Optical Detector Capacity

Sister project: Partial Monitoring Capacity supplies the collective optical capacity benchmark and establishes the qubit comparison between predetermined individual and collective measurements. This repository studies optical detector classes and finite phase-reference resources.

Can reading lost light one mode at a time recover the leading quantum communication rate? Yes, but photon counting and coherent measurements have different limits.

A vacuum optical splitter sends a fraction $a$ of the signal to the receiver, a fraction $b$ to an inaccessible output, and $c=1-a-b$ to a helper. The helper measures its output and sends only a classical record to the receiver. The signal budget is $E$ mean photons per original mode. Both detector classes permit arbitrary sender and receiver block codes.

For fixed $0\lt a\lt 1$, increasing energy and approaching complete collection give, uniformly with $L=\min\{E,1/b\}$ and $1/0=\infty$,

$$Q_{\rm cnt}(E)=\frac12\log_2 L+O_a(1), \qquad Q_1(E)=Q_{\rm coll}(E)+O(1)=\log_2 L+O_a(1).$$

Here $Q_{\rm cnt}$ allows exact photon counting and arbitrary processing of the entire count record. $Q_1$ permits predetermined individual-mode POVMs, including coherent superpositions of different photon numbers. $Q_{\rm coll}$ permits joint measurements of all collected modes. The uniform domain keeps $a$ fixed and includes $b=0$.

At every finite energy, not just in this limit,

$$0\le Q_{\rm coll}(E)-Q_1(E)\le\log_2 e.$$

Thus joint processing across collected modes improves on the best individual-mode detector by at most about $1.443$ qubits per mode. Counting instead loses an unbounded amount as $L$ grows. The distinction is between coherence across photon numbers within a mode and processing across modes. Counting itself is non-Gaussian; this is not a general Gaussian-versus-non-Gaussian comparison.

The coherent detector can use one preselected quadratic phase followed by a finite Fock-space Fourier measurement, with fewer than $2d$ occupied outcomes at cutoff dimension $d$. A three-outcome example already separates coherent detection from optimized counting under a two-photon input cutoff. The full leading-rate theorem concerns the larger, mean-energy-only input class.

How much phase reference is needed?

The reference-energy extension charges the apparatus that enables coherent detection. Let $M$ be its average initial mean photon number per channel use. The initial apparatus is independent of the encoded signal, and each effective helper measurement preserves every local total $N_{D_i}+N_{W_i}$. Classical records remain unrestricted. Under this separate-local-invariance contract,

$$Q_{\rm loc}^{\rm mean}(E,M) =\min\!\left\{\log_2E, \max\!\left\{\tfrac12\log_2E,\log_2(M+1)\right\}\right\}+O_a(1).$$

The formula holds uniformly for fixed $0\lt a\lt 1$, $E\ge1$, $M\ge0$, $0\le b\le\frac12\min\{a,1-a\}$, and $bE\le1$. It characterizes capacity to an additive constant depending only on $a$.

Counting supplies the half-logarithmic rate without a coherent reference. A bounded additive gap to the full logarithm is attainable with $M=O(E)$ and requires $M=\Omega(E)$. The construction uses fresh finite sine references; the converse includes arbitrary endpoint codes and references with infinite variance. Merely preserving the sum of all modes' photon numbers is a broader operation class, outside this profile.

The resource account separately quantifies the required classical record, cutoff, and control accuracy. The support-width bound distinguishes reference spread from mean stock, and the conserving realization makes the detector's photon exchange explicit. These results price prepared photons and transmitted records; they do not determine preparation work, reset cost, or sustained coherent reuse.

Finite-energy separation

At $a=1/4$, $b=1/10000$, and $E=10^6$:

Quantity	Rigorous comparison, qubits per mode
Counting capacity, optimized over all allowed inputs and codes	Less than 7.90
Achievable capacity with coherent individual-mode measurements	Greater than 10.51
Collective helper capacity	Between 11.28 and 11.29

The certificate uses rational logarithm enclosures. These are capacity bounds, not an implemented code or a laboratory operating point. The counting upper bound includes rare, arbitrarily large photon numbers under the mean budget.

