Algorithm guide · primitive
Rényi-2 entropy
Interfere two independently prepared copies to estimate subsystem purity and entanglement entropy from one ancilla.
Circuit size: 5 qubits; 1 measured ancilla
Density matrices and partial trace · Controlled SWAP · Logarithms base two
Review quantum fundamentals →Notation used in this lesson
- |ψ⟩ and ⟨ψ|
- A state vector (ket) and its conjugate transpose (bra). ⟨a|b⟩ is their inner product.
- α, β and i
- α and β are complex amplitudes; i²=−1. A probability is the squared magnitude |α|².
- XOR, · and ⊗
- XOR adds bits modulo two. In s·x, multiply corresponding bits and XOR them. ⊗ combines registers or operations on different wires.
- U†, Σ and Tr
- U† is the adjoint (inverse for a unitary). Σ sums terms. Tr is the matrix trace, the sum of diagonal entries.
- Probability, shots and angles
- Ideal probabilities describe the model; measured shot frequencies fluctuate. Angles inside trigonometric expressions are in radians unless explicitly labeled turns.
1. The problem
Estimate the purity of subsystem A of a prepared pure two-qubit state, then compute its Rényi-2 entropy. Compare with the purity of the whole AB state.
2. How it works
Interfere two independently prepared copies with a SWAP controlled by one ancilla. Swapping only A probes its reduced state; swapping both A and B probes the whole state. Entangled pure pairs have mixed subsystems even though the whole pair is pure.
Prepare two copies
Each pair is √(1−p)|00⟩+√p|11⟩. This repeats a known preparation; it does not clone an unknown state.
Interfere
H on the ancilla, controlled SWAP of the selected subsystem(s), then H.
Measure ancilla
Repeat to estimate P(0).
Decode
Compute purity=2P(0)−1 and S2=−log2(purity).
3. The mathematics, one idea at a time
Tracing out B removes the cross terms; A’s eigenvalues are 1−p and p.
The ancilla encodes purity, and the logarithm reports entropy in bits.
Whole-state purity remains one even when the reduced entropy is nonzero.
Work this through by hand
4. A Bell pair has one bit of reduced Rényi entropy
Five fixed qubits: an ancilla and two prepared two-qubit copies. Start at p=0 as a zero-entropy baseline; p=1/2 is the simplest entangled example.
Choose p=0.5 and Subsystem A.
| Stage | Calculation or state | What it means |
|---|---|---|
| Reduced state | ρ_A=diag(1/2,1/2) | Either local computational outcome is equally likely. |
| Purity | (1/2)²+(1/2)²=1/2 | The subsystem is mixed. |
| Ancilla and entropy | P(0)=3/4; P(1)=1/4; S2=−log2(1/2)=1 | A probability measurement gives the entropy estimate after classical decoding. |
Expected outcomes for this example
These are ideal predictions for the stated configuration, not a result already measured by your browser. Rows describing alternative modes or combined events are labeled explicitly.
| Outcome or event | Ideal probability | Interpretation |
|---|---|---|
| 0 | 75% ideal | Ancilla outcome used for purity. |
| 1 | 25% ideal | Complementary ancilla outcome. |
5. Load the worked example in Composer
The worked example can use smaller or simpler inputs than the gallery defaults. The button below loads the exact circuit used in this lesson.
- Schmidt weight p
- 0.5
- Readout
- Subsystem A
- Save any current circuit you want to keep, then open the worked example. Loading it does not execute a paid hardware job.
- Trace the preparation, main operations and measurements against the worked table. Check the gate editor for controls, targets and angles.
- Start with an ideal noiseless simulator compatible with the circuit. Mid-circuit measurement, reset and feed-forward require support from the chosen backend.
- Inspect the live preview. For a recorded experiment, open Execute, set the available backend options and shots, and Run. Sign in when prompted to retain it in My Jobs.
- Read the output as described below before changing one setting and repeating. The guide describes the original example; edits can change the answer.
6. Read the output correctly
Read c0 only. With an observed zero fraction f0, calculate purity≈2f0−1, then the entropy.
Finite samples can give an unphysical purity estimate outside the allowed range. Do not take a logarithm of a nonpositive raw estimate or interpret it as a physical entropy.
Switch to Whole AB: the ideal ancilla is always 0, purity is 1 and whole-state S2 is 0.
A bit label is a register value; its bar height is a probability or count. For S shots and observed count k, the sampled frequency is k/S. A typical sampling scale for probability p is √[p(1−p)/S]; this is not a strict error bound. Finite samples do not have to match the ideal percentages exactly.
7. Common mistakes & limits
- Reduced mixedness implies entanglement here because AB is pure; it is not a general entanglement certificate for mixed AB states.
- Nonidentical or correlated copies do not measure the advertised purity; the overlap signal changes.
8. Check your understanding
What are reduced purity and S2 for p=0?
Reveal the reasoning
ρ_A=|0⟩⟨0|, purity=1 and S2=0. The pair is the product |00⟩.
9. Further study
The lesson’s register mappings, supported inputs and loadable example follow Quantket’s implemented circuit. A reference may use different wire names, bit order or resource conventions.
Gallery reference: defaults, definitions & implementation notes
Inputs & default settings
Tune entanglement and compare a subsystem SWAP with a whole-state SWAP; inspect ancilla counts in Classical bits.
These are the gallery defaults. The worked example above may override them. Configure the card to change the inputs before loading its editable circuit.
- Schmidt weight p
- Default: 0.5 · A probability from 0 to 1 inclusive.
Probability of |11> in each copy: 0 or 1 gives a product state; 0.5 gives a Bell state.
- Readout
- Default: subsystem · Choices for the current settings: Subsystem A (Reduced purity; entanglement for pure AB.); Whole AB comparison (Full pure-state entropy stays zero.).
Implementation notes
Ancilla q0 probes AB on q1,q2 and a separately prepared copy on q3,q4. Both preparations produce sqrt(1-p)|00> + sqrt(p)|11>; this repeats a known preparation, not cloning an unknown state. Subsystem mode swaps only A (q1,q3), measuring Tr(rho_A^2). Whole-state mode also swaps B (q2,q4), measuring Tr(rho_AB^2)=1 for these pure states, even when A and B are entangled. Repeated ancilla shots estimate purity as 2P(0)-1 and Renyi-2 entropy as -log2(purity), in bits. Reduced entropy measures entanglement here because AB is pure; mixedness alone does not establish entanglement for a mixed AB state. Copies must be identical and independent, or the signal is Tr(rho sigma), not purity. Finite samples fluctuate, and circuit edits can invalidate these assumptions.
Default example: predictions & circuit facts
These values describe the unedited default setup. They are predictions and circuit properties, not measured results from your saved execution.
- Ideal purity
- 0.5000
- Ideal subsystem S2
- 1.0000 bits
- Ancilla P(0)
- 75.00%
- Ancilla P(1)
- 25.00%
Read your saved result
Check the submitted circuit, measured register and execution settings in Run details. Exact results give computed probabilities; sampled results show frequencies from the recorded shots. Edited circuits and noise can change the expected outcome. The algorithm label records the circuit's origin; it does not certify an edited circuit or a measured result.