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Bell State

A Bell state is one of four maximally entangled two-qubit states that form an orthonormal basis and serve as fundamental resources in quantum information.

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A Bell state is one of four maximally entangled pure states of two qubits in quantum mechanics. Together, they form the Bell basis, a standard basis for describing two-qubit systems. They are named after John Stewart Bell and provide elementary examples of quantum entanglement: the joint state cannot be expressed as a product of separate state vectors for the individual qubits. (quantum.cloud.ibm.com)

Mathematical definition

For qubits labeled AA and BB, the four Bell states are

∣Φ+⟩=∣00⟩+∣11⟩2,∣Φ−⟩=∣00⟩−∣11⟩2,|\Phi^+\rangle=\frac{|00\rangle+|11\rangle}{\sqrt2}, \qquad |\Phi^-\rangle=\frac{|00\rangle-|11\rangle}{\sqrt2},
∣Ψ+⟩=∣01⟩+∣10⟩2,∣Ψ−⟩=∣01⟩−∣10⟩2.|\Psi^+\rangle=\frac{|01\rangle+|10\rangle}{\sqrt2}, \qquad |\Psi^-\rangle=\frac{|01\rangle-|10\rangle}{\sqrt2}.

Here ∣ij⟩|ij\rangle abbreviates the tensor product ∣i⟩A⊗∣j⟩B|i\rangle_A\otimes|j\rangle_B. Each state is normalized, and distinct Bell states have zero inner product. They therefore constitute an orthonormal basis of the four-dimensional Hilbert space of two qubits. Any two-qubit pure state can be written as a linear combination of these four vectors. (quantum.cloud.ibm.com)

The plus and minus signs specify a relative phase between the two components of the superposition. This phase is physically significant, even though it does not alter the probabilities obtained by measuring both qubits in the computational basis. (quantum.cloud.ibm.com)

Maximal entanglement

For a Bell state ∣β⟩|\beta\rangle, the joint density matrix is

ρAB=∣β⟩⟨β∣.\rho_{AB}=|\beta\rangle\langle\beta|.

Tracing out either qubit gives

ρA=Tr⁡B(ρAB)=I22,ρB=Tr⁡A(ρAB)=I22,\rho_A=\operatorname{Tr}_B(\rho_{AB})=\frac{I_2}{2}, \qquad \rho_B=\operatorname{Tr}_A(\rho_{AB})=\frac{I_2}{2},

where I2I_2 is the two-dimensional identity matrix. Thus the joint system is in a pure state, while each qubit separately is maximally mixed. The von Neumann entropy of either reduced state, using logarithms to base two, is

S(ρA)=−Tr⁡(ρAlog⁡2ρA)=1.S(\rho_A)=-\operatorname{Tr}(\rho_A\log_2\rho_A)=1.

This is the maximum entanglement entropy possible for a pure state of two qubits. A shared Bell pair consequently represents one ebit, the standard unit of bipartite entanglement. (preskill.caltech.edu)

The four Bell states have the same amount of entanglement. Applying suitable single-qubit unitary operations converts any one into another; more generally, every maximally entangled pure state of two qubits is locally unitarily equivalent to a Bell state. (preskill.caltech.edu)

Measurement correlations

When both qubits are measured in the computational basis:

State Possible joint outcomes Probability of each
∣Φ+⟩\vert \Phi^+\rangle, ∣Φ−⟩\vert \Phi^-\rangle 0000, 1111 1/21/2
∣Ψ+⟩\vert \Psi^+\rangle, ∣Ψ−⟩\vert \Psi^-\rangle 0101, 1010 1/21/2

Thus the Φ\Phi states give matching outcomes, whereas the Ψ\Psi states give opposite outcomes. Each individual outcome remains uniformly random. Measurements in other bases reveal the relative phases that computational-basis measurements cannot distinguish. (quantum.cloud.ibm.com)

Perfect computational-basis correlation alone does not establish entanglement. The separable mixed state

