Quantum Communication and Entanglement Distribution

Quantum communication asks whether an unknown state, including its phase coherence with an inaccessible reference, survives a physical sender–field–receiver protocol. Entanglement distribution asks a related but distinct operational question: after the protocol, do separated laboratories share useful entanglement, at what success probability, and after consuming which references or prior resources? A visible detector response answers neither question by itself.

Required background. Field communication constructs the localized sender–field–receiver channel.

Helpful background. Energy-constrained channel distances supplies error measures whose input optimization is restricted to physically admissible bosonic states.

The chapter’s task map, protocol comparison, and failure controls place this coherence test beside signaling, harvesting, and capacity without duplicating their common diagrams.

What is being distributed?

Let $Q$ be a logical input, $A$ the physical sender system, and $B$ the receiver output. An encoder $\mathcal E_{Q\to A}$, physical channel $\mathcal N_{A\to B}$, and decoder $\mathcal D_{B\to Q'}$ define the effective logical channel

$$\mathcal M_{Q\to Q'} =\mathcal D_{B\to Q'}\circ\mathcal N_{A\to B}\circ\mathcal E_{Q\to A}.$$

To test coherence, purify an input state $\rho_Q$ as $|\psi\rangle_{RQ}$ and leave the reference $R$ untouched. Its entanglement fidelity is

$$F_e(\rho_Q,\mathcal M) =\langle\psi| (\operatorname{id}_R\otimes\mathcal M) (|\psi\rangle\langle\psi|) |\psi\rangle .$$

This number is independent of which purification represents $\rho_Q$ Schumacher 1996, § IV.A, pp. 2618–2620. For a two-dimensional code, taking $\rho_Q=I/2$ makes $|\psi\rangle$ a Bell state and probes every matrix element of the logical channel at once. A channel can transmit the basis labels $0$ and $1$ perfectly while erasing their relative phase, so a classical success probability is not a substitute for $F_e$.

Several nearby tasks should not be merged:

• Entanglement transmission preserves entanglement between $R$ and the encoded system while the latter crosses the channel.
• Entanglement generation starts without entanglement shared across the two laboratories and aims to create it by channel use.
• Entanglement distribution is the broader resource statement that the laboratories end with entanglement; its accounting must name any entangled state, phase reference, classical communication, or stored quantum system supplied beforehand.
• Heralded distribution reports both a conditional state and the probability and timing of the success record. Postselecting a rare branch changes the instrument, not the deterministic channel.

These definitions are operational: they specify what is prepared, which systems cross the laboratory boundary, what is measured, and what is counted as consumed.

A lossy wavepacket benchmark

Consider one normalized field wavepacket with annihilation operator

$$a_h=\int d\mu(k)\,h(k)a_k, \qquad \int d\mu(k)\,|h(k)|^2=1.$$

Suppose the receiver extracts one matched output packet and all inaccessible orthogonal modes begin in vacuum. The reduced channel is then the pure-loss channel

$$b=\sqrt\eta\,a_h+\sqrt{1-\eta}\,e, \qquad 0\leq\eta\leq1,$$

where $e$ is an environmental vacuum mode. In a field protocol, $\eta$ is not a free plane-wave attenuation symbol: it is obtained from propagation, localized coupling, packet mismatch, and receiver access. Detector-mediated models illustrate how these ingredients define an actual channel rather than merely a two-point correlation Cliche and Kempf 2010, §§ III–V, pp. 4–9.

Use the finite-energy single-rail code

$$|0_L\rangle=|0\rangle, \qquad |1_L\rangle=|1_h\rangle .$$

On this code, pure loss has Kraus operators

$$K_0=|0\rangle\langle0|+\sqrt\eta\,|1\rangle\langle1|, \qquad K_1=\sqrt{1-\eta}\,|0\rangle\langle1|.$$

Send the second half of $|\Phi^+\rangle_{RQ}=(|00\rangle+|11\rangle)/\sqrt2$. In the ordered basis $|00\rangle,|01\rangle,|10\rangle,|11\rangle$, the output is

$$\rho_{RQ'}=\frac12 \begin{pmatrix} 1&0&0&\sqrt\eta\\ 0&0&0&0\\ 0&0&1-\eta&0\\ \sqrt\eta&0&0&\eta \end{pmatrix}.$$

Taking its Bell-state overlap gives the reproducible benchmark

$$F_e=\frac{(1+\sqrt\eta)^2}{4}.$$

The same state has standard negativity $\mathcal N=(\lVert\rho^{T_{Q'}}\rVert_1-1)/2=\eta/2$ and two-qubit concurrence $C=\sqrt\eta$. These quantities answer different questions: $F_e$ compares the full logical state with its target, whereas $\mathcal N$ and $C$ quantify output entanglement.

For $\eta=0.64$,

$$F_e=0.81, \qquad \mathcal N=0.32, \qquad C=0.80.$$

The transmitted half of the Bell pair has mean input photon number $1/2$, so the benchmark states both its code and its energy. It also exposes the limiting cases: $\eta=1$ gives perfect transmission, while $\eta=0$ gives $F_e=1/4$ and a separable output.

