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The Bose--Marletto--Vedral experiment is a family of proposals to test gravitationally induced entanglement between initially independent masses. Sougato Bose and collaborators, and independently Chiara Marletto and Vlatko Vedral, described the idea in 2017. Each mass occupies two distinguishable spatial branches, encoding a qubit. Different branch separations give different Newtonian gravitational potential energies and hence different relative phases.^[ref]

For stationary branches at separations $r_{ab}$ held for duration $\tau$, the ideal model givesThe positive phase follows from negative Newtonian gravitational potential energy in $e^{-iV\tau /\hslash }$. A direct algebraic check separates local rephasing from genuine entanglement: the determinant of the two-by-two coefficient matrix vanishes exactly whenThus a large common phase alone is insufficient. For this ideal pure state, the concurrence is $|\sin (\Delta \varphi /2)|$, reaching one at an odd multiple of $\pi$.

The spin-based proposal uses two Stern-Gerlach interferometers: embedded spins label the spatial branches, the masses interact, and recombination transfers the branch information into measurable spin correlations. Measurements in different spin bases supply an entanglement witness. Unwanted electromagnetic interactions and quantum decoherence must be controlled for the gravitational interpretation.^[ref]

The mediator inference assumes initially unentangled systems, local coupling through a mediator, and no other entangling interaction. Under those assumptions, a mediator described only by commuting observables cannot generate the observed entanglement; at least two incompatible observables are required. This is an indirect test of nonclassical mediation, without selecting a complete high-energy theory of quantum gravity.^[ref]

Take $|L⟩=|0⟩$ and $|R⟩=|1⟩$. Initially the two masses are in the product stateThe branch-dependent Newtonian gravitational potential energy is $-G{m}^2/d_{ab}$. Under the stated approximation, only the $|RL⟩$ branch acquires an appreciable relative phase, so after time $t$,The determinant of its two-by-two coefficient matrix is $(1-e^{i\varphi })/4$, which is nonzero unless $\varphi$ is a multiple of $2\pi$. Thus the state generally has Schmidt rank two: the branch-dependent gravitational phase creates gravitationally induced entanglement.

An entanglement witness has a bound obeyed by every separable quantum state and violated by at least one entangled state. For a product state with Bloch vectors $\mathbf{r}$ and $\mathbf{s}$,by the Cauchy-Schwarz inequality. Convexity gives the same bound for every separable mixed state. Consequently $W>1$ certifies entanglement; in conventional operator form, one of $I\mp (X\otimes Z+Y\otimes Y)$ has negative expectation whenever the absolute-value criterion is violated.

For the state above, direct use of the Pauli matrices givesWith the supplied values,which is close to $\pi$. Hence $W\simeq 2.00$, and to the nearest integerThis is the operating principle of the Bose--Marletto--Vedral experiment.

The Schmidt decomposition of an entangled pure two-qubit state has two nonzero terms. Absorb both complex phases into the local basis vectors and interchange the labels of the second qubit to obtainIn this state the only nonzero same-axis two-qubit Pauli correlators areAll mixed-axis correlators vanish. Expanding $(\mathbf{a}\cdot \mathbf{\sigma })\otimes (\mathbf{b}\cdot \mathbf{\sigma })$ therefore gives

For the four stated vectors this becomesIt follows immediately thatEvery separable quantum state obeys the corresponding CHSH inequality with upper bound two. Set $s=2c_0{c}_1>0$, choose ${\beta }^{\prime }=\pi -\beta$, and take $\tan \beta =s$ with $0<\beta <\pi /2$. The displayed expression is thenThus every entangled pure two-qubit state has local measurement correlations that no separable state can reproduce, which is Gisin's theorem. In an ideal Bose--Marletto--Vedral experiment, optimized local measurements can therefore witness any nonzero pure-state entanglement generated during the gravitational interaction. If gravity is the only interaction between the masses, such a violation shows that the mediator cannot be described by a purely classical local variable under the assumptions of the proposal; experimentally, control of decoherence and nongravitational forces is essential to that inference.