The state vector |Ψ〉 is an abstract mathematical entity; its origin and any underlying sub-quantum structure it might have is unknown. The processes that comprise measurement too are unknown. In our experience, certain physical interactions are recognizably ‘measurements’, but there are unknown others which are not. Their incognito presence makes the building of robust quantum computers difficult because they can surreptitiously destroy the quantum superposition of the state vector |Ψ〉. There is deeply mysterious, unexplained physics.

Entanglement, disdainfully viewed by Einstein, Podolsky, and Rosen (EPR) as spooky action at a distance^{26}, comes from postulates 1, 2 and 4. It implies that information can travel instantly from place to place unmindful of distance rather than at a finite speed capped by the speed of light as Einstein had postulated, unless there exist pre-programable “hidden variables” which control the system. Quantum mechanics was therefore incomplete in their view. This is the famous EPR paradox. Bohr promptly rebutted and promoted his complementarity theory^{27}. The spat between Einstein and Bohr drew a lot of media attention too (see Fig. 4.1).

In 1964, John Stewart Bell brilliantly ruled out the necessity of hidden variables in explaining entanglement. The result is famously known as Bell’s theorem.^{28} Entanglement is a physically observable phenomenon; entangled particles are now routinely produced in experiments.

In 1982, Aspect, et al^{29} provided experimental evidence of entanglement. In February 2017, even more stringent experimental evidence of entanglement was provided by Handsteiner, J., et al^{30}. Thus, when two or more particles are entangled, a measurement on any one particle or a combined measurement on a subgroup of particles will cause a ‘collapse’ to occur instantly on the remaining particles no matter where they are in the Universe. A group of entangled particles thus have a distributed existence yet function as a single unit.

### Entangled states are not factorizable states

The existence of entangled states in a composite system remains an enigma. The word ‘entanglement’ (in German, ‘Verschrankung’) was introduced by Schrödinger (among the first people to realize its strangeness) in the early days of quantum mechanics. To get an inkling of entanglement, consider a two-qubit system with the qubits labeled 1 and 2 with respective state vectors |Ψ1〉 = α1|0〉 + β1|1〉 and |Ψ2〉 = α2|0〉 + β2|1〉 . Therefore, any state of the form |Ψ1〉 ⊗ |Ψ2〉 ≡ |Ψ1Ψ2〉 in this composite two-qubit system can be written as

|Ψ1Ψ2〉 = α1α2|00〉 +α1β2 |01〉 +α2β1 |10〉 + β1β2 |11〉 ,

where the coefficient of each of the four basis vectors — |00〉, |01〉, |10〉, |11〉 — is interpreted as a probability amplitude. For example, |α1β2|^{2} gives the probability of finding qubit 1 in state |0〉 and qubit 2 in state |1〉 if the two-qubit system is measured. In this two-qubit system, we notice something unusual. For example, the so-called Bell state

has the remarkable property that there is no single qubit states |a〉 and |b〉 such that |Ψ〉 = |a〉 ⊗ |b〉 . A state of a composite system which cannot be written as a product of states (factorized) of its component systems is called an entangled state. Entanglement is a form of quantum superposition.

Factorizable states can be explained as follows. Let Alice and Bob each possess a qubit. Suppose, now, that Alice and Bob prepare their qubits independently, where both operations may depend on the output of a classical random number generator, e.g., a dice. Therefore, the source of the correlations is a classical random number generator. The two-qubit states which Alice and Bob can prepare in this way using their respective dice are called classically correlated or separable. All other two-qubit states are called entangled. Separable states occur because the dice are not related to each other. If the dice were related, (e.g., if one dice showed 2 the other would invariably show 5, etc.) the result would have been entangled states. Clearly, related dice function as a unit and can only acquire a group state.

Since an entangled state cannot be factorized, it is impossible to specify a pure state of its constituent components; we can only specify a group state. Interestingly, this group state is a pure state in the Hilbert space! In other words, we have knowledge of the correlation between measurement outcomes on qubits 1 and 2 but we cannot, in principle, identify a pure state with each of the qubits 1 and 2 individually. Entanglement is a fundamental difference between classical and quantum physics.

### 4.3 Interpretations of quantum mechanics

Abstract mathematical statements, if they are to describe the world to a human mind, need an isomorphic interpretation that establishes a correspondence between mathematical symbols and the rules by which those symbols are aggregated on the one hand and entities and mental concepts about our Universe on the other. The correspondences we discover may be many. When that happens, we seek a more unified view of our Universe by drawing analogies among the correspondences. In this Section we provide three well-known interpretations of quantum mechanics. The conceptual diversity of these interpretations is remarkable. In Part III, a sub-quantum level view of the quantum world is presented.

^{[26] }Einstein, Podolsky & Rosen (1935).

** ^{[27] }Bohr (1935).**

^{[28] }Bell (1964).

^{[29] }Aspect, Dalibart & Roger (1982). See also Aspect (1991).

^{[30] }Handsteiner, et al (2017). See also: Hensen, et al (2015); and Merali (2015).