Communication Through Quantum Fields

A field does not define a communication channel by itself. The channel is induced by a localized encoding, field propagation, a localized receiver, and a declared readout. Its performance depends on the coupling supports and schedules, field state, input cost, mode matching, and error criterion. Vacuum correlations can correlate two records without carrying the sender’s choice, so causal influence must be extracted from an intervention contrast.

Required background. Algebraically localized operations defines supported encoders and receivers. Infinite-dimensional channel–state methods supplies the domain restrictions needed for bosonic fields.

Helpful background. Thermal states and the KMS condition supplies thermal-noise structure. Localized detector models supplies a concrete transducer.

The sender–field–receiver map

Let $S$ be a sender system, $F$ the field, and $R$ a receiver. In the tensor-product regulator used in this formula, begin with the product density operator $\rho_S\otimes\rho_F\otimes\sigma_R$; initial correlations can be allowed, but they then become additional protocol data and may prevent the reduced evolution from being a channel on arbitrary $\rho_S$. Let $U_A$ be generated by a coupling supported in a compact region $K_A$, and let $U_B$ be generated by a later coupling supported in $K_B$. The induced receiver channel is

$$\mathcal N_{A\to B}(\rho_S) =\operatorname{tr}_{S,F}\!\left[ U_BU_A(\rho_S\otimes\rho_F\otimes\sigma_R) U_A^*U_B^* \right].$$

Every element of this formula is operational: the sender alphabet or code space, the field state, both switching and smearing profiles, the detector gaps, and the traced degrees of freedom. Changing the receiver mode or readout changes $\mathcal N_{A\to B}$ even in the same spacetime.

For a classical message $a$ encoded in states $\rho_a$ and a receiver POVM $\{E_y\}$,

$$p(y\mid a) =\operatorname{tr}[E_y\mathcal N_{A\to B}(\rho_a)]$$

is the actual classical channel. The one-shot Bayes error, the mutual information for one input distribution, and an asymptotic constrained capacity are different quantities. Similarly, coherent information or entanglement fidelity is needed for a quantum-message claim. The chapter task map separates these tasks, the canonical protocol table records their distinct outputs, and the failure-control map locates support tails and mode mismatch.

A compact probe coupling is more than notation. Under the causal-factorization hypotheses of Fewster and Verch 2020, §§ 3–5, it produces a localized operation whose composition follows the causal order. Perturbative Unruh–DeWitt realizations of the induced two-detector channel are constructed by Cliche and Kempf 2010, §§ II–V; their receiver channel has zero classical and quantum capacity at spacelike separation even though the probes can become correlated.

A compact scalar-wavepacket encoder

The causal signal can be seen without choosing a particle basis. Let $\phi$ be a free real scalar field and use real smooth compactly supported test functions $f_A$ and $f_B$, with supports $K_A$ and $K_B$. Define

$$\Phi(f)=\int d^4x\,f(x)\phi(x), \qquad [\Phi(f_A),\Phi(f_B)] =i\Delta(f_A,f_B)\mathbf1.$$

Here $\Delta$ is the causal propagator in the sign convention fixed by the displayed commutator. The sender encodes a symmetric bit $x\in\{-1,+1\}$ by the local Weyl unitary

$$U_x=\exp[i xg\Phi(f_A)].$$

For smooth compact $f_A$, this creates a finite-energy coherent field disturbance under the usual free-field domain assumptions. Its classical wavepacket is the causal solution sourced by $f_A$; it is not a plane wave and does not require a global positive-frequency mode to define the encoding.

Because the commutator is a multiple of the identity, the Baker–Campbell–Hausdorff series terminates:

$$U_x^*\Phi(f_B)U_x =\Phi(f_B)+xg\,c_{AB}\mathbf1, \qquad c_{AB}=\Delta(f_A,f_B).$$

If $K_B$ lies to the causal future of $K_A$, $c_{AB}$ is equivalently the retarded response pairing for this schedule. It contains the spacetime propagation and mode overlap in one number. Its sign depends on the test-function and propagator conventions, while distinguishability depends on $|c_{AB}|$.

