The Bell states, also known as EPR pairs (Einstein-Podolsky-Rosen pairs), are a set of four entangled quantum states of two qubits. They are important in quantum computing and quantum information theory, playing critical roles in applications such as quantum teleportation, quantum cryptography, superdense coding, and tests of the foundations of quantum mechanics. To understand Bell states, it’s essential first to grasp the concept of a qubit and superposition.
A qubit is the basic unit of quantum information, similar to a classical bit. While a classical bit can be in one of two definite states, 0 or 1, a qubit can exist in a superposition of both states simultaneously. The state of a qubit is defined by a linear combination of the basis states |0⟩ and |1⟩, written in Dirac notation as |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex numbers called amplitudes, and |α|² + |β|² = 1. The amplitudes control the probability of measuring the qubit in the |0⟩ state (|α|²) or the |1⟩ state (|β|²).
Entanglement is another quantum singularity that differentiates quantum computing from classical computing. When two or more qubits are entangled, their quantum states are linked together in such a way that they share the same. Measuring the state of one entangled qubit immediately influences the state of the other(s), even if they are separated by large distances. This connection is stronger than any classical correlation.
4 Bell states
Bell states are specific examples of these maximally entangled two-qubit states. There are four Bell states, typically denoted as:
- |Φ⁺⟩ = (|00⟩ + |11⟩) / √2
- |Φ⁻⟩ = (|00⟩ – |11⟩) / √2
- |Ψ⁺⟩ = (|01⟩ + |10⟩) / √2
- |Ψ⁻⟩ = (|01⟩ – |10⟩) / √2
In these notations, |ab⟩ represents the state where the first qubit is in state |a⟩ and the second qubit is in state |b⟩. For example, |00⟩ means both qubits are in the |0⟩ state. The factor of 1/√2 ensures that the states are normalized (the sum of the probabilities of all possible measurement outcomes is 1).
Let’s examine the properties of one of the Bell states, say |Φ⁺⟩ = (|00⟩ + |11⟩) / √2, to illustrate the concept of entanglement. If we were to measure the first qubit of a system in the |Φ⁺⟩ state, we would find it in the |0⟩ state with a probability of |1/√2|² = 1/2, and in the |1⟩ state with a probability of |1/√2|² = 1/2. Immediately upon measurement of the first qubit, the state of the second qubit is also determined. If the first qubit was measured to be |0⟩, the second qubit is also instantaneously |0⟩. Similarly, if the first qubit was |1⟩, the second qubit is also |1⟩. The individual state of each qubit is completely undefined before measurement (it’s in a superposition), but they are connected. The same principle applies to the other Bell states, although the connection patterns differ. For example, in |Ψ⁺⟩ = (|01⟩ + |10⟩) / √2, if the first qubit is measured as |0⟩, the second is |1⟩, and vice versa.
Bell state Matrix
The Bell states, also known as EPR states, are a set of four entangled two-qubit states. They are denoted as β₁, β₂, β₃, and β₄, or |↑↑〉, |↑↓〉, |↓↑〉, and |↓↓〉. These states are defined as follows in Dirac notation:
- β₁ = |Φ⁺〉 = |↑↑〉 = 1/√2 (|00〉 + |11〉)
- β₂ = |Φ⁻〉 = |↑↓〉 = 1/√2 (|00〉 − |11〉)
- β₃ = |Ψ⁺〉 = |↓↑〉 = 1/√2 (|01〉 + |10〉)
- β₄ = |Ψ⁻〉 = |↓↓〉 = 1/√2 (|01〉 − |10〉)
To represent these Bell states as matrices (column vectors), we use the standard computational basis for two qubits, which is {|00〉, |01〉, |10〉, |11〉}. These basis states correspond to the following 4-dimensional column vectors:
|00⟩=
|01⟩=
|10⟩=
|11⟩=
Using these basis vectors, the matrix representations of the Bell states are:
|Φ⁺〉 = β₁ = 1/√2+ 1/√2
|Φ⁻〉 = β₂ = 1/√2 – 1/√2
|Ψ⁺〉 = β₃ = 1/√2 + 1/√2
|Ψ⁻〉 = β₄ = 1/√2 – 1/√2
These Bell states form an orthonormal basis for the two-qubit Hilbert space and are critical in many quantum protocols due to their entangled nature.
How to generate a bell state?
Bell states can be created using quantum circuits starting from two qubits initialized in a separable state, such as |00⟩. A circuit to generate Bell states involves two quantum gates: the Hadamard gate (H) and the Controlled-NOT gate (CNOT).
The Hadamard gate acts on a single qubit and puts it into an equal superposition:
- H|0⟩ = (|0⟩ + |1⟩) / √2
- H|1⟩ = (|0⟩ – |1⟩) / √2
The CNOT gate is a two-qubit gate where the first qubit acts as the control and the second as the target. It flips the state of the target qubit (0 becomes 1, and 1 becomes 0) only if the control qubit is in the |1⟩ state:
- CNOT |00⟩ = |00⟩
- CNOT |01⟩ = |01⟩
- CNOT |10⟩ = |11⟩
- CNOT |11⟩ = |10⟩
To create the |Φ⁺⟩ Bell state, we can apply these gates as follows:
- Start with two qubits in the state |00⟩.
- Apply a Hadamard gate to the first qubit: H⊗I |00⟩ = ((|0⟩ + |1⟩) / √2) ⊗ |0⟩ = (|00⟩ + |10⟩) / √2, where I is the identity operator acting on the second qubit.
- Apply a CNOT gate with the first qubit as control and the second qubit as target: CNOT ((|00⟩ + |10⟩) / √2) = (|00⟩ + |11⟩) / √2 = |Φ⁺⟩.
By applying different single-qubit gates before the CNOT, the other Bell states can be generated as well.
Applications of Bell States
- Quantum Teleportation: Bell states are a critical resource in quantum teleportation, a process that allows the transfer of an unknown quantum state from one location to another without physically moving the qubit itself. Teleportation banks on the entanglement shared by a Bell pair and classical communication.
- Superdense Coding: Bell states can be used to transmit two classical bits of information using only one qubit. This is achieved by encoding the classical bits onto one qubit of an entangled Bell pair and sending it to the receiver, who can then perform a Bell measurement on their two qubits to decode the original two classical bits.
- Quantum Cryptography: Entangled states, including Bell states, are fundamental to some quantum key distribution (QKD) protocols. The entanglement allows for the detection of any eavesdropping attempts, assuring the security of the key exchange based on the laws of quantum mechanics.
- Tests of Quantum Foundations: Bell states were central to the formulation and experimental tests of Bell’s theorem, which demonstrates the non-local nature of quantum mechanics and differentiates it from classical physics. Trials violating Bell inequalities, using entangled photons in states similar to Bell states, provide strong evidence against local practicality, a classical worldview that local properties and actions are independent of each other.
- Quantum Algorithms and Error Correction: While not always openly stated, the principles behind entanglement and the operation of multi-qubit states, as represented by Bell states, are fundamental to the operation of many quantum algorithms and quantum error correction codes. Creating and managing entanglement is a requirement for achieving quantum advantage in computation.
Bell states are a foundation of quantum computing and quantum information science. As entangled states of two qubits, they exhibit strong quantum connections that have no classical counterparts. Their generation through basic quantum gates and their critical role in quantum protocols highlight the unique capabilities and potential of quantum technologies. Bell states provide a fundamental insight into the nature of quantum entanglement and its power for information processing and communication.