Calculator Inputs
Example Data Table
This example uses direct average-state entropy input with base 2 units.
| State | Probability pᵢ | State entropy S(ρᵢ) |
|---|---|---|
| Signal A | 0.50 | 0.10 |
| Signal B | 0.30 | 0.35 |
| Signal C | 0.20 | 0.80 |
Average-state entropy S(ρ): 1.420 bits
Weighted component entropy: 0.315 bits
Holevo χ: 1.105 bits
Formula Used
Holevo quantity: χ = S(ρ) − Σ pᵢ S(ρᵢ)
Average state: ρ = Σ pᵢ ρᵢ
Source entropy: H(X) = − Σ pᵢ log(pᵢ)
In this page, you enter the entropies of the component states directly. The calculator then subtracts the weighted component entropy from the entropy of the average state to estimate the Holevo information ceiling for the ensemble.
How to Use This Calculator
- Choose whether you want results in bits or nats.
- Enter an optional Hilbert-space dimension if you want a log(d) comparison.
- Select direct entropy mode or eigenvalue mode for the average state.
- Fill in probabilities and component-state entropies for at least two states.
- Enable normalization if your listed probabilities do not sum exactly to 1.
- Press the calculate button to show the result above the form.
- Review validation notes, then export the results as CSV or PDF.
Frequently Asked Questions
1. What does the Holevo bound measure?
It gives an upper bound on the classical information that can be extracted from a quantum ensemble after the best possible measurement. The result is a limit, not a guaranteed communication rate.
2. Why do I enter state entropies instead of matrices?
This tool is designed for fast analytical work. If you already know each component state entropy and the average-state entropy or eigenvalues, you can evaluate χ without entering full density matrices.
3. When should I use eigenvalue mode?
Use eigenvalue mode when you know the spectrum of the average density matrix but not its entropy yet. The calculator computes S(ρ) from those eigenvalues using your selected logarithm base.
4. Can χ ever be negative?
For a physically consistent ensemble, χ is nonnegative. A negative result usually means the supplied entropies or eigenvalues are inconsistent, use different bases, or contain a data-entry mistake.
5. Why compare χ with H(X)?
The accessible information cannot exceed the source entropy of the classical labels. If your computed χ is larger than H(X), the input ensemble data is likely inconsistent.
6. What does the dimension field do?
It lets the calculator compare the result against log(d), the maximum possible entropy for a d-dimensional system. That extra check helps validate the average-state entropy you supplied.
7. Should the probabilities sum to exactly one?
Yes. Probabilities define the ensemble weights, so they should total one. Automatic normalization is included for convenience when you have rounded or unnormalized values from intermediate calculations.
8. Are bits and nats interchangeable?
They represent the same information measure with different logarithm bases. Bits use base 2, while nats use the natural logarithm. Keep every entropy input in the same unit system.
This calculator is intended for educational and analytical use. It does not derive density matrices or optimize measurements directly.