Equation field guide
Quantum phase and two-path entanglement
A quantum phase is an angle attached to a complex amplitude. One path’s phase can be changed by convention; phase differences between alternative paths can change interference and entanglement.
Why a phase is an angle
A complex amplitude can be represented as a point rotating around a circle. Its phase measures the rotation in radians. If a system has energy U for time t, quantum evolution adds a phase proportional to −Ut/ħ. Energy × time and ħ have the same units, leaving a dimensionless angle.
A common rotation of every amplitude cannot change interference. What matters is the angle between alternatives that can later be recombined.
Four alternatives for two masses
Each mass has two possible spatial branches, L and R. Together they have four alternatives: LL, LR, RL, and RR. Because the distances between the masses can differ across those alternatives, gravity can give the four amplitudes different phases.
The branch labels are part of the model, not four separately observed classical journeys. The calculation assumes coherent superpositions so their amplitudes can still interfere.
Which difference belongs to both masses?
Adding a phase to A’s L branch changes both LL and LR by the same amount. Adding a phase to B’s R branch changes both LR and RR. In the combination LL + RR − LR − RL, every such local addition appears once with each sign and cancels.
For an ideal balanced pure state, the remaining angle Δ gives concurrence |sin(Δ/2)|. This is zero at Δ = 0 modulo 2π and reaches one at Δ = π modulo 2π. Decoherence and unequal amplitudes need a different calculation.
What a witness adds
A calculated phase difference is a theoretical quantity. To make an experimental claim, researchers must preserve coherence, exclude nongravitational interactions, and measure correlations that certify entanglement.
Changing a quantum reference frame raises a separate question about which subsystems carry the entanglement. The algebraic cancellation of local phases alone does not establish the full frame-transformation result.