Assumptions, results and research status
An exact finite example with declared preparation, operators, parameter domain and numerical diagnostics.
Assumptions
- The state, projector or transfer operator specified on this page is a model input.
- Numerical results use ordinary floating-point arithmetic in the stated units.
Results
- The default matrices and tables are available without JavaScript.
- Interactive results and exports implement the stated finite formulas.
Open questions
- Which source dynamics and physical observable correspondence justify applying this example to a specified system?
The declared model
The example starts from |Φ+⟩=(|00⟩+|11⟩)/√2 on quantum carriers A and B. A separate classical label is independent of that prepared pair. In the label-erasure protocol, the channel acts only on the label; the AB density matrix remains exactly the Bell projector. In the reference-depolarization protocol, a channel instead acts on carrier A. Its retained coefficient is s=exp(−γt), and the joint state becomes
TrAρ=TrBρ=I₂/2.
The second operation changes the shared quantum reference even though neither marginal changes for this preparation. The distinction develops the reference-resource question in Completed records, v21, Proposition 4.7, PDF p. 8. The matrix example is an explicit illustration; a statement about one prepared state's marginals is weaker than the source theorem's hypothesis that a channel fixes full local carrier algebras for all required inputs.
Run the calculation
A shared reference under two completion protocols
Compare erasing an independent classical label with depolarizing quantum reference A. Both local marginals stay fixed for this Bell preparation, but the joint state can change.
Time uses a chosen channel-evolution unit; γ is its inverse. States, correlations, purity, trace distance and negativity are dimensionless.
Retention 0.3678794; purity 0.3515015; negativity 0.02590958. Both reduced states remain I₂/2 for this prepared input.
- Retained coefficient s
- 0.367879441
- Purity Tr ρ²
- 0.351501462
- Negativity
- 0.0259095809
- Both reduced states
- I₂/2
- Distance to initial Bell state
- 0.474090419
- Distance to maximally mixed I₄/4
- 0.275909581
Depolarize reference A; retain B. The xx and zz correlations are 0.36787944; the yy correlation is -0.36787944. The negativity threshold occurs at 1.0986123 chosen time units.
| Eigenvalue | ρ | Partial transpose on B |
|---|---|---|
| 1 | 0.525909581 | 0.34196986 |
| 2 | 0.15803014 | 0.34196986 |
| 3 | 0.15803014 | 0.34196986 |
| 4 | 0.15803014 | -0.0259095809 |
The spectral columns list their own eigenvalues; rows do not identify a common eigenvector. Trace distance includes the factor one half. A zero negativity here can occur before exact depolarization.
Inspect the full density matrices and marginals
| Row | 00 | 01 | 10 | 11 |
|---|---|---|---|---|
| 00 | 0.34196986 | 0 | 0 | 0.18393972 |
| 01 | 0 | 0.15803014 | 0 | 0 |
| 10 | 0 | 0 | 0.15803014 | 0 |
| 11 | 0.18393972 | 0 | 0 | 0.34196986 |
| Row | 00 | 01 | 10 | 11 |
|---|---|---|---|---|
| 00 | 0.34196986 | 0 | 0 | 0 |
| 01 | 0 | 0.15803014 | 0.18393972 | 0 |
| 10 | 0 | 0.18393972 | 0.15803014 | 0 |
| 11 | 0 | 0 | 0 | 0.34196986 |
| Row | 1 | 2 |
|---|---|---|
| 1 | 0.5 | 0 |
| 2 | 0 | 0.5 |
| Row | 1 | 2 |
|---|---|---|
| 1 | 0.5 | 0 |
| 2 | 0 | 0.5 |
Read the time-series values
| Time | Retention s | Purity | Negativity | Distance to Bell |
|---|---|---|---|---|
| 0 | 1 | 1 | 0.5 | 0 |
| 1 | 0.36787944 | 0.35150146 | 0.025909581 | 0.47409042 |
| 1.098612 | 0.33333333 | 0.33333333 | 0 | 0.5 |
| 1.9 | 0.14956862 | 0.26677808 | 0 | 0.63782354 |
| 2.9 | 0.05502322 | 0.25227067 | 0 | 0.70873258 |
| 3.9 | 0.020241911 | 0.2503073 | 0 | 0.73481857 |
| 4.9 | 0.0074465831 | 0.25004159 | 0 | 0.74441506 |
| 5.9 | 0.0027394448 | 0.25000563 | 0 | 0.74794542 |
| 6.9 | 0.0010077854 | 0.25000076 | 0 | 0.74924416 |
| 7.9 | 0.00037074354 | 0.2500001 | 0 | 0.74972194 |
| 8.9 | 0.00013638893 | 0.25000001 | 0 | 0.74989771 |
| 9.9 | 0.000050174682 | 0.25 | 0 | 0.74996237 |
| 10 | 0.00004539993 | 0.25 | 0 | 0.74996595 |
At positive rate and time this reference-A channel does not fix the full A algebra, even though both displayed marginals remain unchanged for the supplied Bell input. This distinction is part of the protocol definition.
