Quantum Entanglement Fidelity Calculator

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Overview

This calculator estimates how the fidelity of an entangled quantum state decays over time, assuming a simple exponential decoherence model. By entering an initial fidelity, an effective coherence time, and the elapsed time, you obtain a quick, order-of-magnitude prediction of the current entanglement quality.

Use it as a planning aid for laboratory experiments, simulations, or protocol design in quantum communication and quantum computing. It is not a substitute for detailed experimental analysis or full density-matrix simulations.

What Is Entanglement Fidelity?

In quantum mechanics, entanglement describes nonclassical correlations between two or more particles. Measurements on one particle are statistically correlated with measurements on another, in a way that cannot be explained by classical local hidden variables. These correlations underlie many key applications:

Multi-qubit gates in quantum computers
Quantum teleportation of unknown quantum states
Quantum key distribution (QKD) and other secure communication protocols
Entanglement swapping and quantum repeaters

To quantify how closely a prepared state matches a target entangled state, physicists use a metric called fidelity. Intuitively, fidelity measures "how similar" two quantum states are.

For a prepared state $ρ$ and a target pure state $| ψ ⟩$ , the state fidelity is

$F = ⟨ ψ ∣ ρ ∣ ψ ⟩$

where $F = 1$ means the prepared state is identical to the target, and $F = 0$ means they are completely orthogonal. In typical experiments with noisy hardware, entanglement fidelities might range from around 0.6 up to 0.99 or higher, depending on the platform and protocol.

Modeling Entanglement Decay

Real entangled states are never perfectly isolated. Coupling to the environment, control errors, and material defects cause decoherence. Over time, this decoherence reduces entanglement and drives the system toward a more classical or mixed state.

For many systems, especially in the Markovian approximation, the decay of coherence and fidelity can be modeled as a simple exponential process. If $F_{0}$ is the fidelity immediately after entanglement generation, $τ$ is an effective coherence time, and $t$ is the elapsed time, a common approximation is

$F (t) = F_{0} e^{- \frac{t}{τ}}$

Here:

$F_{0}$ is the initial fidelity at $t = 0$ , typically between 0 and 1.
$τ$ is an effective coherence time (often quoted in microseconds $μ^{s}$ ), setting the characteristic scale over which the state loses coherence.
$t$ is the elapsed time since the state was created or last refreshed.

A larger $τ$ means a slower decay: for fixed $F_{0}$ , the fidelity remains high for a longer time. A smaller $τ$ corresponds to faster decoherence and more rapid loss of entanglement.

In experimental practice, the effective coherence time might be related to more familiar quantities such as $T_{1}$ (energy relaxation time) and $T_{2}$ (dephasing time), or to channel parameters in noisy quantum communication links. The calculator does not require you to specify these separately; instead, you provide a single effective $τ$ that captures the dominant decay behavior under your conditions.

How to Use the Calculator

Enter the initial fidelity $F_{0}$ .

This is the fidelity measured (or targeted) immediately after entanglement generation, before significant storage or transmission. It should be a dimensionless number between 0 and 1. Typical experimental values might be in the range 0.6–0.99.
Specify the coherence time $τ$ in microseconds.

Use an effective coherence time for your entangled state, expressed in $μ^{s}$ . This value can come from an independent fit to your decay curves, from known $T_{2}$ times of your qubits, or from channel characterization in a communication setup.
Provide the elapsed time $t$ in microseconds.

This is the time interval since the state was created or since it last underwent an operation that effectively "refreshes" its coherence (such as entanglement purification or swapping).
Run the estimate.

When you submit the form, the calculator applies the exponential model $F (t) = F_{0} e^{- \frac{t}{τ}}$ and returns the predicted fidelity at time $t$ .

Interpreting the Result

The output is a single number between 0 and $F_{0}$ , representing the estimated entanglement fidelity at the specified time. To make sense of this value, it is useful to relate it to qualitative categories and protocol-specific thresholds:

High fidelity (roughly $F ≳ 0.9$ ) – Often considered suitable for demanding quantum communication tasks, high-fidelity gate operations, and entanglement-based protocols with limited error correction.
Moderate fidelity (roughly 0.7–0.9) – May still be usable, especially if combined with entanglement distillation or quantum error correction, but performance margins are smaller and protocol success probabilities may degrade.
Low fidelity (roughly $F ≲ 0.7$ ) – Frequently indicates that the state is too noisy for many entanglement-based applications without substantial error-mitigation or distillation overhead. At very low values, the state may no longer be useful as an entanglement resource.

These ranges are only rough guidelines; the exact thresholds depend strongly on the specific protocol (for example, quantum teleportation versus QKD), the amount of classical post-processing available, and whether you can perform entanglement purification or use error-correcting codes.

In experimental design, the calculator can help you answer questions such as:

How long can I store an entangled pair in memory before its fidelity drops below my target threshold?
Given a known coherence time, what is the latest time at which I should perform a gate, measurement, or entanglement swapping operation?
How sensitive is my protocol performance to modest improvements in coherence time or initial fidelity?

