Q043 · Entanglement / classical messages

Quantum Teleportation & Swapping

Prepare a state, share a Bell pair and inspect all four measurement branches. Hold back the two classical bits, restore the correction, then join two independent pairs by entanglement swapping. Acquire receiver measurements and compare conditional records with the unconditioned state.

Configured final target fidelity $\text{—}$

Bell pairs consumed per trial $\text{—}$

MAKE A PREDICTION

Without the two classical bits, can Bob recover the input?

Compare message delivery with a blocked line. Inspect a Bell branch, then measure many independently prepared trials.

Choose a prediction, compare the cases, then acquire.

Preview step; acquisition runs to the configured waiting budget

Allow classical messageApply correction after arrivalUse four-qubit entanglement swappingReceiver measurementInput polar angle

\theta

65^{\circ}

Input azimuth

\varphi

35^{\circ}

Bell resource visibility (each pair)

v

1

Input angles affect teleportation only. Swap mode starts with two Bell pairs. Axis changes retain records; source or channel changes start a new record.

Timing, errors and sampling

Classical message delay

d

2\,\mathrm{ticks}

Receiver waiting budget

t

3\,\mathrm{ticks}

Flip probability per message bit

e

0

Trials per measurement basis

N

400

Simulation seed

s

1984

Bell branch to inspect (preview only)

Exact three- or four-qubit classical simulation with Bell-diagonal resource noise and independent classical message flips. Timing uses abstract protocol ticks, not distance or a signal velocity. Ideal complete Bell measurements and local gates are assumed. Repeated state preparation enables verification; the original input is consumed.

Recorded evidence and resource cost

SIMULATED TRIALS

What the receiver measured

Colored bars use trials; gray marks are model predictions. Pointwise 95% Wilson intervals describe verification fractions, not a process certificate or optional-stopping guarantee. Bell fidelity needs all three axes and can fluctuate outside its physical range as a linear sample estimate.

Current verification fraction / 95% range

\text{—}

Bell fidelity from three measured axes

\text{—}

Trials / Bell pairs consumed

\text{—}

Start with the essentials

Focus question: What changes at Bob when Alice measures, before any message is available?
One-sentence intuition: Every Bell branch needs a different correction. Ignoring the branch gives the same fully mixed receiver state for every input.

Core mathematical model

Prepared input

|\psi\rangle=\cos(\theta/2)|0\rangle+e^{i\varphi}\sin(\theta/2)|1\rangle

The simulator exposes the preparation angles for teaching. A recipient of a single unknown state would not be given those angles.

Noisy resource

\rho_{AB}=v|\Phi^+\rangle\langle\Phi^+|+(1-v)I_4/4

Visibility ranges from a pure Bell pair to the maximally mixed pair. It is implemented as a mixture of four Pauli-related Bell states, not a shortened pure-state arrow.

Bell measurement branches

|\psi\rangle|\Phi^+\rangle\longmapsto\frac12\sum_{a,b}|ab\rangle X^bZ^a|\psi\rangle

The first displayed bit is the phase-correction bit; the second is the bit-flip correction bit. Each branch has probability one quarter. Correction applies a bit flip, then a phase flip; global phase is irrelevant.

Before the classical message

\rho_B=\sum_{a,b}p_{ab}\rho_{B|ab}=I_2/2

Without branch information or correction the receiver is fully mixed. The branch preview is an omniscient conditional view, not information accessible to Bob.

Noise limits after correct messages

F_{\mathrm{transfer}}=\frac{1+v}{2},\qquad F_{\mathrm{swap}}=\frac{1+3v^2}{4}

For swapping, both independently prepared pairs have the same visibility. These limits assume ideal gates, correct messages and no additional memory noise.

Measured swapped-pair fidelity

\widehat F_{\Phi^+}=\frac{1+\widehat C_{XX}-\widehat C_{YY}+\widehat C_{ZZ}}4

Correlations use independently prepared trials measured along three common Pauli axes. The linear estimate is not projected onto physical states, so finite samples can give values outside the physical interval. No joint confidence interval is supplied.

Common difficulties

Unknown input versus verifier

Typical misconceptionThe verifier somehow knows an arbitrary unknown state.

Better mental modelThis teaching experiment repeats a known preparation. Its inverse-projector test checks that specific input only, not a universal process.

Conditional state versus message

Typical misconceptionA colored conditional branch allows instant communication.

Better mental modelThe branch label is unavailable to the receiver until a classical message arrives. Displaying it in the simulator does not transmit information.

Abstract circuit versus optics

Typical misconceptionEvery optical Bell analyzer succeeds deterministically.

Better mental modelThis is a gate-level protocol with ideal complete Bell measurement. Optical collection, partial Bell discrimination, loss and hardware timing are not modeled.

Run the experiment

01
Stop the message
Choose Block the message, acquire the known-input verification and inspect all four branches.
What to observe: Conditional groups differ but the unsorted receiver success probability is one half.
02
Spend the resource
Restore the message and acquire again. Then change the input angles.
What to observe: The ideal correction works for each input. Repeated preparation supplies new inputs and Bell pairs; none are cloned.
03
Let the message arrive too late
Compare the wait budget with the classical delay.
What to observe: Both are abstract protocol ticks. Arrival gates the correction; there is no inferred physical speed or memory decoherence.
04
Swap two pairs
Choose Swap two pairs, acquire all three axes and inspect the measured fidelity.
What to observe: The two outer qubits share a conditional Bell state. Individually each stays fully mixed; without the message the outer pair is fully mixed too.