Partial Monitoring Capacity

Even the best predetermined individual measurements can leave quantum transmission capacity unused.

A lossy quantum signal reaches a receiver, an inaccessible environment and a collected field. A helper measures the collected field and sends only a classical record to the receiver. How much quantum information can that record recover, and when do joint measurements recover more than the best measurements made one output at a time?

Read next	Purpose
Physical picture · Preskill reading guide	Start from the physical question and one tutorial, then work through four local bridges
Capacity theorem · Exact product optimum	Follow both sides of the individual-versus-collective measurement comparison
Model and claims · Proof dependencies	Check the resources, supporting claims and source attribution
Prior-work comparison · Contribution assessment	Separate established ingredients from the measurement optimization
Verification · LLM guide	Inspect evidence and locate authoritative material by question

What the helper can access

A memoryless isometry $V:A\to B\otimes E\otimes D$ divides each input into three outputs.

Output	Who holds it	Available operation
$B$	Receiver	Arbitrary quantum decoding with the helper's record
$E$	Inaccessible environment	No access
$D$	Helper	Quantum processing and measurement; a classical message only to the receiver

The sender and receiver may use arbitrary block codes. The unrestricted helper may store and jointly measure many collected outputs. The product benchmark permits every single-use POVM, with choices fixed in advance and allowed to vary between uses. Outcome-adaptive local measurements and general separable block POVMs are outside that benchmark.

The rate is asymptotic, unconditional entanglement transmission per original channel use. No preshared entanglement, sender feedback, coherent helper link or postselection is supplied.

Counting is already the best individual measurement

For qubit splitting, let $a,b,c$ be the received, inaccessible and collected fractions, with $a+b+c=1$. Throughout $a>b>0,\ c>0$, photon counting attains the largest capacity over all predetermined product helper POVMs and all inputs, including coherent inputs. The strict-gap corollary establishes $0<Q_{\rm prod}<Q_{\rm meas}$ throughout this region, with a positive parameter-dependent lower bound on the difference.

A certified numerical illustration at $(a,b,c)=(0.2,0.08,0.72)$:

Helper measurement resource	Capacity, qubits/use
Best predetermined individual measurements, attained by counting	$0.18621044$
Unrestricted joint measurements	$0.30570954$
Certified difference	$0.11949910$

Both rows allow arbitrary sender/receiver block codes. A better predetermined single-use detector cannot close the gap anywhere in the stated interior region. Counting already reaches the positive-capacity boundary $a>b$; the collective benefit is a higher rate. The exact product proof supplies the optimization over all measurements. It establishes counting optimality after input optimization, rather than for every fixed coherent input.

Why the capacity is exact

The receiver must be able to simulate the inaccessible output while preserving its correlations with the helper. The required joint channel identity is

$$\mathcal N_{ED} =(\mathcal T_{B\to E}\otimes\mathrm{id}_D)\mathcal N_{BD}.$$

It holds for every input, including inputs entangled with a reference. Marginal degradability alone is insufficient. Under this condition,

$$\boxed{ Q_{\rm meas} =\max_{\rho_A}\left[\min\{S(B),S(BD)\}-S(E)\right]. }$$

Both finite-dimensional upper bounds follow from the established one-way distillation theorem for degradable states after regrouping registers. A known assistance ensemble, followed by ordinary channel coding, attains their minimum. The proof dependencies give the register reduction and source attribution. The separate product optimum still requires its determinant and convexity argument.

Inaccessible optical loss imposes an energy ceiling

For vacuum optical splitting, an average incident budget of $N$ photons per original mode gives

$$Q_{\rm optical}(N)= \begin{cases} g(aN)-g(bN),&a>b,\\\ 0,&a\le b, \end{cases} \qquad g(x)=(x+1)\log_2(x+1)-x\log_2x.$$

For $a>b>0$, increasing signal energy approaches $\log_2(a/b)$ qubits per mode. At positive energy the positive-rate boundary is $a>b$. The budget is an average signal-energy constraint, not a peak photon-number or apparatus-energy constraint; the environmental inputs remain vacuum. The optical proof includes arbitrary correlated, non-Gaussian inputs. The qubit counting-optimality result does not extend to optical measurements here.

One tutorial, then these results

The sole external teaching anchor is:

John Preskill, Quantum Shannon Theory, Chapter 10, 2025 revision.

Versioned text, arXiv:1604.07450v5 · Author chapter

The reading guide selects the entropy, classical-record, degradability and coding sections. Four local bridges then explain the extra steps:

Bridge	What the repository adds
Measurement record	A worked counting example, the copied classical flag and the actual complement
Exact unrestricted rate	The two cuts, their common input, established state-converse attribution and matching helper construction
Optimal individual measurement	Arbitrary input coherence, the convex chord bound and counting saturation
Optical energy limit	Thermal entropy, the all-input converse and the order of cutoff and coding limits

The chapter supplies the common language. Original research papers retain their attributions; the guide explains their role locally without adding another required tutorial. The chapter is linked, not redistributed.

Boundaries and prior work

The helper's quantum memory, circuit complexity, phase references and classical message length are unrestricted. An efficient collective helper, a finite-block implementation and a necessity theorem against all adaptive local strategies are not established. The model and claim map states these boundaries precisely.

Environmental assistance and collective-assistance gains have established predecessors. The capacity consequence and exact product benchmark are distinguished in the contribution assessment.

Evidence and reproduction

The supplied environment is Python 3.13.5. Install the pinned dependencies and use a fresh evidence directory:

python -m pip install -r requirements.txt
python verify.py --integrity-only
python -m unittest discover -s tests -v
python verify.py --output-dir local-evidence-001
python checks/certify_product_capacity.py --output local-evidence-001/product-capacity-certificate.json

Four preserved scientific suites contain 23 monitoring groups: 6 threshold/counting, 7 exact qubit rate, 6 optical audit and 4 consolidation checks. The eight infrastructure tests and three high-precision strict-gap regression tests are separate. The rational certificate encloses the example's scalar product rate and gap; it does not prove the analytical optimization over POVMs.

The runner retains logs, environments, source hashes and every comparison difference without refreshing reference reports. Verification policy distinguishes passing assertions, numerical agreement and exact-byte reproduction. Provenance identifies all 70 protected source files and excerpts.

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.

The LLM guide gives relevant questions, search phrases and authoritative reading links. This repository is available under the MIT license.