Now realize a finite-time receiver with a pointer having $[Q_R,P_R]=i$. Use the compact interaction

$$U_B=\exp[-i\kappa P_R\Phi(f_B)].$$

It gives

$$U_B^*Q_RU_B=Q_R+\kappa\Phi(f_B).$$

Suppose the field state is centered and quasifree, the initial pointer is centered Gaussian, and the two are initially independent. The pointer outcome $Y$ conditioned on $x$ is then Gaussian:

$$p(y\mid x) =\frac1{\sqrt{2\pi V}} \exp\!\left[-\frac{(y-x\mu)^2}{2V}\right],$$

with

$$\mu=\kappa g c_{AB}, \qquad V=V_Q+\kappa^2V_F, \qquad V_F=\omega_F(\Phi(f_B)^2).$$

$V_Q$ is detector readout noise and $V_F$ is the field’s local fluctuation in the receiver profile. The sender changes the mean through the commutator, while the field state controls much of the noise through its symmetric two-point function. In a free field, the commutator is state independent, so moving from vacuum to a thermal KMS state leaves $c_{AB}$ unchanged but generally increases $V_F$.

For equal priors and equal variances, thresholding at $y=0$ is the minimum-error hard decision. Its error probability is

$$P_{\mathrm e} =\frac12\operatorname{erfc}\!\left( \frac{|\mu|}{\sqrt{2V}} \right).$$

The thresholded record is a binary symmetric channel with achievable rate

$$R_{\mathrm{hard}} =1-h_2(P_{\mathrm e})$$

bits per independent use, where $h_2$ is binary entropy. This is the capacity of the hard-decision channel. It is a lower bound on what an optimized decoder using the full analog outcome could achieve, not the unconstrained capacity of the quantum field.

First QFT benchmark: matched finite-time reception

For a concrete quadrature, work with a massless scalar in $3+1$ Minkowski spacetime and measure all coordinates in one chosen length unit. Define the normalized compact bump

$$b(u)=\frac1{\mathcal Z} \begin{cases} \exp[-1/(1-u^2)],&|u|\lt1,\\ 0,&|u|\geq1, \end{cases} \qquad \mathcal Z=\int_{-1}^{1}e^{-1/(1-u^2)}du.$$

Use

$$\begin{aligned} f_A(t,\mathbf x) &=b(t)b(x_1)b(x_2)b(x_3),\\ f_B(t,\mathbf x) &=f_A(t-4,\mathbf x-4\mathbf e_1). \end{aligned}$$

The supports are compact, the receiver is later, and their centers are null separated. With

$$G_{\mathrm{ret}}(t,\mathbf x) =\frac{\theta(t)}{4\pi|\mathbf x|} \delta(t-|\mathbf x|),$$

the response magnitude is obtained from the finite-support integral

$$|c_{AB}| =\left|\int d^4x\,d^4y\, f_B(y)G_{\mathrm{ret}}(y-x)f_A(x)\right|.$$

For the Minkowski vacuum, an independent momentum-space quadrature gives

$$V_F =\int\frac{d^3\mathbf k}{2(2\pi)^3|\mathbf k|} \left|\widetilde f_B(|\mathbf k|,\mathbf k)\right|^2,$$

where $\widetilde f(k^0,\mathbf k)=\int d^4x\,e^{i(k^0t-\mathbf k\cdot\mathbf x)}f(x)$. These formulas completely specify the profile, support, Fourier convention, causal kernel, and integrations. Restoring a physical length rescales $f$, $g$, and $\kappa$ with their canonical dimensions but leaves the dimensionless pointer ratios below unchanged.

