The Bell-State Measurement (BSM) is a fundamental operation in quantum information processing, playing a crucial role in communication, computation, and quantum teleportation.
The BSM is a joint measurement on two qubits that projects them onto one of the four Bell sates.
What are the four Bell states?
There are four maximally entangled two-qubit states, known as the Bell states, that form a complete basis for two-qubit entanglement.
$$ \begin{matrix} \ket{\phi^{+}}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11}),& &\ket{\phi^{-}}=\frac{1}{\sqrt{2}}(\ket{00}-\ket{11}), \\ \ket{\psi^{+}}=\frac{1}{\sqrt{2}}(\ket{01}+\ket{10}),& & \ket{\psi^{-}}=\frac{1}{\sqrt{2}}(\ket{01}-\ket{10}). \end{matrix} $$
What is special about the Bell states?
In each Bell state, the two qubits are perfectly correlated: measuring one qubit immediately determines the outcome of measuring the other. Individually, each qubit’s measurement outcome is completely random. The correlations appear only when the two results are compared.
What is a Bell-state measurement?
The purpose of the BSM is to discriminate between the four Bell states. They are the simultaneous eigenstates of two joint Pauli measurements: one two-qubit Pauli measurement comparing the qubits in the computational basis (ZZ), and the other comparing them in the complementary basis (XX). Each Bell state corresponds to a specific pair of ±1 eigenvalues associated with these two Pauli operators. This makes the Bell state perfectly identifiable by an ideal Bell-state measurement.
Are Bell-state measurements easy in photonic?
Unfortunately, a fully deterministic linear-optical BSM is fundamentally impossible because photons do not naturally interact. When no additional resources are used, only two of the four Bell states can be unambiguously distinguished. Overcoming this limitation requires extra photons or nonlinear interactions, making the BSM a key technical challenge in photonic quantum technologies.
How are Bell-state measurements used in photonic quantum computing?
Despite this limitation, linear-optical BSM is a central tool for photonic fault-tolerant quantum computing.
- It can be used in fusion-based quantum computing, to perform entanglement and measurement simultaneously.
- It can be used to perform entangling gates in hybrid spin-photon quantum computing architectures.