for experimental imperfections by extending the framework to mixed Gaussian states, including photon loss.
• To improve success probabilities for key quantum states like cat states, weak cubic phase states, and NOON states compared to prior methods.

Proposed method

• Derives the Wigner function of the conditional output state after measuring (N−M) modes of an N-mode Gaussian state using photon-number-resolving detectors (PNRDs).
• Expresses the output state as a Gaussian operation applied to a finite Fock-superposition non-Gaussian state, enabling analytical characterization.
• Uses the displacement vector and covariance matrix of the input Gaussian state to compute the conditional output state and measurement success probability.
• Applies the formalism to both pure and mixed Gaussian input states, allowing modeling of realistic experimental losses.
• Optimizes the optical circuit (squeezing, displacements, interferometers) to maximize success probability for a given target non-Gaussian state.
• Validates the framework through explicit examples, including single-mode and multimode non-Gaussian states, using analytical and numerical methods.

Experimental results

Research questions

• RQ1 How can the conditional generation of non-Gaussian photonic states from multimode Gaussian states be systematically characterized using analytic expressions?
• RQ2 What is the optimal optical circuit configuration (squeezing, displacements, interferometers) to maximize the success probability for generating a specific target non-Gaussian state?
• RQ3 How does the framework account for experimental imperfections such as photon loss in the input Gaussian state?
• RQ4 Can the framework improve the success probability for generating key non-Gaussian states like cat states, weak cubic phase states, and NOON states compared to existing methods?
• RQ5 What is the relationship between the input Gaussian state parameters and the resulting non-Gaussian output state in terms of fidelity and probability?

Key findings

• The framework achieves a success probability of approximately 10–20% for generating cat states, significantly improving upon previous schemes.
• For weak cubic phase states, the method achieves a success probability of 4–6%, a substantial improvement over the prior machine learning-based method that reached 1–2%.
• The W state (|W_M⟩) can be generated with 100% fidelity and a maximum success probability of 25%, independent of the number of modes M.
• NOON states with N=2,3,4 are generated with success probabilities of 6.25%, 1.54%, and 0.55% respectively, exceeding previous results by factors of 3–25.
• The GKP state is generated with a fidelity of 0.818 and a success probability of 1.1% using a three-mode Gaussian state with two measured modes.
• The formalism generalizes to mixed Gaussian states, enabling analysis of photon loss and other experimental imperfections in realistic setups.