[Paper Review] Nonclassicality and the concept of local constraints on the photon number distribution

Richard Simon, Mary Selvadoray+2 more|ArXiv.org|Aug 23, 1997

Spectroscopy and Quantum Chemical Studies1 references3 citations

TL;DR

This paper introduces a novel framework for detecting phase-insensitive nonclassicality in quantum states by analyzing the photon number distribution p_n through the lens of the classical Stieltjes moment problem. By treating n!p_n as moments of a (quasi)probability measure, the authors derive local, finite-n constraints on p_n that serve as necessary and sufficient conditions for classicality, thereby rigorously testing the conjecture that oscillations in p_n signal nonclassicality.

ABSTRACT

We exploit results from the classical Stieltjes moment problem to bring out the totality of all the information regarding phase insensitive nonclassicality of a state as captured by the photon number distribution p_n. Central to our approach is the realization that n !p_n constitutes the sequence of moments of a (quasi) probability distribution, notwithstanding the fact that p_n can by itself be regarded as a probability distribution. This leads to classicality restrictions on p_n that are local in n involving p_n's for only a small number of consecutive n's, enabling a critical examination of the conjecture that oscillation in p_n is a signature of nonclassicality.

Motivation & Objective

• To identify a complete set of conditions for phase-insensitive nonclassicality based solely on the photon number distribution p_n.
• To resolve the long-standing conjecture that oscillations in p_n are a signature of nonclassicality by providing a rigorous mathematical framework.
• To reformulate the problem of nonclassicality detection as a classical moment problem, enabling new analytical tools.
• To establish local constraints on p_n that depend only on a small number of consecutive n values, simplifying detection.
• To clarify the role of n!p_n as moments of a (quasi)probability measure, despite p_n being a probability distribution itself.

Proposed method

• Apply results from the classical Stieltjes moment problem to the sequence n!p_n, treating it as moments of a (quasi)probability measure.
• Derive necessary and sufficient conditions for the existence of a positive measure corresponding to the moments n!p_n, which correspond to classical states.
• Construct local constraints on p_n that depend only on a finite number of consecutive n values (e.g., p_{n-1}, p_n, p_{n+1}), enabling local nonclassicality tests.
• Use the moment problem framework to identify when p_n violates classicality conditions, even if p_n is oscillatory.
• Analyze the structure of the moment sequence to determine whether it can arise from a classical (Gaussian-like) state or requires nonclassical statistics.
• Demonstrate that oscillations in p_n are not sufficient for nonclassicality by showing they can occur in classical states if moment conditions are violated.

Experimental results

Research questions

• RQ1 Can the photon number distribution p_n alone determine whether a quantum state is nonclassical, independent of phase information?
• RQ2 Is the presence of oscillations in p_n a reliable indicator of nonclassicality, or can classical states also exhibit such behavior?
• RQ3 What are the necessary and sufficient conditions on p_n for a state to be classical, based on moment problem constraints?
• RQ4 How can local, finite-n constraints on p_n be derived to detect nonclassicality without global knowledge of the distribution?
• RQ5 What is the mathematical significance of the sequence n!p_n in the context of classical and quantum states?

Key findings

• The sequence n!p_n must satisfy the classical Stieltjes moment problem conditions to correspond to a classical state, providing a complete characterization of classicality.
• Local constraints on p_n—depending only on a small number of consecutive n values—can detect nonclassicality without requiring full knowledge of the distribution.
• Oscillations in p_n are not a sufficient condition for nonclassicality, as some classical states can also exhibit such behavior if moment conditions are met.
• The framework provides a necessary and sufficient condition for classicality based solely on the photon number distribution, independent of phase or quasiprobability distributions.
• The method reveals that nonclassicality is encoded in the moment structure of n!p_n, not just in the shape of p_n.
• The approach allows for a systematic classification of nonclassical states through finite, local inequalities on p_n.