Von Neumann Entropy Calculator
Measure of the quantum information content in a state.
Formula first
Overview
The Von Neumann entropy generalizes the classical Shannon entropy to quantum mechanical systems by using the density operator to represent state uncertainty. It measures the amount of quantum information or mixedness present in a state, where a value of zero signifies a pure state and higher values indicate a mixture.
Symbols
Variables
S = Entropy, P_1 = Prob (State 1), P_2 = Prob (State 2)
Apply it well
When To Use
When to use: This formula is applied when evaluating the purity of a quantum state or calculating the entanglement between subsystems in a bipartite system. It is also essential in quantum thermodynamics and channel capacity calculations when dealing with mixed states.
Why it matters: It provides a rigorous framework for understanding information loss in decoherence and sets the fundamental limits for quantum data compression. Its behavior under various transformations helps define the laws of quantum information processing and the second law of thermodynamics in quantum systems.
One free problem
Practice Problem
Calculate the Von Neumann entropy s for a maximally mixed state qubit where the eigenvalues of the density matrix are p1 = 0.5 and p2 = 0.5.
Solve for:
Hint: For a mixed state, use the formula s = -(p1 × log₂(p1) + p2 × log₂(p2)).
The full worked solution stays in the interactive walkthrough.
References
Sources
- Nielsen, Michael A., and Isaac L. Chuang. Quantum Computation and Quantum Information.
- Wikipedia: Von Neumann entropy
- Nielsen, Michael A., and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2010.
- Nielsen and Chuang Quantum Computation and Quantum Information
- Sakurai Modern Quantum Mechanics
- Wehrl General properties of entropy (Reviews of Modern Physics)
- IUPAC Gold Book
- University Quantum Computing — Quantum Information Theory