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What Are Bell States & How To Express Quantum Systems Using Them

What Are Bell States & How To Express Quantum Systems Using Them

Let's explore Bell states popularly used in Quantum Teleportation and Superdense Coding.

Dr. Ashish Bamania's avatar
Dr. Ashish Bamania
Aug 02, 2025
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Into Quantum
Into Quantum
What Are Bell States & How To Express Quantum Systems Using Them
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Photograph of John Stewart Bell, a physicist best known for formulating Bell’s theorem, which demonstrated that quantum entanglement cannot be explained by any local hidden variable theory. (Source: Wikipedia)

Bell states are maximally entangled two-qubit states, named after the physicist John Stewart Bell.

‘Entanglement’ means that the measurement of just one qubit instantaneously affects the other, regardless of how far apart they are placed.

‘Maximally entangled’ means that the two qubits are fully correlated, or knowing everything about one tells us everything about the other.

Take an example of a Bell state |Φ+> as follows:

It is a maximally entangled state of two qubits (let’s call them A and B).

When we measure |Φ+>, the system either collapses into |00> or |11> with a 50% (1/2) chance for each since their probability amplitude is 1/√2.

If we get the state |00>, this is a combination as shown below:

This means that both qubits A and B will be |0>.

Similarly, if we get the state |11>, this is a combination as shown below:

This means that both qubits A and B will be |1>.

Both of the qubits are perfectly correlated, or maximally entangled.


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The Four Bell States

There are four Bell states, and their Dirac notations are shown below:

  1. Φ⁺ (Phi plus)

2. Φ⁻ (Phi minus)

3. Ψ⁺ (Psi plus)

4. Ψ⁻ (Psi minus)

Bell states are used in:

  • Quantum Key Distribution (QKD) protocols, such as the BBM92 protocol used in Cryptography

  • Superdense Coding that allows someone to send two classical bits of information to another by transmitting only one qubit

  • Quantum Teleportation that transfers a particle’s quantum state to another location without moving the particle itself


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