Q043 · Entanglement / classical messages

Quantum Teleportation & Swapping

If the classical message is blocked, does a teleported quantum state still arrive? No. Without the two classical bits, the receiver does not know which Bell measurement outcome occurred and cannot pick the right correction. Averaged over the four outcomes, the qubit in hand is completely mixed, whatever the input. With the bits and the correction the state arrives, and the original is used up.

Prepare a state, share a Bell pair and look at all four measurement branches. Block the classical message, then compare fourfold optical coincidence records, delayed-choice grouping, and dense coding with three or four messages. Every failed attempt still counts toward the resources used.

Configured final target fidelity $\text{—}$

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

Make a prediction

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

How to read the scene

Compare a delivered message with a blocked one. Look at one Bell branch, then measure many trials, each prepared fresh.

Choose a prediction, compare the cases, then acquire.

Preview step; acquisition runs to the configured waiting budget

Allow classical messageApply correction after arrivalInput 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

Use four-qubit entanglement swappingReceiver measurementClassical message delay

d

2\,\mathrm{ticks}

Receiver waiting time

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)

The ideal protocol uses complete Bell measurements and abstract time steps. The separate optical workbench uses partial Bell projections, schematic nanosecond paths and independent detector loss. The exact classical simulation of a few qubits leaves out vacuum from the source, multipair emission and background counts. Quality given success and yield over all attempts are divided by different totals.

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{—}

Inspect Bell branches and individual trials

Try partial optical detection, delayed choice and dense coding

Optical records · Q043

Keep the attempts that failed

The apparatus above previews the original protocol with an ideal, complete Bell measurement. This separate optical workbench uses a partial analyzer, missed photons and fixed timing rules. Its counts are simulated pulse records, not a reconstruction of a historical experiment.

ProtocolReceiver analyzer (teleport / coding)Single-photon efficiency

0.85

Pauli channel error probability

0

Coincidence residual window / ns

3

Uniform timing half-width / ns

0.5

Central optical delay / ns

485

Import saved JSON

Attempts / accepted / failures

\text{—}

One-input pass fraction or decoding

\text{—}

Information per attempt / per accepted attempt

\text{—}

Prepared pairs / transmitted qubits

\text{—}

A pulse identity follows every path

\delta t_i=t_i-t_{i,\mathrm{ref}}

\max_i\delta t_i-\min_i\delta t_i\leq W

The gate uses calibrated path offsets within the same pulse; delayed photons need not arrive simultaneously with earlier detections. All failed pulses stay in the denominator. No cross-pulse matching, dark counts, multipair emission or detector dead time are modeled.

Sort the same earlier outcomes

Only the grouping changes. The earlier file is fixed before the later random choice. All observed outer pairs have uncorrelated outcomes before later conditioning. This table excludes outer detector misses; the full record of attempts keeps them. Conditional subsets can have different correlations. This demonstrates conditional probabilities, not backward causation or an experimental entanglement certificate.

E_j=\frac{N_{00}+N_{11}-N_{01}-N_{10}}{N_j}

Earlier outcomes, later grouping
Axis	Retained	First outcome zero	Correlation

I(M;D)=\sum_{m,d}p_{md}\log_2\!\frac{p_{md}}{p_m p_d}

Empirical mutual information includes erasure as an output and all attempted symbols. It assumes a known indexed trial clock and the prescribed exact pair preparation. Message-dependent erasure can itself carry information; this does not mean every erased symbol can be decoded. This finite-sample plug-in statistic has upward bias and is not historical delivered rate or channel capacity. The accepted-only value uses a different, possibly message-biased input distribution. Every attempt budgets one already shared Bell pair and one transmitted member; source vacuum, background, pair distribution and optical stabilization costs are omitted.

Latest 12 completed pulse records; CSV includes every attempt
Pulse	Basis / choice	Outcome / message	Pattern	Accepted	Reason

Latest recorded timestamps / ns; a dash is a missed or unused detector
Pulse	Trigger	Outer / receiver A	Outer B	Central 1	Central 2	Choice

Measurement assumptions and primary sources

Polarization basis: horizontal is zero, vertical is one. Bell codes are phase then parity; local encodings act on the travelling member. The original ideal workbench remains independently available above.