Evaluate the two integrals by adaptive quadrature, choose $\kappa=\sqrt{0.50/V_F}$, absorb the sign of $c_{AB}$ into the bit labeling, and set $g=0.80/(\kappa|c_{AB}|)$. With pointer variance $V_Q=0.14$, this gives

$$\kappa g c_{AB}=0.80, \qquad \kappa^2V_F=0.50, \qquad V_Q=0.14.$$

All three quantities are dimensionless in pointer units. Thus $V=0.64$, $\sqrt V=0.80$, and the signal-to-noise ratio for either symbol is $|\mu|/\sqrt V=1$. Direct substitution gives

$$P_{\mathrm e} =\frac12\operatorname{erfc}\!\left(\frac1{\sqrt2}\right) =0.158655\ldots,$$

and

$$R_{\mathrm{hard}} =1-h_2(0.158655\ldots) =0.3689\ldots\ \text{bits/use}.$$

A numerical implementation should reproduce both values, integrate the compact profiles to obtain $c_{AB}$, and separately report $V_F$ and $V_Q$. Combining the two variances before validation can hide an incorrect field normalization or detector calibration.

The quadrature has several independent convergence tests. Refine the real-space grid until the compact-support integral for $c_{AB}$ stabilizes, and refine the momentum grid and ultraviolet cutoff independently for $V_F$. Translate both profiles together to check Poincaré-translation invariance, reverse the encoded sign to check that the mean changes sign while the variance does not, and move the receiver to a spacelike compact support to obtain the exact-zero control. Numerical leakage in that last test should decrease with the quadrature error; a persistent value signals a kernel, support, or interpolation mistake.

Repeated pointer measurements estimate more than the hard decision. The likelihood ratio for the two Gaussian laws is linear in $y$, so the sign threshold is optimal for minimum error, but the magnitude $|y|$ still carries reliability information. Keeping the full analog record produces the binary-input mutual information

$$I(X;Y) =\frac12\sum_{x=\pm1} \int dy\,p(y\mid x) \log_2\frac{2p(y\mid x)}{p(y\mid+1)+p(y\mid-1)}.$$

This integral is directly reproducible from $\mu$ and $V$ and is no smaller than $R_{\mathrm{hard}}$, because thresholding is a classical post-processing step. It is still tied to the fixed binary alphabet and equal prior. Optimizing over a larger alphabet under an energy constraint defines a different communication problem.

Finite-time implementation also requires a sampling protocol. The field and pointer must be re-prepared for each nominally independent use, or temporal correlations between uses must be included in a memory-channel model. The duration used in a bits-per-second claim includes sender switching, propagation, receiver coupling, readout, and reset. Dividing the bits-per-use benchmark only by the interaction time would overstate the operational rate.

The input cost is also part of the benchmark. The two symbols use opposite coherent displacements and therefore have the same mean energy cost in a quadratic free Hamiltonian. Its value is proportional to $g^2$ times the one-particle norm of the classical solution sourced by $f_A$ and must be quoted when comparing codes. A rate per use is not a rate per unit energy or time. Energy-resolved detector channels and their constrained capacities are treated by Barcellos and Landulfo 2021, §§ III–VI.

Causal and mode-matching controls

Spacelike support. Move $K_B$ so that it is spacelike separated from $K_A$, keeping the initial field state and receiver unchanged. Microcausality gives $c_{AB}=0$, hence $\mu=0$, $P_{\mathrm e}=1/2$, and $R_{\mathrm{hard}}=0$. Receiver noise remains, and the sender and receiver may still have correlated records, but neither depends on the encoded sign at the receiver alone.

If the receiver is entirely earlier than the sender, its record is produced before $U_x$ and cannot be changed retrocausally. One must compose the operations in that physical order; conjugating an earlier observable by a later sender unitary would describe a different, nonoperational question. Thus “outside the causal future” includes both the spacelike null test and the requirement not to reverse a timelike schedule.

Mode mismatch. Keep $K_B\subset J^+(K_A)$ but choose a receiver profile with $\Delta(f_A,f_B)=0$. The regions are causally connected, yet this particular linear receiver is symplectically orthogonal to the signal. Again $P_{\mathrm e}=1/2$. Timelike separation permits communication; it does not guarantee that an arbitrary mode or detector gap captures it.

Support tails. Compact $f_A,f_B$ make the causal null exact. Gaussian profiles never vanish, so a nominally spacelike calculation has tail-mediated overlap. It can still be a useful finite-resolution model, but it must report a tail norm and an induced bound on $|c_{AB}|$ rather than call the result exact locality.

