PACS number: O3.65.Bz Abstract: Two or more quantum systems are said to be in an entangled or nonfactorisable state if their joint (supposedly pure) wave-function is not expressible as a product of individual wave functions but is instead a superposition of product states. It is only when the systems are in a factorisable state that they can be considered to be separated (in the sense of Bell). We show that whenever two quantum systems interact with each other, it is impossible that all factorisable states remain factorisable during the interaction unless the full Hamiltonian does not couple these systems so to say unless they do not really interact. We also present certain conditions under which particular factorisable states remain factorisable although they represent a bipartite system whose components mutually interact. We identify certain quasi-classical regimes that satisfy these conditions and show that they correspond to classical, pre-quantum, paradigms associated to the concept of particle
PACS number: O3.65.Bz Abstract: Two or more quantum systems are said to be in an entangled or nonfactorisable state if their joint (supposedly pure) wave-function is not expressible as a product of individual wave functions but is instead a superposition of product states. It is only when the systems are in a factorisable state that they can be considered to be separated (in the sense of Bell). We show that whenever two quantum systems interact with each other, it is impossible that all factorisable states remain factorisable during the interaction unless the full Hamiltonian does not couple these systems so to say unless they do not really interact. We also present certain conditions under which particular factorisable states remain factorisable although they represent a bipartite system whose components mutually interact. We identify certain quasi-classical regimes that satisfy these conditions and show that they correspond to classical, pre-quantum, paradigms associated to the concept of particle
Two or more quantum systems are said to be in an entangled or non-factorisable state if their joint (supposedly pure) wave-function is not expressible as a product of individual wave functions but is instead a superposition of product states. It is only when the systems are in a factorisable state that they can be considered to be separated (in the sense of Bell). We show that whenever two quantum systems interact with each other, it is impossible that all factorisable states remain factorisable during the interaction unless the full Hamiltonian does not couple these systems so to say unless they do not really interact. We also present certain conditions under which particular factorisable states remain factorisable although they represent a bipartite system whose components mutually interact and pay a particular attention to the case where the two particles interact mutually through an action at a distance in the three dimensional space.
Entanglement can be understood as an extraordinary degree of correlation between states of quantum systems – a correlation that cannot be given an explanation in terms of something like a common cause. Entanglement can occur between two or more quantum systems, and the most interesting case is when these correlations occur between systems that are space-like separated, meaning that changes made to one system are immediately correlated with changes in a distant system even though there is no time for a signal to travel between them. In this case one says that quantum entanglement leads to non-local correlations, or non-locality. More precisely, entanglement can be defined in the following way. Consider two particles, A and B, whose (pure) states can be represented by the state vectors ψA and ψB. Instead of representing the state of each particle individually, one can represent the composite two-particle system by another wavefunction, Ψ AB. If the two particles are unentangled , then the composite state is just the tensor product of the states of the components: Ψ AB = ψ A ⊗ ψ B; the state is then said to be factorable (or separable). If the particles are entangled, however, then the state of the composite system cannot be written as such a product of a definite state for A and a definite state for B. This is how an entangled state is defined for pure states: a state is entangled if and only if it cannot be factored: Ψ AB ≠ ψ A ⊗ ψ B.
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.
Three paradigms commonly used in classical, pre-quantum physics to describe particles (that is: the material point, the test-particle and the diluted particle (droplet model)) can be identified as limit-cases of a quantum regime in which pairs of particles interact without getting entangled with each other. This entanglement-free regime also provides a simplified model of what is called in the decoherence approach islands of classicality, that is, preferred bases that would be selected through evolution by a Darwinist mechanism that aims at optimising information. We show how, under very general conditions, coherent states are natural candidates for classical pointer states. This occurs essentially because, when a (supposedly bosonic) system coherently exchanges only one quantum at a time with the (supposedly bosonic) environment, coherent states of the system do not get entangled with the environment, due to the bosonic symmetry.
Introduction The first realization that the validity of the quantum superposition principle in the Hilbert space describing a composite quantum system may give rise to fundamentally new correlations between the constituent subsystems came in the landmark 1935 paper by Einstein, Podolsky, and Rosen (EPR), where it was shown how the measurement statistics of observables in certain quantum states could not be reproduced by assigning definite wavefunctions to individual subsystems. It was in response to the EPR paper that Schrödinger, in the same year, coined the term entanglement ( Verschränkung ) to acknowledge the failure of classical intuition in describing the relationship between the “parts” and the “whole” in the quantum world: Whenever one has a complete expectation catalog – a maximum total knowledge – a ψ function – for two completely separated bodies, …then one obviously has it also for the two bodies together. But the converse is not true. The best possible knowledge of a total system does not necessarily include total knowledge of all its parts, not even when these are fully separated from each other and at the moment are not influencing each other at all. While Bell's strengthening of the original EPR-paradox setting and the subsequent experimental verification of Bell inequalities irreversibly changed the perception of entanglement from a property of counterintuitive “spookiness” to (beyond reasonable doubt) an experimental reality, the concept and implications of entanglement continue to be associated with a host of physical, mathematical, and philosophical challenges. In particular, investigation of entanglement in both its qualitative and quantitative aspects has intensified under the impetus of quantum information science (QIS).
Quantum entanglement describes superposition states in multi-dimensional systems-at least two partite-which cannot be factorized and are thus non-separable. Non-separable states also exist in classical theories involving vector spaces. In both cases, it is possible to violate a Bell-like inequality. This has led to controversial discussions, which we resolve by identifying an operational distinction between the classical and quantum cases.This article is part of the theme issue 'The quantum theory of light'.