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Quantum Entanglement Calculator

Calculate quantum entanglement, Bell states, and quantum correlations

Parameters

ⓘ
ⓘ
radⓘ
radⓘ
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Controls

xⓘ

Calculated Values

Entanglement Entropy:
NaN;NaN;
Concurrence:
0.00;0.00;
Bell Violation:
0.00;0.00;
Fidelity with |Φ⁺⟩:
1.00;1.00;
Fidelity with |Ψ⁺⟩:
0.50;0.50;
Normalized Alpha:
0.71;0.71;
Normalized Beta:
0.71;0.71;
State Norm:
1.00;1.00;

Examples

Example 1: Bell State |Φ⁺⟩

Maximally entangled Bell state with equal superposition.

  • Entanglement Entropy: 1.001.00
  • Concurrence: 1.001.00
  • Bell Violation: 2.832.83
  • Fidelity with |Φ⁺⟩: 1.001.00

Example 2: Product State

Unentangled product state |00⟩.

  • Entanglement Entropy: 0.000.00
  • Concurrence: 0.000.00
  • Bell Violation: 0.000.00
  • Fidelity with |Φ⁺⟩: 0.500.50

Example 3: Partially Entangled State

Partially entangled state with phase.

  • Entanglement Entropy: 0.720.72
  • Concurrence: 0.480.48
  • Bell Violation: 1.361.36
  • Fidelity with |Φ⁺⟩: 0.820.82

Visualization

Quantum Entanglement

Quantum entanglement is a fundamental phenomenon in quantum mechanics where two or more particles become correlated in such a way that the quantum state of each particle cannot be described independently. This 'spooky action at a distance,' as Einstein called it, is one of the most counterintuitive aspects of quantum theory.

A Bell state is a maximally entangled quantum state of two qubits. The four Bell states are: |Φ⁺⟩ = (|00⟩ + |11⟩)/√2, |Φ⁻⟩ = (|00⟩ - |11⟩)/√2, |Ψ⁺⟩ = (|01⟩ + |10⟩)/√2, and |Ψ⁻⟩ = (|01⟩ - |10⟩)/√2. These states are fundamental to quantum computing and quantum communication.

The entanglement entropy measures the degree of entanglement between subsystems. For a pure state of two qubits, the entanglement entropy is given by S = -Tr(ρ₁log₂ρ₁), where ρ₁ is the reduced density matrix of one subsystem. Maximally entangled states have entropy S = 1.

Bell's inequality provides a way to test whether quantum mechanics violates local realism. The CHSH inequality states that |⟨A₁B₁⟩ + ⟨A₁B₂⟩ + ⟨A₂B₁⟩ - ⟨A₂B₂⟩| ≤ 2 for any local hidden variable theory, but quantum mechanics can violate this bound up to 2√2.

Quantum entanglement has applications in quantum computing, quantum cryptography, quantum teleportation, and quantum sensing. It's essential for quantum algorithms like quantum key distribution and quantum error correction.

Key Concepts

  • Bell States: Four maximally entangled two-qubit states
  • Entanglement Entropy: Measure of quantum correlations
  • Bell Inequality: Test for local hidden variables
  • Quantum Correlation: Non-local correlations between particles
  • Reduced Density Matrix: Describes subsystem state
  • CHSH Inequality: Specific form of Bell's inequality

Real-World Applications

  • Quantum Computing: Entangled qubits for algorithms
  • Quantum Cryptography: Secure key distribution
  • Quantum Teleportation: Transfer of quantum states
  • Quantum Sensing: Enhanced measurement precision
  • Quantum Communication: Long-distance quantum networks

Explore Further

More quantum mechanics tools

Physics Equations

Bell State |Φ⁺⟩:
∣Φ+⟩=12(∣00⟩+∣11⟩)|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)
Bell State |Ψ⁺⟩:
∣Ψ+⟩=12(∣01⟩+∣10⟩)|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)
Entanglement Entropy:
S=−Tr(ρ1log⁡2ρ1)S = -\text{Tr}(\rho_1\log_2\rho_1)
CHSH Inequality:
∣⟨A1B1⟩+⟨A1B2⟩+⟨A2B1⟩−⟨A2B2⟩∣≤2|\langle A_1B_1\rangle + \langle A_1B_2\rangle + \langle A_2B_1\rangle - \langle A_2B_2\rangle| \leq 2
Quantum State:
∣ψ⟩=α∣00⟩+β∣11⟩|\psi\rangle = \alpha|00\rangle + \beta|11\rangle

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Normalize the Quantum State

First, we normalize the quantum state to ensure it's properly normalized:

Equation:

∣ψ⟩=α∣00⟩+β∣11⟩α2+β2|\psi\rangle = \frac{\alpha|00\rangle + \beta|11\rangle}{\sqrt{\alpha^2 + \beta^2}}

Calculation:

∣ψ⟩=0.7071∣00⟩+0.7071∣11⟩0.70712+0.70712|\psi\rangle = \frac{0.7071|00\rangle + 0.7071|11\rangle}{\sqrt{0.7071^2 + 0.7071^2}}

Explanation:

This ensures the state has unit norm and is physically valid.

