Published equation contexts
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.
None of this makes the experiment ambiguous, and this is the paper’s second and sharper result. The branch separations that enter the gravitational phase each mass’s superposition picks up are relational quantities — distances between mass A’s branches and mass B’s branches — and relational quantities do not change when you change which system you have designated the reference frame. Write = G t / ( ) for the phase a pair of branches i,j \{L,R\} accumulates over interaction time t at separation , with G Newton’s constant and the reduced Planck constant. Three of the four possible combinations of these branch-pair phases can always be removed…
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.
Read this term in its guide →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.
Read this term in its guide →The gravitational phase when A takes its R branch and B takes its R branch. It also enters with a plus sign.
Read this term in its guide →The phase of the crossed branch pair. It is subtracted so phases belonging only to one local branch cancel out.
Read this term in its guide →The phase of the other crossed branch pair. It is also subtracted in the local-phase-invariant combination.
Read this term in its guide →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.
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.
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.
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.
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.
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.
A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.
Equation 7 · Quantum Relativity
This alternating sum isolates the branch-phase difference that local phase choices cannot remove. In the ideal balanced state, it controls whether the two path qubits are entangled.
None of this makes the experiment ambiguous, and this is the paper’s second and sharper result. The branch separations that enter the gravitational phase each mass’s superposition picks up are relational quantities — distances between mass A’s branches and mass B’s branches — and relational quantities do not change when you change which system you have designated the reference frame. Write = G t / ( ) for the phase a pair of branches i,j \{L,R\} accumulates over interaction time t at separation , with G Newton’s constant and the reduced Planck constant. Three of the four possible combinations of these branch-pair phases can always be removed…