\Phi^{\pm}=\frac{\lvert HH\rangle\pm\lvert VV\rangle}{\sqrt{2}}

\Psi^{\pm}=\frac{\lvert HV\rangle\pm\lvert VH\rangle}{\sqrt{2}}

(0,1,2,3)\leftrightarrow(\Phi^+,\Psi^+,\Phi^-,\Psi^-)

The passive analyzer identifies the two opposite-polarization Bell states; equal prior states therefore succeed at most half the time before loss. The delayed-choice version uses a locally rotated analyzer that selects the two same-polarization Bell states and compares the negative branch with HH / VV product outcomes. These are declared ideal polarization projectors with independent threshold-channel efficiency; not a wave-optics or SPDC hardware model.

Three-message coding sends only Psi-plus, Psi-minus and Phi-minus. The analyzer still cannot tell the two Phi signs apart. An extra beam splitter at just one of the original detectors registers the third category with a quarter of the probability, before shared detector losses, following the ideal rate ratio stated by Mattle et al. The noise and timing sliders are teaching parameters, not efficiencies fitted to the 1996 experiment.

Start with the essentials

Focus question: What changes at Bob when Alice measures, before any message is available?
One-sentence intuition: Each Bell branch needs a different correction. If you ignore the branch, the receiver's state is fully mixed, and it is the same 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 shows the preparation angles for teaching. Someone handed a single unknown state would not be given them.

A noisy entangled pair

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

The visibility runs from a pure Bell pair to a maximally mixed pair. The noisy pair is a mixture of the four Bell states, which are related by Pauli operations; it is a mixed state, not a pure state with a shorter arrow.

Bell measurement branches

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

The first bit shown controls the phase correction, and the second controls the bit-flip correction. Each branch occurs with probability one quarter. The correction applies the bit flip first, then the phase flip; the global phase does not matter.

Before the classical message

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

Without the branch information or a correction, the receiver's qubit is fully mixed. The branch preview shows what an all-knowing observer would see; Bob has no access to it.

Noise limits after correct messages

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

For swapping, the two independently prepared pairs have the same visibility. These limits assume ideal gates, correct messages and no extra noise while the qubits are stored.

Measured swapped-pair fidelity

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

Each correlation comes from separately prepared trials with both qubits measured along the same Pauli axis, for three axes. The linear estimate is not forced back onto a physical state, so a finite sample can give a value outside the physical range. No joint confidence interval is given.

Common difficulties

How the verifier checks an unknown state

Typical misconceptionThe verifier somehow knows an arbitrary unknown state.

Better mental modelThe teaching experiment repeats a known preparation. Its test projects back onto that specific input, so it checks only that input, not the process for every possible state.

A colored branch is not a message

Typical misconceptionThe colored conditional branch allows instant communication.

Better mental modelThe receiver cannot know the branch label until the classical message arrives. Showing it in the simulator does not send any information.

Ideal circuit versus real optics

Typical misconceptionEvery optical Bell analyzer succeeds every time.

Better mental modelThis gate-level protocol uses an ideal, complete Bell measurement. Optical collection, partial Bell discrimination, loss and hardware timing are not modeled in it.

Run the experiment

01
Block the message
Choose Block the message, acquire the known-input verification and inspect all four branches.
What to observe: The groups sorted by branch differ, but without sorting, the receiver passes the test with probability one half.
02
Use up the entanglement
Restore the message and acquire again. Then change the input angles.
What to observe: The ideal correction works for every input. Each repetition uses a freshly prepared input and a new Bell pair; nothing is cloned.
03
Let the message arrive too late
Compare the wait budget with the classical delay.
What to observe: Both are counted in abstract protocol ticks. Bob can correct only after the message arrives. No physical speed is implied, and memory decoherence is not modeled.
04
Swap two pairs
Choose Swap two pairs, acquire all three axes and inspect the measured fidelity.
What to observe: Given the branch, the two outer qubits share a Bell state. Each one alone stays fully mixed, and without the message the outer pair as a whole is fully mixed too.