2

Step 2: Calculate Normalized Coefficients

We calculate the normalized coefficients:

Equation:

αnorm=αα2+β2,βnorm=βα2+β2\alpha_{norm} = \frac{\alpha}{\sqrt{\alpha^2 + \beta^2}}, \quad \beta_{norm} = \frac{\beta}{\sqrt{\alpha^2 + \beta^2}}

Calculation:

αnorm=0.7071,βnorm=0.7071\alpha_{norm} = 0.7071, \quad \beta_{norm} = 0.7071

Explanation:

These are the properly normalized coefficients of the quantum state.

3

Step 3: Calculate Eigenvalues of Reduced Density Matrix

We find the eigenvalues of the reduced density matrix:

Equation:

λ1,2=12(1±1−4∣αnormβnorm∣2sin⁡2ϕ)\lambda_{1,2} = \frac{1}{2}(1 \pm \sqrt{1 - 4|\alpha_{norm}\beta_{norm}|^2\sin^2\phi})

Calculation:

λ1=1.0000,λ2=0.0000\lambda_1 = 1.0000, \quad \lambda_2 = 0.0000

Explanation:

These eigenvalues determine the entanglement properties.

4

Step 4: Calculate Entanglement Entropy

We calculate the entanglement entropy:

Equation:

S=−λ1log⁡2λ1−λ2log⁡2λ2S = -\lambda_1\log_2\lambda_1 - \lambda_2\log_2\lambda_2

Calculation:

S=−1.0000log⁡2(1.0000)−0.0000log⁡2(0.0000)=NaNS = -1.0000\log_2(1.0000) - 0.0000\log_2(0.0000) = NaN

Explanation:

This measures the degree of entanglement between the two qubits.

5

Step 5: Calculate Concurrence

We calculate the concurrence as a measure of entanglement:

Equation:

C=2∣αnormβnormsin⁡ϕ∣C = 2|\alpha_{norm}\beta_{norm}\sin\phi|

Calculation:

C=2∣0.7071×0.7071×sin⁡(0.0000)∣=0.0000C = 2|0.7071 \times 0.7071 \times \sin(0.0000)| = 0.0000

Explanation:

Concurrence is another measure of entanglement that ranges from 0 to 1.

6

Step 6: Calculate Bell Inequality Violation

We calculate the Bell inequality violation:

Equation:

Bell Violation=22C\text{Bell Violation} = 2\sqrt{2}C

Calculation:

Bell Violation=22×0.0000=0.0000\text{Bell Violation} = 2\sqrt{2} \times 0.0000 = 0.0000

Explanation:

This shows how much the state violates local realism.

Frequently Asked Questions (FAQ)

What is quantum entanglement?

Quantum entanglement is a phenomenon where two or more particles become correlated in such a way that the quantum state of each particle cannot be described independently. The particles remain correlated even when separated by large distances.

What are Bell states?

Bell states are four maximally entangled quantum states of two qubits. They are fundamental to quantum computing and include |Φ⁺⟩ = (|00⟩ + |11⟩)/√2, |Φ⁻⟩ = (|00⟩ - |11⟩)/√2, |Ψ⁺⟩ = (|01⟩ + |10⟩)/√2, and |Ψ⁻⟩ = (|01⟩ - |10⟩)/√2.

What is entanglement entropy?

Entanglement entropy measures the degree of entanglement between subsystems. For a pure state of two qubits, it's calculated from the reduced density matrix and ranges from 0 (unentangled) to 1 (maximally entangled).

What is Bell's inequality?

Bell's inequality provides a way to test whether quantum mechanics violates local realism. The CHSH inequality states that certain correlations must satisfy |⟨A₁B₁⟩ + ⟨A₁B₂⟩ + ⟨A₂B₁⟩ - ⟨A₂B₂⟩| ≤ 2 for any local hidden variable theory.

How is entanglement used in quantum computing?

Entanglement is essential for quantum computing algorithms. Entangled qubits can perform certain calculations exponentially faster than classical computers. It's used in quantum teleportation, quantum error correction, and quantum key distribution.

Practice MCQs

  1. A Bell state is:
  2. The entanglement entropy of a maximally entangled state is:
  3. Bell's inequality tests for:
  4. The CHSH inequality bound for local theories is:
  5. Quantum entanglement allows:

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