Bell State Measurement
The Bell measurement is an important concept in quantum information science: It is a joint quantum-mechanical measurement of two qubits that determines in which of the four Bell states the two qubits are in.
If the qubits were not in a Bell state before, they get projected into a Bell state (according to the projection rule of quantum measurements), and as Bell states are entangled, a Bell measurement is an entangling operation.
Bell-state measurement is the crucial step in quantum teleportation. The result of a Bell-state measurement is used by one's co-conspirator to reconstruct the original state of a teleported particle from half of an entangled pair (the "quantum channel") that was previously shared between the two ends.
Experiments which utilize so-called "linear evolution, local measurement" techniques cannot realize a complete Bell state measurement. Linear evolution means that the detection apparatus acts on each particle independently from the state or evolution of the other, and local measurement means that each particle is localized at a particular detector registering a "click" to indicate that a particle has been detected. Such devices can be constructed, for example, from mirrors, beam splitters, and wave plates, and are attractive from an experimental perspective because they are easy to use and have a high measurement cross-section.
For entanglement in a single qubit variable, only three distinct classes out of four Bell states are distinguishable using such linear optical techniques. This means two Bell states cannot be distinguished from each other, limiting the efficiency of quantum communication protocols such as teleportation. If a Bell state is measured from this ambiguous class, the teleportation event fails.
Entangling particles in multiple qubit variables, such as (for photonic systems) polarization and a two-element subset of orbital angular momentum states, allows the experimenter to trace over one variable and achieve a complete Bell state measurement in the other. Leveraging so-called hyper-entangled systems thus has an advantage for teleportation. It also has advantages for other protocols such as superdense coding, in which hyper-entanglement increases the channel capacity.
In general, for hyper-entanglement in variables, one can distinguish between at most classes out of Bell states using linear optical techniques.
Bell measurements of ion qubits in ion trap experiments, the distinction of all four states is possible.
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