Equation 7 · How Much of Gravitationally Induced Entanglement Is in the Eye of the Frame
What does this equation mean?
Each of the four joint paths has a gravitational phase. A phase added only to one branch of mass A or B can be changed locally and cannot create entanglement between them. Add the two matching-path phases and subtract the two crossed-path phases: the local additions cancel, leaving the physically relevant two-body phase combination.
What the alternating signs remove
Suppose a change of local phase convention adds α_L or α_R to A’s paths and β_L or β_R to B’s paths. The four joint phases then gain α_L+β_L, α_L+β_R, α_R+β_L, and α_R+β_R. Add LL and RR and subtract LR and RL: the added terms cancel exactly. The result therefore tracks a two-body relationship rather than an arbitrary choice of local phase origin.
Read it piece by piece
Entangling phase difference
The phase combination left after local phases of A and B cancel. It compares matching branch pairs (LL and RR) with crossed pairs (LR and RL). In an ideal balanced, pure two-branch state, its value modulo 2π determines the generated two-qubit entanglement.
Explore this idea →Both masses on L
The gravitational phase when A takes its L branch and B takes its L branch. It enters with a plus sign in the invariant combination.
Explore this idea →Both masses on R
The gravitational phase when A takes its R branch and B takes its R branch. It also enters with a plus sign.
Explore this idea →A on L, B on R
The phase of the crossed branch pair. It is subtracted so phases belonging only to one local branch cancel out.
Explore this idea →A on R, B on L
The phase of the other crossed branch pair. It is also subtracted in the local-phase-invariant combination.
Explore this idea →How to interpret it
Let φ_{ij} change by α_i + β_j, where α_i is any phase applied locally to A’s branch i and β_j is any phase applied locally to B’s branch j. In φ_{LL}+φ_{RR}−φ_{LR}−φ_{RL}, every α and β appears once with a plus and once with a minus, so all cancel. For the ideal initial state (|L⟩+|R⟩)⊗(|L⟩+|R⟩)/2, the concurrence is |sin(φ_ent/2)|: zero when φ_ent is 0 modulo 2π and maximal when it is π modulo 2π. Real experiments can lose coherence and need an actual witness measurement.
Try the four branch phases
This is an idealized geometry: mass A and mass B each have a left and right branch. Matching branches are separated by a gap d; crossed branches are farther apart by a horizontal offset Δx. Change the values to see which phase difference survives.
Illustrative symmetric geometry. The phase and concurrence calculations assume coherent, stationary branches and no other interactions. The article’s physical experiment has additional constraints.
From a phase difference to entanglement
In a balanced, coherent two-branch state, φ_ent = 0 modulo 2π means the four amplitudes can be separated into one state for A times one state for B. A nonzero invariant phase generally prevents that factorization. The pure-state concurrence |sin(φ_ent/2)| is a precise way to quantify this in the idealized two-qubit model; it does not by itself describe decoherence or measurement noise.
A compact derivation for advanced readers
Give the four normalized branch amplitudes the values L = exp(iφ_LL)/2, R = exp(iφ_LR)/2, L = exp(iφ_RL)/2, and R = exp(iφ_RR)/2. For a pure two-qubit state, concurrence is C = 2|L R − R L|. Substituting the phases yields C = |sin((φ_LL + φ_RR − φ_LR − φ_RL)/2)|. The determinant is zero exactly when the two-by-two amplitude matrix has rank one and the state factorizes. This derivation requires equal branch amplitudes and retained coherence.
What this does and does not prove
The cancellation above proves invariance under local rephasing. The article discusses a further quantum-reference-frame calculation, which is a separate claim requiring its own transformation. A positive experimental entanglement witness also depends on ruling out nongravitational interactions and on the assumptions used to infer the nature of the mediator.
Sources and further reading
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