The term “entanglement” was first coined in 1935 by Erwin Schrödinger in a letter to Albert Einstein while describing the correlation between a pair of particles which had interacted before and then separated. Previously, such a correlation had played key role in the argument put forth in the seminal paper by Einstein, Podolsky, and Rosen in 1935 (EPR Paradox) where they questioned the completeness of quantum mechanics based on a thought experiment (discussed below). Back then, Niels Bohr’s ideas concerning interpretation of quantum theory and complementarity dominated the conceptual understanding of the quantum theory. According to complementarity principle, canonically conjugate variables such as position and momentum do not have simultaneous reality in the quantum theory.
Einstein’s main criticism to the Copenhagen Interpretation of quantum mechanics was its inability to predict the outcome of a simple experiment. For instance, the position measurement of a quantum system (or any other observable property), and the view that the system doesn’t possess a definite value corresponding to it, until the measurement has been performed, leads to the quantum indeterminism. However, Einstein was of the realist view and contrarily believed that quantum systems do possess definite values independent of the measurement and the act of measurement merely reveals it. He famously told the philosopher Hilary Putna “Look, I don’t believe that when I am not in my bedroom my bed spreads out all over the room, and whenever I open the door and come in, it jumps into the corner.”
Einstein believed that the description of quantum mechanics was incomplete and some extra information (local hidden variables) was missing which would make it deterministic. In this line of thought, EPR envisaged the following thought experiment. They considered a situation where a pair of quantum systems interacts in such a way that their position and momentum get correlated and this correlation is maintained even when the two particles are widely separated in space. Consequently, the measurement of position (or momentum) for one of the systems would determine the position (or momentum) of the other particle located far away; simultaneously maintaining the conditions of locality. On this basis, they argued that locality and the completeness of the quantum description are not maintained by the wave function based description of a system in quantum theory.
Later in 1965, John S. Bell came up with a mathematical formulation in the form of an inequality, known as Bell’s inequality, to test the local hidden variable theories. It puts a bound of "2" on the correlations captured by such theories. This inequality is violated (with some assumptions) by entanglement and from this one can infer that quantum mechanics is not a local hidden variable theory. Thus, quantum indeterminism is intrinsic to the quantum systems.
Quantum entanglement is a type of non-classical correlation shared among quantum systems. We see correlations in our daily lives; for example, the colour of a pair of socks, position and momentum correlations of a projectile which split into two pieces in its trajectory, etc. Quantum entanglement is unique in the sense that quantum systems offer a choice of basis for the measurement and entanglement is a correlation which is maintained in more than one basis.
\( \left|{\psi}\right\rangle_{AB} \neq \left|{\psi}\right\rangle_{A} \otimes \left|{\psi}\right\rangle_{B} \)
Mathematically speaking, for pure states \((Tr(\rho^2)=1)\), if the state of a joint quantum system can’t be written as the product of the states from subsystem Hilbert spaces then the state is said to be entangled, else separable.
Whereas, for mixed states \((Tr(\rho^2)<1)\), if the state of joint quantum system can’t be written as the convex sum of the states from its subsystem Hilbert spaces then the joint state is said to be entangled, else separable.
\( \rho_{AB} \neq \sum_i p_i \rho_A \otimes \rho_B \)
A good measure of entanglement \(E (\hat{\rho})\) must assign a value zero for separable state and one for maximally entangled state. It should be non-increasing under local operation and classical communication (LOCC). Negativity is such a measure of entanglement based on the positive partial transpose (PPT) criterion. It is defined as the twice of the absolute some of the negative eigen values of the partially transposed density matrix with respect to one of the subsystems.