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A device implementing a Bell-basis measurement on two qubits. A complete analyzer distinguishes all four Bell states. A basic passive linear optics analyzer for two polarization qubits uses a balanced beam splitter, polarizing beam splitters and single-photon detectors. Its ideal unambiguous success probability is $1/2$: the two opposite-polarization Bell states have distinct click patterns, while the two same-polarization Bell states remain mutually ambiguous. Additional physical resources can change this restriction. A destructive analyzer absorbs the measured photons; it need not preserve the detected Bell state.

In entanglement swapping, start with independent Bell pairs $|{\varphi }^{+}{⟩}_{12}|{\varphi }^{+}{⟩}_{34}$. Send qubits 2 and 3 to a central Bell state analyzer, retaining 1 and 4 at separate endpoints. Reordering the tensor product factors to group 23 and 14 gives the identityConsequently each ideal Bell-basis measurement outcome has probability $1/4$ and projects 14 onto the corresponding Bell state, even though those two photons have never interacted. Communicating the two-bit result permits an endpoint Pauli gate to convert every outcome into the same chosen Bell pair. Without that record, the endpoints have ${\rho }_{14}=I_4/4$, a product state, rather than a known entangled state.

The optical Bell state analyzer overlaps photons 2 and 3 at a balanced beam splitter, with matched spatial modes, spectra and arrival times, and resolves the output photon polarizations using polarizing beam splitters and single-photon detectors. Two-photon interference maps Bell states to different output patterns. The antisymmetric $|{\psi }^{-}⟩$ gives one photon in each output with opposite polarizations. The symmetric $|{\psi }^{+}⟩$ gives two opposite-polarization photons in the same output; the two polarization detectors there can resolve this pattern. The $|{\varphi }^{\pm }⟩$ states give same-polarization pairs and remain mutually ambiguous in this basic setup. The detected pattern therefore heralds a particular endpoint Bell state. With ideal detection this arrangement succeeds unambiguously on two of four equally likely inputs, with probability $1/2$; the successful events can be selected. A complete analyzer would accept all four outcomes. The endpoint quantum entanglement is conditional on the measurement record, so it enables no instantaneous signalling.^[ref]

Write $R_z(\theta )=e^{-i\theta Z/2}$ and $R_x(\theta )=e^{-i\theta X/2}$, using the Pauli operators. The Hadamard gate obeys $HZH=X$, so unitary conjugation of the matrix exponential gives $H{R}_z(\beta )H=R_x(\beta )$. Therefore the symmetric z-x-z beam splitter factorization isThe rightmost operator acts first. Each factor is unitary, which also proves the given matrix is unitary. Choosinggives $U_{\mathrm{BS}}=-iH$, so it is exactly a Hadamard gate up to global phase.

For the specified identical splitters with $\alpha =0$, put $c=\cos (\beta /2)$ and $s=\sin (\beta /2)$. Here $U_{\mathrm{BS}}=R_x(\beta )$, and direct multiplication givesThus the upper-port probability iswith $P_1={\sin }^2\beta {\cos }^2(\varphi /2)$. These identical beam splitters in a two-path interferometer give a nonconstant interference fringe whenever $\sin \beta \ne 0$. With this matrix and port convention, two identical balanced splitters swap the bright and dark ports relative to the two-Hadamard model: at $\varphi =0$, $R_x(\pi /2{)}^2=-iX$. The reflection phases must remain in the calculation.

The upper-port maximum and minimum are $1$ and ${\cos }^2\beta$. Hence the peak-to-peak modulation is ${\sin }^2\beta$ and its normalized interferometric visibility isBoth measures of contrast are maximal precisely forThese are balanced beam splitters, with equal single-input transmission and reflection probabilities. When $\beta \in \pi \mathbb{Z}$, only one route contributes to the output and the fringes disappear.

Interference between indistinguishable alternatives involving two photons. The Hong-Ou-Mandel effect is an example: for identical photons arriving at opposite inputs of a balanced beam splitter, the two alternatives for one photon in each output cancel. A Bell state analyzer exploits the dependence of the output statistics on the symmetry of the joint photon polarization state. Matching arrival time, spectrum and spatial mode prevents those alternatives from being distinguished by unobserved degrees of freedom.