Perturbative control. In detector expansions, signal and local excitation noise can occur at comparable coupling orders. Verify positivity and normalization of the reduced receiver state and show that omitted terms are smaller than the claimed error or rate margin. Rapid-interaction channels can sometimes be evaluated nonperturbatively; Tjoa and Gallock-Yoshimura 2022, §§ III–VI provides an explicit example, but its capacity remains tied to its declared interaction model.

From a bit channel to quantum communication

The Weyl benchmark sends one classical bit alphabet and reads one quadrature. It does not establish coherent qubit transmission. A quantum code must preserve superpositions and be evaluated by entanglement fidelity, coherent information, or a proved quantum-capacity bound. A wavepacket code must also specify the physical mode: packet normalization, bandwidth, mean energy, localization tails, receiver overlap, and decoding operation.

The induced quantum channel may be lossy and noisy even when causal propagation is perfect. Tracing inaccessible field modes produces loss; finite switching distorts the matched mode; a KMS state adds noise; and a phase-reference mismatch can destroy coherence while leaving classical amplitude signaling. These are channel properties, not violations of causality. Channel–state reconstruction from the preceding page can help characterize a regulated bosonic channel, provided its finite-energy reference and topology are retained.

Common pitfalls

Calling a correlator a channel. A Wightman function helps determine noise and correlations. Only the sender-dependent conditional law or induced CP map defines communication.

Quoting “the capacity.” State whether the number is one-shot mutual information, hard-decision capacity, Holevo information, classical capacity, quantum capacity, or an assisted capacity, and state the energy and bandwidth constraint.

Using an ideal global mode as a receiver. A plane-wave annihilation operator is not a compact apparatus. Replace it by a supported detector or give a quantitative localization and mode-mismatch error.

Exercises

1. Derive the sender-induced shift

Use the commutator convention on this page to derive $U_x^*\Phi(f_B)U_x$.

Solution

Set $A=-ixg\Phi(f_A)$ and $B=\Phi(f_B)$. Then

$$e^ABe^{-A}=B+[A,B] =B+xg\Delta(f_A,f_B)\mathbf1.$$

All higher nested commutators vanish because the first commutator is a scalar multiple of the identity.

2. Reproduce the communication benchmark

For $|\mu|=0.80$ and $V=0.64$, calculate the optimum threshold error and the hard-decision rate.

Solution

The normalized separation is $|\mu|/\sqrt V=1$. Therefore

$$P_{\mathrm e}=\Phi(-1)=0.158655\ldots.$$

Using $h_2(p)=-p\log_2p-(1-p)\log_2(1-p)$ gives $h_2(P_{\mathrm e})=0.6311\ldots$ and $R_{\mathrm{hard}}=0.3689\ldots$ bits/use.

3. Separate causal access from mode matching

Give two physically distinct reasons for $c_{AB}=0$, and state what each one proves.

Solution

If the compact supports are spacelike separated, microcausality forces $c_{AB}=0$ for every such pair of profiles; this certifies a causal null. If the supports are timelike related but the chosen $f_B$ is symplectically orthogonal to the propagated sender solution, only this receiver mode has zero response. The latter proves mode mismatch, not absence of every possible timelike communication channel.

4. Add thermal noise

Suppose the commutator response remains $\mu=0.80$ but thermal field noise raises the total variance from $0.64$ to $1.44$. Find the new hard-decision error qualitatively and explain why causality is unchanged.

Solution

The normalized separation becomes $0.80/1.20=2/3$, so

$$P_{\mathrm e}=\frac12\operatorname{erfc}\!\left(\frac{2}{3\sqrt2}\right) =\Phi(-2/3)\approx0.2525.$$

The error increases because the symmetric two-point function contributes more receiver noise. The commutator and its causal support are unchanged, so the light-cone restriction on influence is the same.

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.
• Fewster, C. J., and Verch, R. (2020). “Quantum Fields and Local Measurements.” Communications in Mathematical Physics 378, 851–889. 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.