The classical capacity of a relaxing qubit

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Family 276 · Mathematical physics
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

A qubit can carry a classical message, but relaxation makes different signals harder to distinguish. Ordinary amplitude damping describes decay toward a ground state. Generalized amplitude damping also permits upward transitions, with a nonzero stationary excited-state population.

The communication question is how many classical bits per channel use can be transmitted with error tending to zero. Optimizing a single qubit is only the beginning. A sender may encode a message in an entangled state across many channel uses, and a receiver may measure the entire output jointly. Such collective coding can outperform a one-use calculation for some quantum channels.

Family 276 reports that this additional input entanglement gives no capacity advantage for any qubit generalized amplitude-damping channel. The capacity equals a maximum over one real variable, attained at the level of Holevo information by a pair of pure signals. The manuscript also claims additivity when this channel is paired with any finite-dimensional quantum channel.

2. Background and essential definitions

Two parameters for relaxation

Write a qubit state as

$$\rho=\begin{pmatrix}1-q&z\\\overline z&q\end{pmatrix}, \qquad 0\leq q\leq1,\quad |z|^2\leq q(1-q).$$

The population $q$ is the probability of finding the excited state $|1\rangle$; $z$ describes coherence between the basis states. For damping strength $\gamma\in[0,1]$ and stationary excited-state population $\nu\in[0,1]$, the channel is

$$\mathcal A_{\gamma,\nu}(\rho)= \begin{pmatrix}1-t&\sqrt{1-\gamma}\,z\\ \sqrt{1-\gamma}\,\overline z&t\end{pmatrix}, \qquad t=(1-\gamma)q+\gamma\nu.$$

Population moves toward $\nu$, while coherence shrinks. This explicit convention avoids ambiguity about which thermal parameter denotes excited-state occupation. Ordinary downward damping is $\nu=0$. At $\gamma=0$ the channel is the identity; at $\gamma=1$ it replaces every input by the same diagonal state.

A quantum channel is a completely positive trace-preserving linear map. Complete positivity means positivity survives when the channel acts on part of a larger system. Independent, memoryless uses are represented by tensor powers $\mathcal A^{\otimes n}$, even when the input to those uses is entangled.

Holevo information and capacity

For a state $\sigma$, let $S(\sigma)=-\operatorname{Tr}\sigma\ln\sigma$. The entropy of an ensemble’s average output, minus its average individual output entropy, gives its Holevo information. Optimizing finite input ensembles yields

$$\chi(\mathcal N)=\frac1{\ln2}\sup_{\{p_a,\rho_a\}} \left[S\!\left(\sum_a p_a\mathcal N(\rho_a)\right) -\sum_a p_aS(\mathcal N(\rho_a))\right].$$

The factor converts natural-log entropy to bits. Operationally, the unassisted classical capacity allows arbitrary block inputs and collective output measurements, but no preshared entanglement or feedback. The coding theorem gives

$$C(\mathcal N)=\sup_{n\geq1}\frac1n\chi(\mathcal N^{\otimes n}).$$

This optimization over all block lengths is called regularization. Showing $C=\chi$ requires an upper bound on every block, not just a good one-use ensemble.

3. Development history and previous partial results

The coding theorems of Hausladen and collaborators, Holevo, and Schumacher–Westmoreland established the role of collective measurements and Holevo information. Hastings subsequently showed that general Holevo additivity fails. Consequently a simple-looking one-use optimizer cannot, by itself, settle unrestricted capacity.

For ordinary amplitude damping, Bennett, Shor, Smolin and Thapliyal described a two-signal scalar optimization, and Giovannetti and Fazio explicitly derived the one-use expression and attaining ensemble. Cortese and Berry studied the geometry of two-state qubit optimizers. Hou and Fang characterized the generalized channel’s one-use optimum, while Khatri, Sharma and Wilde placed it alongside bounds on unrestricted capacity.

The historical gap was regularization: could entangled blocks exceed the known one-use value? Approximate-additivity work of Leditzky, Kaur, Datta and Wilde addressed bounds rather than an exact general formula. Fang’s September 2026 preprint, in the version cited by the manuscript, supplied a close converse throughout the generalized family but generally left a gap.

An important direct predecessor is Tang, Zhu, Bai and Wang. They established capacity additivity with arbitrary finite-dimensional partners for qubit channels admitting a pure output, including ordinary amplitude damping. Their triangular block-matrix entropy inequality supplies the zero-block estimate used here. The present manuscript credits that result; ordinary damping is a recovered special case, not its new unresolved target.

At strictly interior thermal occupation and positive damping, generalized amplitude damping has no pure output, so that criterion does not apply. Other regions were already covered by King’s unital-qubit theorem and Shor’s entanglement-breaking theorem. The claimed advance extends the entropy argument across the entire parameter square, including the remaining thermal channels.

4. What the new manuscript claims

Define binary entropy in nats by $h(q)=-q\ln q-(1-q)\ln(1-q)$, with endpoint values understood continuously. A two-by-two state of determinant $u$ has entropy

$$g(u)=h\!\left(\frac{1+\sqrt{1-4u}}2\right), \qquad 0\leq u\leq\frac14.$$

For a pure input with excitation probability $p$, the output determinant is

$$v_{\gamma,\nu}(p)=\gamma\nu(1-\nu) +\gamma(1-\gamma)(p-\nu)^2.$$

The manuscript states, for every $\gamma,\nu\in[0,1]$,

$$C(\mathcal A_{\gamma,\nu})=\chi(\mathcal A_{\gamma,\nu}) =\frac1{\ln2}\max_{0\leq p\leq1} \left[h((1-\gamma)p+\gamma\nu)-g(v_{\gamma,\nu}(p))\right].$$

More strongly, every finite block obeys $\chi(\mathcal A_{\gamma,\nu}^{\otimes n})=n\chi(\mathcal A_{\gamma,\nu})$. If $p$ maximizes the formula, the equiprobable pure states

$$|\phi_\pm\rangle=\sqrt{1-p}\,|0\rangle\pm\sqrt p\,|1\rangle$$

attain the one-use optimum. Their independent product ensemble attains each finite-block Holevo optimum. Reliable communication at rates approaching capacity still uses the coding theorem and collective decoding; optimal Holevo information is not a claim of zero-error single-use distinguishability.

