[Paper Review] Photon Antibunching, Sub-Poisson Statistics and Cauchy-Bunyakovsky and Bell's Inequalities

И. В. Волович|arXiv (Cornell University)|Jun 9, 2011

Quantum Mechanics and Applications2 references3 citations

TL;DR

This paper rigorously demonstrates that sub-Poisson photon statistics and photon antibunching—key signatures of non-classical light—cannot be described by classical probabilistic models. By showing that the variance-to-mean difference $ K $ is negative for quantum states (e.g., $ K = -n $ for $ n $-particle states), and that this violates the Cauchy-Bunyakovsky inequality, the paper proves the impossibility of classical hidden variable or random field representations for such quantum states.

ABSTRACT

We discuss some mathematical aspects of photon antibunching and sub-Poisson photon statistics. It is known that Bell's inequalities for entangled states can be reduced to the Cauchy-Bunyakovsky inequalities. In this note some rigorous results on impossibility of classical hidden variables representations of certain quantum correlation functions are proved which are also based on the Cauchy-Bunyakovsky inequalities. The difference K between the variance and the mean as a measure of non-classicality of a state is discussed. For the classical case K is nonnegative while for the n-particle state it is negative and moreover it equals -n. The non-classicality of quantum states discussed here for the sub-Poisson statistics is different from another non-classicality called entanglement though both can be traced to the violation of the Cauchy-Bunyakovsky inequality.

Motivation & Objective

• To establish a rigorous mathematical framework for sub-Poisson photon statistics and photon antibunching as non-classical phenomena.
• To prove the impossibility of classical probabilistic representations (hidden variables) for quantum correlation functions in these cases.
• To clarify the distinction between non-classicality due to sub-Poisson statistics and that due to entanglement, despite both arising from Cauchy-Bunyakovsky inequality violation.
• To introduce and analyze the measure $ K = \langle \Delta n^2 \rangle - \langle n \rangle $ as an indicator of non-classicality.
• To extend the connection between quantum non-classicality and foundational inequalities, linking sub-Poisson statistics to Bell-type nonlocality via the Cauchy-Bunyakovsky inequality.

Proposed method

• Defines sub-Poisson statistics via the variance-to-mean difference $ K = \langle a^{*2}a^2 \rangle - \langle a^*a \rangle^2 $, showing $ K < 0 $ implies non-classicality.
• Uses the commutation relation $[a, a^*] = 1$ to derive $ K = ||a^2\psi||^2 - ||a\psi||^4 $, enabling explicit computation for Fock states.
• Applies the Cauchy-Bunyakovsky inequality to show that any classical random variable representation would require $ K \geq 0 $, contradicting quantum results.
• Proves that $ n $-particle states yield $ K = -n $, a negative value incompatible with classical probability.
• Analyzes the two-time correlation function $ P(\tau) $ to define antibunching as $ P(\tau) > P(0) $, which violates classical stationarity constraints.
• Demonstrates that if $ P(\tau) > P(0) $, then no classical stochastic process can reproduce the correlation, thus proving non-classicality.

Experimental results

Research questions

• RQ1 Can sub-Poisson photon statistics be described by a classical hidden variable model?
• RQ2 What is the role of the Cauchy-Bunyakovsky inequality in establishing the non-classical nature of quantum states?
• RQ3 How does the quantity $ K = \langle \Delta n^2 \rangle - \langle n \rangle $ serve as a quantitative measure of non-classicality?
• RQ4 Is there a fundamental difference between non-classicality due to sub-Poisson statistics and that due to entanglement?
• RQ5 Can the spacetime dependence of non-classical states like antibunched light be systematically analyzed using this framework?

Key findings

• For $ n $-particle Fock states, the quantity $ K $ is exactly $ -n $, demonstrating a clear quantum signature incompatible with classical statistics.
• The Cauchy-Bunyakovsky inequality implies that classical models must satisfy $ K \geq 0 $, but quantum states violate this, proving the impossibility of classical hidden variable representations.
• Photon antibunching, defined by $ P(\tau) > P(0) $ for $ \tau > 0 $, cannot be modeled by any classical stationary stochastic process.
• The measure $ K $ is negative for sub-Poisson states and vanishes for Poisson (coherent) states, providing a quantitative distinction between classical and quantum light.
• Non-classicality in sub-Poisson statistics and in entanglement both stem from violation of the Cauchy-Bunyakovsky inequality, but represent distinct physical phenomena.
• The paper establishes a rigorous mathematical link between foundational quantum inequalities and observable non-classical light properties like antibunching and sub-Poisson statistics.