Enter density-matrix eigenvalues
Leave unused fields empty. Entries must be real and nonnegative.
Formula used
S = −k Tr(ρ ln ρ)
S = −k Σi λi ln(λi), where λi are density-matrix eigenvalues.
For a valid density matrix, every eigenvalue is nonnegative and their sum equals one. The calculator uses the spectral formula, so diagonalizing a matrix is unnecessary once its eigenvalues are known. When automatic normalization is active, the submitted values are scaled to make their trace equal one.
How to use this calculator
- Enter the nonnegative eigenvalues of the density matrix.
- Set the Hilbert-space dimension, including zero-probability basis states when relevant.
- Select dimensionless entropy, Boltzmann units, or a custom positive constant.
- Enable normalization only when your submitted weights do not already sum to one.
- Press Calculate entropy and review entropy, purity, effective rank, and exported results.
Example data
| Eigenvalues | Dimension | k | Expected S/k | Interpretation |
|---|---|---|---|---|
| 0.5, 0.5 | 2 | 1 | 0.693147 nats | Maximally mixed qubit |
| 1, 0 | 2 | 1 | 0 nats | Pure state |
| 0.7, 0.2, 0.1 | 3 | 1 | 0.801819 nats | Partly mixed qutrit |
Quantum uncertainty from state spectra
Von Neumann entropy measures uncertainty within a quantum state. It extends Shannon entropy to density matrices. The calculation begins with a density operator, written as rho. Its eigenvalues describe the probabilities of orthogonal quantum outcomes. These values fully determine the entropy. Eigenvectors do not change the numerical result.
A pure state has one eigenvalue equal to one. Every other eigenvalue is zero. Its entropy is zero. The state contains no classical mixing across orthogonal possibilities. A mixed state has several positive eigenvalues. Its entropy becomes positive. Greater spreading usually creates greater uncertainty.
Why eigenvalues matter
Density matrices can contain complex off-diagonal elements. Those elements represent coherence in a chosen basis. Entropy does not require calculating each matrix element after diagonalization. The spectral theorem converts the problem into eigenvalues. Apply minus lambda times its natural logarithm for every positive eigenvalue. Then add all terms. Finally multiply by k.
Zero eigenvalues are safe. Their contribution is defined as zero. This follows from the limiting behavior of x ln x as x approaches zero. Negative eigenvalues are not valid for a physical density matrix. A trace other than one also needs attention. It may reflect incomplete normalization or a nonphysical input.
Interpreting the calculated values
The calculator reports entropy in nats when k equals one. It also converts the dimensionless value into bits. Divide nats by ln two to get bits. Select the Boltzmann constant for thermodynamic units. Then entropy has units of joules per kelvin. This is useful when connecting quantum statistical states with thermal physics.
Purity provides a useful companion measure. It equals Tr of rho squared. A pure state has purity one. Mixed states have lower purity. Linear entropy is one minus purity. Effective rank is exp of S divided by k. It estimates how many equally weighted states would produce the same entropy.
Bounds, dimension, and normalization
For a system with dimension d, the maximum dimensionless entropy is ln d. It occurs when every eigenvalue equals one divided by d. The normalized entropy compares your result against this upper bound. It ranges from zero to one for valid normalized states. This comparison is especially helpful across systems with different dimensions.
Check the trace before interpreting results. A proper density matrix has trace one. Automatic normalization rescales submitted nonnegative weights. It is convenient for probability weights or measured populations. Do not use it to hide a modeling error. Confirm the physical preparation, basis dimension, and data source first. Good inputs make entropy comparisons meaningful, reproducible, and scientifically defensible.
Entropy also differs from energy. A low-energy quantum state can still be mixed. A high-energy state can remain pure. Entropy describes spectral uncertainty, not energy magnitude alone. In open systems, coupling to an environment often turns a pure subsystem into a mixed reduced state. Tracking entropy then helps describe decoherence, entanglement, thermalization, and information flow. Use consistent dimensions and constants for comparisons. State valid, transparent, repeatable physical assumptions. This supports fair scientific interpretation.
Frequently asked questions
1. What does von Neumann entropy measure?
It measures uncertainty or mixedness in a quantum state described by a density matrix. It is zero for pure states and increases as probability spreads across orthogonal quantum states.
2. Why are eigenvalues used instead of matrix entries?
The entropy depends only on the spectrum of the density matrix. Eigenvalues are basis independent. Once they are known, the full matrix is unnecessary for this calculation.
3. Can an eigenvalue be zero?
Yes. A zero eigenvalue contributes zero entropy. The limit of x ln x is zero as x approaches zero, so no special numerical correction is required.
4. Why must eigenvalues sum to one?
A physical density matrix has unit trace. Its eigenvalues therefore sum to one. This condition ensures they act as valid probabilities after diagonalization.
5. What does automatic normalization do?
It divides every entered nonnegative value by their total. The resulting values sum to one. Use it for unnormalized weights, not for correcting uncertain or invalid experimental data.
6. What are nats and bits?
Nats use natural logarithms. Bits use logarithms with base two. One nat equals approximately 1.4427 bits. The calculator displays both for easier comparison.
7. When should I use the Boltzmann constant?
Use it when entropy needs thermodynamic units of joules per kelvin. For quantum information comparisons, dimensionless entropy with k equal to one is usually clearer.
8. What is the maximum possible entropy?
For dimension d, the maximum is k ln d. It occurs when every available state has equal probability, meaning every eigenvalue equals one divided by d.
9. What does purity show?
Purity is Tr of rho squared. It equals one for pure states. Lower values indicate a more mixed state. It complements entropy but is not measured in bits or nats.
10. Can I enter more basis states than positive eigenvalues?
Yes. Set the Hilbert-space dimension higher than the entered positive eigenvalues when zero-probability states exist. This changes the maximum entropy comparison, not the calculated state entropy.
11. Does entropy depend on the chosen basis?
No. Von Neumann entropy is invariant under unitary basis changes. The density matrix may look different, but its eigenvalues and entropy remain unchanged.