Entanglement is a quantum phenomenon in which there is a certain relationship between the whole quantum system and its parts. Consider a two-qubit system in an arbitrary state. That is,
∣Ψ⟩ = a∣0⟩∣0⟩ + b∣0⟩∣1⟩ + c∣1⟩∣0⟩ + d∣1⟩∣1⟩,
where |a|² + |b|² + |c|² + |d|² = 1.
If a·d ≠ b·c, then the system is entangled.
To explain the intuition behind this definition, consider the two qubits individually. The most general way of describing them is:
∣Ψ₁⟩ = α ∣0⟩ + β ∣1⟩,
where |α|² + |β|² = 1, and
∣Ψ₂⟩ = γ ∣0⟩ + δ ∣1⟩,
where |γ|² + |δ|² = 1.
If the relationship between the whole and the parts was similar to that in classical physics, it would seem natural to represent the entire system as:
∣Ψ₁⟩∣Ψ₂⟩ = (α ∣0⟩ + β ∣1⟩)(γ ∣0⟩ + δ ∣1⟩)
= α·γ ∣0⟩∣0⟩ + α·δ ∣0⟩∣1⟩ + β·γ ∣1⟩∣0⟩ + β·δ ∣1⟩∣1⟩.
However, it turns out that there are states of this system that cannot be represented this way. That is, for given values of a, b, c and d, there is no way that we can assign values to α, β, γ and δ to simultaneously satisfy:
(1) a = α·γ
(2) b = α·δ
(3) c = β·γ
(4) d = β·δ.
A famous example is that a = d = 1 and b = c = 0. Proof: to satisfy equations (1) and (4), we require α, β, γ and δ to all be non-zero; however, to satisfy equation (2), one of α and δ have to be zero; notice that we have already reasoned that α and δ both have to be non-zero; hence, this system of equations does not have a solution.
Equations that have no solutions are nothing new to mathematicians and physicists. Despite this, entanglement has huge implications for quantum technology.
As the analysis of entanglement, first conceived by Einstein in a thought experiment, grew more sophisticated, it began to spur ideas for various forms of quantum technology. Subsequent results, both theoretical and experimental, showed that the role of entanglement in numerous quantum protocols cannot be underestimated. Today, it pervades the research literature. The following is a non-exhaustive list:
Quantum metrology: This is the study of precise measurements. Entanglement allows the precision to achieve so-called Heisenberg scaling.
Quantum communication: Many quantum communication protocols, from quantum key distribution to quantum teleportation, rely heavily on entanglement. For example, the protocols may involve steps like entanglement swapping or entanglement purification.
Quantum simulation: An important long-standing goal of physicists is to gain a greater understanding of the various states of matter. Entanglement is naturally present in some extremely interesting material systems. Simulating these systems on quantum processors is currently producing exciting results.
Quantum computation: Measures of entanglement typically grow to their largest values during the most crucial stages of a quantum algorithm. When there is redundant information in a large superposition, classical computers can reproduce the results of the computation. However, large amounts of entanglement put a stop to this — the information stored in the superposition is too highly and complexly correlated for classical computers to process efficiently.