Read the science

Start with the physical model and claims, then follow the connected proof route.

Question	Authoritative account
What information, controls, and energy are supplied?	Model
Why do arbitrary codes obey the entropy bounds?	Operational reduction
Why do the two detector classes have different leading rates?	Counting bounds and coherent construction
Where does the qubit counting optimum first fail?	Two-photon separation
What is uniform as energy and inaccessible loss vary?	Uniform laws
What does a finite apparatus budget buy?	Mean-reference capacity profile and connected resource proofs
What remains outside the comparison, and which results are inherited?	Interpretation and related work
How are the finite calculations reproduced?	Verification

Sister project: Partial Monitoring Capacity

Partial Monitoring Capacity asks how much communication can be recovered by measuring a partially accessible environment. Its collective-capacity benchmark and its qubit measurement optimum provide two starting points for the optical detector question here.

Account	Question and contribution
Partial Monitoring Capacity	Exact partially monitored capacities, including the optimum over predetermined product helper measurements in the qubit model and its strict separation from unrestricted collective measurements.
Optical Detector Capacity	Counting versus coherent individual-mode measurements in the full optical model, the first failure of qubit counting optimality at a two-photon cutoff, and the stated detector-resource profiles.

The qubit-to-optical bridge makes the comparison concrete. Counting optimality with at most one input excitation does not extend to two excitations or to unrestricted optical inputs. Coherence among photon numbers within one mode and joint measurements across channel uses are different operations; both projects retain arbitrary sender and receiver block codes. The inherited collective benchmark is credited to Partial Monitoring, while the optical detector and reference-budget results have their proofs here.

Each repository supplies its own model, scientific route, and thirty-milestone learning map. The sister links connect the results without adding a prerequisite reading assignment.

Learn the project

The single educational anchor is John Preskill, Quantum Shannon Theory, Chapter 10, arXiv:1604.07450v5 (updated June 2025; posted 8 July 2025). The learning entrance gives a focused route rather than a whole-chapter assignment.

The 30-milestone map connects 14 background/preliminaries milestones to 16 narrative/innovation milestones. It progresses from optical states and entropy through the detector comparison to the finite-reference capacity profile. Every milestone has an understanding target, earlier dependencies, a conceptual connection, and source pointers.

The Preskill bridge translates notation, works through an explicit counting channel, and explains the additional optical, finite-energy, and detector arguments. In particular, Preskill's one-use coherent-information quantity is not this project's individual-mode helper capacity. The science pages remain the authoritative proofs; the learning route does not replace their assumptions or primary-source attribution.

Reproduce

python -m pip install -r verification/requirements.txt
python verification/verify.py
python verification/check_learning.py
python -m unittest discover -s verification -p "test_*.py" -v
python verification/qutrit_certificate.py

The runner checks finite channel identities, complete finite POVMs, entropy inequalities, and the rational rate certificate. The learning checks validate the milestone structure, links, and worked examples. Written proofs establish the arbitrary-input and asymptotic statements. Verification scope distinguishes these roles. The historical learning-route review retains the 28 October checks and display-access limits; subsequent validation is recorded in the merged pull requests.

Scope

The base model assumes vacuum ancillary inputs, memoryless splitting, unlimited reliable helper-to-receiver classical communication, and unrestricted endpoint quantum coding. Its energy budget $E$ prices the signal. The resource extensions separately charge reference mean photons or the transmitted record under their stated contracts. General coherent control remains allowed; physical synthesis, preparation work, and reset costs are not optimized. No sender feedback or outcome postselection is supplied.

Purpose and contact

This repository serves as a record of the work and a guide for the author’s self-directed learning. For discussion or potential collaboration, please contact Ruge Lin at gogoko699@gmail.com.