Equations and conventions used
ρA = ρB = I₂/2; ⟨XX⟩ = ⟨ZZ⟩ = s; ⟨YY⟩ = −s.
Tr ρ² = (1+3s²)/4; negativity = max[0,(3s−1)/4].
Trace distance to Bell = 3(1−s)/4; trace distance to I₄/4 = 3s/4.
A supplied Bell pair and two specified channels. Equal marginals for this one input do not establish that a channel fixes both full carrier algebras. The classical label is independent of AB in the label-erasure protocol.
Downloads include the chosen inputs, units and model scope. Values use standard floating-point arithmetic. Read the source and its version notes.
Try these comparisons
- Start with reference depolarization at γ=1 and t=1. Compare the reduced states with the joint xx, yy and zz correlations.
- Choose “Erase only the classical label.” The AB reference remains a pure Bell state for every displayed rate and time.
- Choose the negativity threshold γt=log 3. The partially transposed state has a zero lowest eigenvalue, but the retained coefficient is still 1/3.
- Set γ=0 in the depolarization protocol. The channel becomes the identity for every finite time. Then restore γ=1 and use t=log 2 to reproduce s=1/2.
The chart varies time from zero to ten at the selected rate. A label-erasure curve is constant because its operation acts on the independent label, not on either member of the Bell pair.
Understand the calculated quantities
The Bell-basis eigenvalues are (1+3s)/4 and three copies of (1−s)/4. The partial transpose has three eigenvalues (1+s)/4 and one (1−3s)/4. These give purity (1+3s²)/4 and negativity max[0,(3s−1)/4]. The two trace distances have different reference states: distance to the initial Bell state is 3(1−s)/4; distance to the maximally mixed product state is 3s/4.
At γt=log 2, purity is 7/16, negativity is 1/8 and each of the two trace distances is 3/8. The reduced states still equal I₂/2. A finite entanglement threshold does not imply exact channel erasure at that time: exp(−γt) remains positive for every finite γt.
Scope and interpretation
This example specifies the preparation and the subsystem on which each channel acts. It does not claim that every possible record-completion process is depolarizing, or that a label can never be correlated with a reference in another protocol. A different preparation needs its own calculation.
To apply the source theorem, describe the map on the complete algebra of observables. Fixing expectation values for one state does not establish an identity of channels. The completion-channel derivation and reference protocols supply that larger context.
Reproduce the result
Download JSON to retain the complete model inputs, matrices, units, analytic quantities and numerical diagnostics. Download CSV for the chart's numerical family; its opening rows record the inputs and definitions. The default example remains readable with JavaScript disabled.
The worked exercises provide hand-checkable values. The package also includes a verifier that can be run independently with Node.js. For a research contribution, use the research note template and state which part of the model you have checked or changed.
Continue reading
Sources for this guide
Source manuscripts, version notes, and standard references are listed in the reference library.