Quantum Purification Calculator

Explore mixed-state structure using purity, entropy, and rank. Compare qubit and eigenvalue-based inputs with ease. Get fast insights from formulas, tables, graphs, and exports.

Calculator Inputs

State label

Input mode

State dimension

Eigenvalue λ1

Eigenvalue λ2

Eigenvalue λ3

Eigenvalue λ4

Auto-normalize eigenvalues

Tolerance

Bloch component r_x

Bloch component r_y

Bloch component r_z

Formula Used

Purity from eigenvalues:
\( P = \mathrm{Tr}(\rho^2) = \sum_i \lambda_i^2 \)

Linear entropy:
\( L = 1 - P \)

Normalized purity:
\( P_n = \dfrac{dP - 1}{d - 1} \)

Normalized linear entropy:
\( L_n = \dfrac{d(1 - P)}{d - 1} \)

Von Neumann entropy:
\( S(\rho) = -\sum_i \lambda_i \log_2(\lambda_i) \)

Participation ratio:
\( R = \dfrac{1}{P} \)

Bloch-vector qubit purity:
\( \rho = \dfrac{I + \vec{r}\cdot\vec{\sigma}}{2}, \quad P = \dfrac{1 + |\vec{r}|^2}{2} \)

Minimum ancilla dimension for purification:
It equals the spectral rank of the mixed state. The required ancilla qubits are \( \lceil \log_2(\text{rank}) \rceil \).

How to Use This Calculator

Select Eigenvalue spectrum for a general mixed state, or Bloch vector for a qubit.
Enter your state label, values, dimension, and tolerance.
Use auto-normalization if your eigenvalues are proportional but not yet normalized.
Press the calculate button to see purity, entropy, rank, ancilla size, a result table, and a spectrum graph.

Example Data Table

Example	Mode	Inputs	Purity	Entropy	Rank	Ancilla Dimension
Pure qubit	Bloch	r = (0, 0, 1)	1.000000	0.000000	1	1
Maximally mixed qubit	Eigenvalues	0.5, 0.5	0.500000	1.000000	2	2
Partially mixed qutrit	Eigenvalues	0.7, 0.2, 0.1	0.540000	1.156780	3	3
Four-level thermal-like state	Eigenvalues	0.55, 0.25, 0.15, 0.05	0.390000	1.601014	4	4

FAQs

1. What does purity measure?

Purity measures how close a quantum state is to being pure. A pure state has purity 1, while mixed states have lower values. It is computed from the density matrix as Tr(ρ²).

2. What is quantum purification?

Quantum purification represents a mixed state as part of a larger pure state. You introduce an ancilla system so that tracing out the ancilla reproduces the original mixed density matrix.

3. Why does the ancilla dimension equal the rank?

The smallest ancilla needed for purification must support one orthogonal component for each nonzero eigenvalue. That count is exactly the spectral rank of the mixed state.

4. When should I use Bloch-vector mode?

Use Bloch-vector mode only for qubits. It is convenient when your state is described by r_x, r_y, and r_z rather than directly by eigenvalues.

5. Why can entropy be high when purity is low?

Low purity means the state is strongly mixed. Mixed states typically spread probability across several eigenvalues, which increases uncertainty and therefore raises the von Neumann entropy.

6. What happens if my eigenvalues do not sum to one?

If auto-normalization is enabled, the calculator rescales them to form a valid probability spectrum. If disabled, the tool requires the sum to equal one within the chosen tolerance.

7. Can negative eigenvalues be entered?

No. Negative eigenvalues make the density matrix unphysical. A valid density matrix must be positive semidefinite, so all eigenvalues must be nonnegative and sum to one.

8. What does participation ratio tell me?

Participation ratio is 1 divided by purity. It estimates how many basis components contribute effectively to the mixed state. Larger values indicate broader spectral spreading and stronger mixedness.