Classical capacity and entropy inequalities for generalized amplitude damping

1 Introduction

Amplitude damping models a qubit that loses an excitation to its surroundings. At nonzero stationary excited-state population, transitions in both directions are possible; generalized amplitude damping describes this relaxation together with the decay of coherence. We determine how much classical information can be transmitted through repeated, independent uses of this channel when a sender may encode each message into an arbitrary state of the entire input block. We also determine how its one-shot and regularized classical capacities behave when it is used in parallel with an arbitrary finite-dimensional partner channel.

This permission is essential. Although a channel acts independently at each site, an input signal can be entangled across sites, and the receiver can measure the outputs collectively. The pure-signal coding theorem of Hausladen, Jozsa, Schumacher, Westmoreland, and Wootters [8] established the role of collective decoding. The mixed-signal coding theorems of Holevo and Schumacher–Westmoreland [10, 16], applied to arbitrary input blocks, express the unassisted classical capacity as a regularization of the Holevo information over arbitrarily long blocks. Optimizing a single use gives an achievable rate, but does not by itself rule out an advantage from entangled signals. General Holevo additivity fails [7], so removing regularization requires an argument specific to the channel.

For ordinary amplitude damping, Bennett, Shor, Smolin, and Thapliyal [2, preprint, Section III B, Eq. (89)] described the one-parameter optimization over two equiprobable pure signals with opposite phases. Giovannetti and Fazio [6, Section III A, Eqs. (42)–(43)] explicitly derived the optimized one-use expression and its attaining ensemble. Leditzky, Kaur, Datta, and Wilde [15] identified its unrestricted classical capacity as an open problem in their 2018 study of approximate additivity. The geometry of two-signal qubit optimizers was developed by Cortese [4, Section 7.1] and Berry [3]. Hou and Fang [11] characterized the generalized channel’s one-use Holevo optimum and evaluated the resulting scalar optimization numerically. Khatri, Sharma, and Wilde [12, Section VI] placed that characterization alongside upper bounds on the unrestricted capacity. More recently, Fang [5, Theorem 1] derived a close converse bound throughout the generalized family; that bound leaves a gap from the one-use value in general.

Tang, Zhu, Bai, and Wang [18, Corollary 5.1, Lemma 5.2, and Theorem 5.5] established Holevo and classical-capacity additivity with arbitrary finite-dimensional partners for qubit channels admitting a pure output, including ordinary amplitude damping. Their triangular block entropy inequality [18, Theorem 4.2] is the zero-block estimate recalled below. At positive damping and strictly interior thermal occupation, the generalized channel has no pure output, so that channel criterion does not apply. We extend the triangular estimate to the thermal block structure by a convex mixture that preserves its scalar entropy term. Both zero-block factorizations in this mixture have the same scalar contribution; entropy concavity therefore preserves the lower bound needed for the thermal channel.

Some parameter regimes already have exact capacity formulas from broader additivity theorems. King’s theorem for unital qubit channels [13] applies at stationary occupation 1/2, and Shor’s entanglement-breaking theorem [17] applies in the corresponding region of the generalized family; see [12, Section VI] for that specialization. Those theorems permit an arbitrary partner channel. The entropy argument below covers every parameter pair. Its rectangular form also gives additivity of one-shot Holevo capacity and minimum output entropy with every finite-dimensional partner, and regularization gives additivity of unassisted classical capacity for the same pair.

1.1 Definitions and result

All vector spaces are finite dimensional, and a star denotes the adjoint. Write \mathcal B(\mathcal H) for the algebra of linear operators on a complex Hilbert space \mathcal H. For a positive semidefinite matrix, not necessarily normalized, write

Thus \ent is entropy in nats on states. Write

for the entropy of a probability vector and the binary entropy, respectively. The determinant entropy function is

A two-by-two state of determinant u has entropy g(u), by its characteristic polynomial.

For \gamma,\nu\in[0,1], define the generalized amplitude-damping channel by

The input is a state when 0\le q\le1 and |z|^2\le q(1-q). Here \nu is the stationary excited-state population; this explicit convention fixes any possible ambiguity in thermal parameters.