For every finite-dimensional partner channel $\mathcal N$, the manuscript additionally claims

$$\begin{aligned} \chi(\mathcal A\otimes\mathcal N)&=\chi(\mathcal A)+\chi(\mathcal N),\\ S_{\min}(\mathcal A\otimes\mathcal N)&=S_{\min}(\mathcal A)+S_{\min}(\mathcal N),\\ C(\mathcal A\otimes\mathcal N)&=C(\mathcal A)+C(\mathcal N). \end{aligned}$$

Here $S_{\min}(\mathcal N)=\min_\rho S(\mathcal N(\rho))$, and $\mathcal A=\mathcal A_{\gamma,\nu}$. The partner retains its own regularization: the assertion does not force $C(\mathcal N)=\chi(\mathcal N)$.

At the unital midpoint $\nu=1/2$, the formula simplifies to

$$C(\mathcal A_{\gamma,1/2})= 1-\frac1{\ln2}h\!\left(\frac{1+\sqrt{1-\gamma}}2\right).$$

The endpoints give one bit per use at zero damping and zero bits at complete replacement. Exchanging the basis states gives symmetry under $\nu\mapsto1-\nu$.

5. Why the advance matters

Importance: 4/5, provisional. The family determines a fundamental noise model’s classical capacity over all parameters and claims a strong arbitrary-partner additivity property. It is a major channel-specific advance rather than a general restoration of additivity, which is known to fail.

The scalar formula makes the distinction between thermal relaxation and input coding explicit. The first entropy term describes the average output; the second measures noise remaining in each pure signal. Their difference is the available information. Entangling signals across uses is permitted in the optimization but, according to the theorem, cannot improve that difference per use.

The partner result is broader than tensor-power additivity alone. It says this relaxation channel cannot produce an extra joint capacity benefit when paired with any finite-dimensional channel, even one whose own capacity requires regularization. The score reflects this significance if the claims hold, without assessing proof correctness.

6. The main idea

Keep track of populations

For a pure $n$-qubit state, let $p_j$ be the probability of an excited-state outcome at site $j$. The central entropy estimate is

$$S\!\left(\mathcal A^{\otimes n}(|\psi\rangle\langle\psi|)\right) \geq\sum_{j=1}^{n}e(p_j), \qquad e(p)=g(v_{\gamma,\nu}(p)).$$

Retaining each population is essential. A lower bound only by the unconstrained minimum output entropy would discard information needed to optimize the ensemble average.

Split the input according to its first qubit. The two normalized conditional vectors on the remaining qubits may be nonorthogonal and entangled. Applying a channel to those branches produces two matrices of Kraus columns; their Gram matrices describe branch outputs and cross terms. This converts the problem into an entropy inequality for block matrices.

Extend the triangular estimate

The predecessor inequality treats a factor with one zero block. Its entropy has a scalar contribution determined by the masses of the corner blocks, in addition to weighted entropies within them. The manuscript reproduces that estimate using an integral of log-determinant differences, averaging over coordinate signs and scalar concavity.

The thermal extension mixes two zero-block factorizations with the same scalar contribution. Entropy concavity preserves that common term. Allowing rectangular matrices is important because a channel’s output dimension and number of Kraus operators need not agree.

The resulting one-step bound separates the first-qubit entropy from conditional branch entropies. Convexity of $e$ combines conditional populations into the original populations. Iteration gives the displayed bound without assuming a product input.

Match the upper bound with two signals

For any input ensemble, the entropy of its average output is at most the sum of binary entropies of its average output populations. The population-sensitive lower bound controls the average entropy of individual outputs. Subtracting leaves one scalar objective per site, bounded by the claimed maximum.

The opposite-phase pair cancels coherence in its average while preserving the same excitation probability in each pure signal. It therefore attains both sides of this comparison. With a general partner channel, the same rectangular argument leaves the partner’s Holevo optimization intact; iterating before regularization yields the capacity identity.

7. Scope, limitations, and open questions

The capacity is unassisted and classical. It is not the quantum capacity, private capacity or entanglement-assisted capacity. Collective decoding remains allowed and may be needed; the conclusion concerns the lack of benefit from entangled input ensembles for the stated rate.

The channel is memoryless, and the arbitrary partner is finite dimensional. Correlated environmental noise or infinite-dimensional partners require separate hypotheses. The maximum is an exact one-variable characterization, not a universal elementary expression for its optimizing population or a claim of unique optimizers.

The extension from a pure-output criterion to thermal block structure suggests further entropy questions for other channel families. It does not establish additivity for general qubit channels beyond those covered by the stated arguments.

8. Sources and further reading

The Tang–Zhu–Bai–Wang predecessor establishes ordinary-damping capacity and the triangular inequality. Khatri, Sharma and Wilde discuss the generalized model and earlier bounds. The source below supplies the full-parameter theorem and arbitrary-partner conclusions, reported as stated.

Source manuscripts

• M1. OpenAI, Classical capacity and entropy inequalities for generalized amplitude damping (September 24, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.