Coherence as a Resource

Coherence is the presence of definite phase relationships in a superposition; it is the resource consumed by interference and destroyed by noise.

What coherence means

Coherence is the existence of well-defined relative phases among the components of a superposition. In the density matrix, coherence lives in the off-diagonal entries. A state with large off-diagonal terms is highly coherent; one with only diagonal entries is a classical mixture with no coherence.

Coherence versus populations

The diagonal entries of a density matrix are populations — the probabilities of each basis state. The off-diagonal entries are coherences — the phase relationships that make a superposition more than a classical mixture. Two states can have identical populations but different coherences, and only the coherent one supports interference.

A resource for computation

Coherence is the fuel that interference burns. Every quantum speedup depends on definite relative phases so that amplitude paths can add and cancel. Without coherence, a superposition collapses to a mere probability distribution and the algorithm degrades to classical random guessing. In this sense coherence, not superposition alone, is the operative resource.

How it is lost

Decoherence is precisely the loss of coherence: environmental coupling randomises relative phases and drives the off-diagonal terms to zero, characterised by the T2 time. The state drifts from pure toward mixed, and its computational value drains away with the coherences.

Basis dependence and measures

Coherence is defined relative to a chosen basis — a state fully coherent in the Z basis may be diagonal in the X basis. Resource theories of coherence quantify it with measures such as the l1-norm of the off-diagonal terms or the relative entropy of coherence. These frameworks let coherence be treated on the same rigorous footing as entanglement, as a quantifiable resource that operations can create, consume, or waste.