Worked Example

Suppose you have an entangled pair of qubits generated in a superconducting circuit. Immediately after state preparation, tomography reports an entanglement fidelity of

$F_{0} = 0.95 .$

From independent measurements, you estimate an effective coherence time of

$τ = 50 μ s .$

You would like to know the expected fidelity after an elapsed time of

$t = 100 μ^{s} .$

Using the exponential model,

$F (t) = F_{0} e^{- \frac{t}{τ}} = 0.95 \times e^{- \frac{100}{50}} = 0.95 \times e^{- 2} .$

Since $e^{- 2} \approx 0.1353$ , the estimated fidelity is

$F (100 μ s) \approx 0.95 \times 0.1353 \approx 0.129 .$

An entanglement fidelity of roughly 0.13 is very low for most protocols. Based on this, you might decide that your usable window for high-quality operations is far shorter than 100 $μ^{s}$ . For instance, you might solve for the time at which the fidelity falls to 0.9:

$0.9 = 0.95 e^{- \frac{t}{50}} \Rightarrow e^{- \frac{t}{50}} = \frac{0.9}{0.95} \approx 0.9474 .$

Taking natural logarithms,

$- \frac{t}{50} = \ln (0.9474) \Rightarrow t \approx - 50 \ln (0.9474) \approx 2.7 μs .$

This indicates that, under this simple model, you would need to perform critical operations within a few microseconds to keep the fidelity above 0.9.

Comparison of Parameters and Effects

The table below summarizes how each input parameter influences the output and how you might interpret different regimes in practice.

Parameter / Regime	Typical Range	Effect on Fidelity Estimate	Practical Interpretation
Initial fidelity $F_{0}$	0.6 to 0.99 (dimensionless)	Sets the starting point of the decay curve; higher $F_{0}$ uniformly raises $F (t)$ .	Reflects state-preparation quality and control accuracy at $t = 0$ .
Coherence time $τ$	From a few $μ^{s}$ to many ms, depending on platform	Controls how quickly fidelity decays with time; larger $τ$ flattens the decay.	Encapsulates decoherence processes such as relaxation ( $T_{1}$ ) and dephasing ( $T_{2}$ ).
Elapsed time $t$	0 to several multiples of $τ$	Increases the decay factor $e^{- \frac{t}{τ}}$ ; when $t ≫ τ$ , $F (t)$ becomes very small.	Represents storage time in memory, transmission delay, or protocol duration.
High-fidelity regime	$F (t) ≳ 0.9$	System remains close to the target entangled state.	Often acceptable for teleportation, QKD, and high-precision gates with modest error correction.
Intermediate regime	$0.7 ≲ F (t) ≲ 0.9$	Noticeable decoherence but still some useful entanglement.	May require entanglement distillation, error mitigation, or relaxed protocol requirements.
Low-fidelity regime	$F (t) ⩽ 0.7$	Strong decay; state may be nearly classical or highly mixed.	Often unsuitable for entanglement-based tasks without heavy error correction or purification.

Assumptions and Limitations

The calculator is intentionally simple and makes several important assumptions. Keep these in mind when interpreting the results:

Single-parameter exponential decay. The model assumes that entanglement fidelity decays as $F (t) = F_{0} e^{- \frac{t}{τ}}$ with a single effective coherence time $τ$ . Many real systems exhibit multi-exponential or stretched-exponential behavior, especially when multiple noise processes are present.
Markovian noise approximation. Non-Markovian effects, such as memory in the environment or strongly time-correlated noise, are not captured. In such cases, deviations from a pure exponential law can be significant.
No explicit gate or control errors. The calculator focuses on decoherence during idle evolution or transmission. It does not separately model additional infidelity from imperfect gates, measurements, or control pulses that may occur during your protocol.
Effective coherence time input. The $τ$ you enter is treated as a single effective parameter. In practice, it may be derived from a combination of $T_{1}$ , $T_{2}$ , dephasing rates, or channel noise parameters, and its value can depend on the specific sequence used to measure it.
Order-of-magnitude guidance only. The result is best viewed as a quick estimate of trends and scales, not a replacement for full density-matrix simulations, master-equation modeling, or detailed experimental characterization.
No automatic validation of physical consistency. While the calculator expects a fidelity between 0 and 1 and non-negative times, it does not enforce all physical constraints that might apply in your particular setup. Users should apply domain knowledge when choosing inputs and interpreting outputs.

When planning critical experiments or designing complex protocols such as multi-hop quantum repeaters, fault-tolerant quantum processors, or long-distance QKD networks, use this tool as a fast sanity check. For precise performance predictions, complement it with more detailed numerical modeling and experimental validation.

Enter parameters to estimate fidelity.

Quantum Entanglement Fidelity Calculator

Overview

What Is Entanglement Fidelity?

Modeling Entanglement Decay

How to Use the Calculator

Interpreting the Result

Worked Example

Comparison of Parameters and Effects

Assumptions and Limitations

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