Entangled states : Classical versus Quantum

TL;DR

This paper provides an elementary proof that entangled states are inherently quantum and cannot occur in classical mechanics, highlighting the fundamental differences in the mathematical structures of classical and quantum states.

Contribution

It offers a rigorous proof distinguishing classical from quantum entanglement and explains the origin of entanglement in quantum systems using mathematical concepts.

Findings

Entangled states cannot arise in classical mechanics.

Classical states are always product states, unlike quantum states.

Entanglement is a non-commutative phenomenon.

Abstract

Quantum mechanics of composite systems, gives rise to certain special states called entangled states. A physical system, that is in an entangled state displays an intricate correlation between its subsystems. There are also some composite quantum states (classically correlated states or separable states) that are not entangled. It is generally claimed, often without a rigorous proof to support, that these intricate correlations of an entangled state cannot occur in a classical system. This expository article, provides an elementary proof that entangled states cannot arise in the setting of classical mechanics. In addition, a detailed description of the origin of entanglement in quantum systems is included. The mathematical concepts that are necessary for this purpose are presented. The absence of entanglement in the classical setting is due to the fact that every pure classical state of a composite system is a product state, that is, a tensor product of two pure states of the subsystems. In contrast, there are pure composite quantum states that cannot be expressed in the form of a product state or even by a convex sum of product states. Roughly speaking, this is because classical states are positive valued functions on the phase-space while quantum states are positive linear operators. The structure of the tensor product between two commutative spaces of scalar valued functions is drastically different from that of the tensor product between two non-commutative spaces of linear operators. In other words, entanglement is a non-commutative phenomenon.