Phase and herald records are physical resources

The single-rail coherence $|0\rangle\langle1|$ transforms under a phase rotation $U_\theta=e^{i\theta a^\dagger a}$. If sender and receiver lack a shared phase reference, the operational input is phase-twirled,

$$\mathcal T(\rho)=\frac1{2\pi}\int_0^{2\pi} d\theta\,U_\theta\rho U_\theta^\dagger.$$

The Bell off-diagonal terms then disappear. After the same loss channel,

$$F_e^{\mathrm{twirl}}=\frac{1+\eta}{4}.$$

At $\eta=0.64$ this is $0.41$, despite unchanged number-basis transition probabilities; even at zero loss it is only $1/2$. Reference-frame uncertainty can therefore convert apparent coherent transmission into an incoherent mixture Vaccaro, Anselmi, and Wiseman 2003, §§ II–III. A protocol may instead distribute or consume a phase reference, but that resource must appear in its specification.

Heralding needs equally explicit accounting. If a dual-rail photon is accepted only when one photon reaches the receiver, ideal loss produces a conditional fidelity of one but success probability $p_s=\eta$. Over $n$ independent trials, the expected number of accepted pairs is $n\eta$, not $n$. A rate must also include the time for the classical success message, detector reset, and rejected trials. Reporting only the conditional fidelity would let $p_s\to0$ masquerade as a perfect distribution protocol.

From the field interaction to the channel claim

A relativistic implementation begins with compact or quantitatively controlled sender and receiver couplings. One derives the receiver channel after tracing inaccessible field modes, then checks that the receiver lies in the sender’s causal future for any claimed transmission. Switching work, background excitations, and communication energy are separate terms; moving energy between them does not increase a code’s quantum performance Barcellos and Landulfo 2021, §§ III–VI, pp. 4–12.

The pure-loss benchmark applies only when one packet is isolated, the environment entering the unused port is vacuum, and encoding and decoding remain inside the declared subspace. Thermal occupation changes the channel; leakage into orthogonal packets changes the decoder; localization tails can change which events are causally connected. Rapid-interaction detector models can yield finite-dimensional or bosonic communication channels, but their capacity and energy statements remain model-dependent Tjoa and Gallock-Yoshimura 2022, §§ III–VI, pp. 5–14.

Three controls make the operational conclusion robust:

1. Remove the shared phase reference while preserving number-basis calibration. A fall from $F_e$ to $F_e^{\mathrm{twirl}}$ identifies reference-dependent coherence.
2. Keep every failed heralding record instead of conditioning on success. This recovers the trace-preserving channel and its honest yield.
3. Replace the field coupling by a classical record with the same basis-state detection statistics. If Bell coherence vanishes, classical signaling alone did not distribute quantum information.

Exercises

1. Lossy Bell pair. Derive the entanglement fidelity for the single-rail pure-loss channel at $\eta=0.36$.

Solution

Only the no-loss Kraus operator has a nonzero Bell-state expectation:

$$\langle\Phi^+|(I\otimes K_0)|\Phi^+\rangle =\frac{1+\sqrt\eta}{2}.$$

Therefore $F_e=(1+\sqrt\eta)^2/4$. With $\sqrt\eta=0.6$, $F_e=0.64$. The negativity is $\eta/2=0.18$, so high state overlap and the amount of surviving entanglement should not be reported as the same metric.

2. A perfect classical bit can be a failed qubit. Let $\mathcal Z(\rho)=\tfrac12(\rho+Z\rho Z)$ be complete dephasing. Show that it transmits the computational-basis bit without error but has Bell-state entanglement fidelity $1/2$.

Solution

$\mathcal Z(|j\rangle\langle j|)=|j\rangle\langle j|$ for $j=0,1$, so the receiver distinguishes the basis states perfectly. Acting on $|\Phi^+\rangle$ gives

$$(I\otimes\mathcal Z)(\Phi^+) =\frac12\bigl(|00\rangle\langle00|+|11\rangle\langle11|\bigr).$$

Its overlap with $|\Phi^+\rangle$ is $1/2$, and the state is separable. Classical contrast therefore does not certify quantum transmission.

3. Conditional quality versus yield. A heralded protocol succeeds with probability $p_s=10^{-3}$, has conditional fidelity $0.999$, and takes $2\,\mathrm{ms}$ per independent trial. Find the accepted-pair rate before any further losses.

Solution

There are $500$ trials per second, so the expected accepted rate is

$$R=p_s\times500=0.5\ \text{pairs per second}.$$

The high branch fidelity does not turn this into a 500-pair-per-second protocol. Classical acknowledgement and reset time would reduce the rate further if they are not already included in the $2\,\mathrm{ms}$ clock.

References

• Barcellos, I. B., and Landulfo, A. G. S. (2021). “Relativistic Quantum Communication: Energy Cost and Channel Capacities.” Physical Review D 104, 105018. DOI. Open PDF.
• Cliche, M., and Kempf, A. (2010). “The Relativistic Quantum Channel of Communication through Field Quanta.” Physical Review A 81, 012330. DOI. Open PDF.
• Schumacher, B. (1996). “Sending Entanglement Through Noisy Quantum Channels.” Physical Review A 54, 2614–2628. DOI. Open PDF.
• Tjoa, E., and Gallock-Yoshimura, K. (2022). “Channel Capacity of Relativistic Quantum Communication with Rapid Interaction.” Physical Review D 105, 085011. DOI. Open PDF.
• Vaccaro, J. A., Anselmi, F., and Wiseman, H. M. (2003). “Entanglement of Identical Particles and Reference Phase Uncertainty.” International Journal of Quantum Information 1(4), 427–441. DOI. Open PDF.