ρmix=12∣00⟩⟨00∣+12∣11⟩⟨11∣\rho_{\mathrm{mix}} =\frac12|00\rangle\langle00| +\frac12|11\rangle\langle11|

has the same computational-basis outcome probabilities as ∣Φ+⟩|\Phi^+\rangle, but lacks the quantum coherence between ∣00⟩|00\rangle and ∣11⟩|11\rangle. Measurements in additional bases distinguish these states. (quantum.cloud.ibm.com)

These correlations do not permit faster-than-light communication. Without receiving the other observer’s outcome through a classical channel, an observer sees only random local results; the classical communication required by teleportation cannot be bypassed using entanglement. (quantum.cloud.ibm.com)

Preparation and Bell-basis measurement

A standard circuit prepares ∣Φ+⟩|\Phi^+\rangle from ∣00⟩|00\rangle using two quantum gates:

  1. Apply a Hadamard gate to qubit AA.
  2. Apply a controlled-NOT gate, with AA as control and BB as target.

The state evolves as

∣00⟩⟶∣00⟩+∣10⟩2⟶∣00⟩+∣11⟩2.|00\rangle \longrightarrow \frac{|00\rangle+|10\rangle}{\sqrt2} \longrightarrow \frac{|00\rangle+|11\rangle}{\sqrt2}.

Local bit-flip and phase-flip operations then produce the other Bell states. (quantum.cloud.ibm.com)

An ideal Bell-basis measurement distinguishes the four states by reversing this circuit: apply controlled-NOT, then Hadamard on AA, and finally measure both qubits in the computational basis. With the ordering used here, the mappings are

Φ+↦00,Φ−↦10,Ψ+↦01,Ψ−↦11.\Phi^+\mapsto00,\quad \Phi^-\mapsto10,\quad \Psi^+\mapsto01,\quad \Psi^-\mapsto11.

The ability to distinguish all four states is a joint-system operation, not a measurement of either qubit alone. (quantum.cloud.ibm.com)

Bell inequalities

Bell states can produce measurement statistics incompatible with local hidden-variable models. In the CHSH inequality, such models satisfy ∣S∣≤2|S|\leq2. With appropriately chosen local measurement settings, an ideal Bell pair attains ∣S∣=22|S|=2\sqrt2, the maximum allowed by quantum mechanics. This provides a quantitative demonstration of Bell’s theorem. A violation depends on the measurement settings; merely preparing a Bell state and measuring both qubits in one fixed basis is not a Bell test. (quantum.cloud.ibm.com)

Quantum-information applications

Two fundamental communication protocols use Bell pairs as a shared resource:

  • Quantum teleportation: one shared Bell pair, a Bell-basis measurement, and two classical bits of communication transfer an unknown qubit state to a receiver. The receiver applies an outcome-dependent correction. The protocol transfers quantum information, not matter. (preskill.caltech.edu)
  • Superdense coding: a sender encodes two classical bits by converting a shared Bell pair into one of the four Bell states through operations on their own qubit. After that qubit is transmitted, the receiver distinguishes the states using a Bell-basis measurement. The protocol requires previously shared entanglement in addition to the transmitted qubit. (quantum.cloud.ibm.com)

References

  1. Quantum information | IBM Quantum Learningquantum.cloud.ibm.com
  2. Introduction | IBM Quantum Learningquantum.cloud.ibm.com
  3. Chapter 5: Quantum Information Theorypreskill.caltech.edu
  4. Ph/CS 219A Quantum Computation, Lecture 8: Superdense Coding and Quantum Teleportationpreskill.caltech.edu
  5. Superdense coding | IBM Quantum Learningquantum.cloud.ibm.com
  6. Quantum Teleportation | IBM Quantum Learningquantum.cloud.ibm.com
  7. Quantum teleportation | IBM Quantum Learningquantum.cloud.ibm.com
  8. CHSH game | IBM Quantum Learningquantum.cloud.ibm.com
  9. CHSH inequality | IBM Quantum Documentationquantum.cloud.ibm.com