How much quantum information can noise transmit?

The question in perspective

A noisy channel can preserve some states well and destroy others. Reliable quantum transmission asks for something stronger: an encoding and decoding that protect arbitrary quantum data, including its entanglement with a reference system. The rate must remain achievable as the number of channel uses grows and the error vanishes.

Define the task before the rate

Let \(\mathcal N\) be a completely positive, trace-preserving map. For \(n\) independent uses, an encoder \(\mathcal E_n\) and decoder \(\mathcal D_n\) aim to transmit a system of dimension \(K_n\). With \(\Phi_{K_n}\) a maximally entangled state of that system and an untouched reference, write the complete encoded channel as \(\Lambda_n=\mathcal D_n\circ\mathcal N^{\otimes n}\circ\mathcal E_n\). Require

\[F_e=\bra{\Phi_{K_n}}(\operatorname{id}\otimes\Lambda_n)(\Phi_{K_n})\ket{\Phi_{K_n}}\longrightarrow1.\]

The unassisted quantum capacity \(Q(\mathcal N)\) is the supremum of achievable asymptotic rates \(\log_2K_n/n\), measured in qubits per use. Shared entanglement and two-way classical assistance define different capacities.

To see why joint coding matters, purify an input \(\rho_A\) as \(\ket\psi_{RA}\), and set \(\omega_{RB}=(\operatorname{id}_R\otimes\mathcal N)(\ket\psi\bra\psi)\). The coherent information is \(I_c(\rho,\mathcal N)=S(B)_\omega-S(RB)_\omega\). For finite-dimensional memoryless channels, the coding theorem states

\[Q(\mathcal N)=\sup_{n\geq1}\frac1n\max_{\rho_{A^n}}I_c(\rho_{A^n},\mathcal N^{\otimes n}).\]

The optimization permits inputs entangled across channel uses. The formula is an operational characterization, not generally a tractable single-use answer.

A qubit erasure channel replaces the input by an orthogonal flag with probability \(p\): \(\mathcal E_p(\rho)=(1-p)\rho\oplus p\ket e\bra e\). For a maximally mixed input, the output entropy is \(h_2(p)+(1-p)\), and the reference–output entropy is \(h_2(p)+p\). Their difference is \(1-2p\). The exact unassisted capacity is

\[Q(\mathcal E_p)=\max\{0,1-2p\}.\]

Evaluating that input alone proves neither optimality nor the converse; those require the channel’s structural properties and a coding argument.

What is established

Erasure-channel capacities give an exact solvable benchmark. Quantum Shannon theory develops achievable rates and converse methods. Smith and Yard’s superactivation construction exhibits two zero-capacity channels whose joint channel has positive capacity. Capacity is therefore not an additive scalar summary of every possible joint use.

What remains unresolved

General channels lack a universal computable single-use capacity expression. For a specified noise family, useful progress may be a better code, a tighter upper bound, or a proof that a regularization is unnecessary in a restricted regime. Correlated noise and finite block lengths require separately stated models.

A concrete next investigation

Reproduce: calculate the erasure-channel entropies and plot the exact capacity. At \(p=0.25\), the unassisted rate is 0.5 qubits per use.

Check: explain why the flagged losses cannot simply be ignored: the environment acquires the erased inputs. Postselection changes the success probability and does not remove its resource cost.

Extend: perturb the erasure model with a stated additional noise process, construct an achievable rate, and compare it with a valid converse. Report the gap rather than presenting a numerical input optimization as an exact capacity.

Selected references

1. Capacities of Quantum Erasure Channels

   Charles H. Bennett, David P. DiVincenzo, John A. Smolin. 1997. Research paper

   Exact erasure-channel capacities, distinguishing assistance models.

2. From Classical to Quantum Shannon Theory

   Mark M. Wilde. 2011. Textbook manuscript

   Coding tasks, coherent information, regularization, and converse bounds.

3. Quantum Communication With Zero-Capacity Channels

   Graeme Smith, Jon Yard. 2008. Research paper

   Superactivation: joint use of two zero-capacity channels can transmit quantum information.

Years refer to first arXiv submission. Publication details are available in the linked records.