We measure Holevo information and capacity in bits:

The supremum runs over all finite ensembles of input states, with q_a\ge0 and \sum_aq_a=1. For a tensor-power channel, the states are unrestricted on the entire input space. An n-use classical code assigns a state on the full input space to each of its M_n equiprobable messages and uses one measurement on all n outputs. The unassisted classical capacity C(\mathcal N) is the supremum of rates R for which such codes have average decoding error tending to zero and \liminf_{n\to\infty}n^{-1}\log_2 M_n\ge R. There is no preshared entanglement or feedback. The channel is memoryless, so its n-use action is \mathcal N^{\otimes n}. The coding theorem gives

Indeed, the mixed-signal coding theorem [10, 16] applies to any fixed finite ensemble of block inputs, treating each block as one letter. It achieves rates below the ensemble’s Holevo information per physical channel use; see [10, preprint, Section 2, Eq. (5)] for the finite-alphabet statement. Conversely, the Holevo bound and Fano’s inequality applied to the uniform output ensemble of any n-use code with average error \epsilon_n give

The normalized asymptotic limit equals the supremum in (1.4): if a_n=\chi(\mathcal N^{\otimes n}), product ensembles give a_{m+n}\ge a_m+a_n, and 0\le a_n\le n\log_2 d_{\mathrm{out}}, where d_{\mathrm{out}} is the output dimension. For fixed k, writing n=qk+r gives a_n\ge q a_k; taking a lower limit and then the supremum over k proves the assertion. This coding result supplies the operational interpretation of the entropy optimization below.

For a finite-dimensional channel, define its minimum output entropy in nats by

where \rho ranges over all input density matrices. The minimum exists because the input state space is compact and entropy is continuous in finite dimensions.

The tensor-power upper bound includes all entangled block signals; a product ensemble attains it. Equation (1.4) turns this finite-block optimum into an asymptotic reliable rate with collective decoding.

The rectangular thermal inequality developed below gives the following stronger consequence.

The partner retains its full regularization: C(\mathcal N) can differ from \chi(\mathcal N).

1.2 Proof strategy

For a pure one-qubit input with excitation probability p, the output determinant is v_{\gamma,\nu}(p) and its entropy is e(p):=g(v_{\gamma,\nu}(p)). The central estimate extends this exact one-site value to every pure n-qubit input \psi:

where p_j is the probability of outcome 1 at site j in the computational basis. A bound only on the unconstrained minimum output entropy would lose these populations and would not suffice for the Holevo optimization.

To prove (1.12), split the input according to its first qubit: \psi=\sqrt{1-p_1}\ket0\otimes\psi_0+ \sqrt{p_1}\ket1\otimes\psi_1. The normalized conditional vectors \psi_0,\psi_1 can be nonorthogonal and can remain entangled on the other sites. For the channel on those sites choose a Kraus representation \mathcal N(P)=\sum_J L_JPL_J^*. Define column J of X as L_J\psi_0 and column J of Y as L_J\psi_1. Their Gram products are the two branch outputs, and XY^* is the image of the cross operator between the branches. The thermal block inequality gives

This one-step application does not require \mathcal N to be a damping tensor power: its Kraus columns may have any finite rectangular shape. For a damping tensor power, convexity of e combines the conditional populations into the original site populations, and iteration gives (1.12) without restricting the input vector.

The matrix proof leading to this recursion is developed in Sections 2 and 3. For a zero-block factor M=\bigl(\begin{smallmatrix}A&B\\ C&0\end{smallmatrix}\bigr), with squared Hilbert–Schmidt masses x=\Tr(AA^*), y=\Tr(BB^*), z=\Tr(CC^*) summing to one, the entropy contribution beyond the weighted block entropies is at least g(yz). We express the entropy deficit as an integral of log-determinant differences. Joint convexity permits simultaneous averaging over coordinate signs; this reduces the estimate to a concave scalar deficit. Zero padding preserves the result for rectangular blocks, which are required by the Kraus construction. The thermal extension uses two such factorizations whose products of corner masses agree, so the same g term survives their convex mixture.

Finally, for an ensemble with mean populations \overline p_j, the entropy of the ensemble-average output is at most \sum_j h((1-\gamma)\overline p_j+\gamma\nu), by the diagonal-entropy bound in the computational basis and classical subadditivity. Entropy concavity and convexity of e first extend (1.12) to mixed inputs. Averaging this bound and using convexity of e bounds the average of the individual output entropies from below by \sum_j e(\overline p_j). The difference gives the upper bound in Theorem 1.1; products of the phase pair (1.8) attain it. For an arbitrary partner, the branch outputs form an ensemble whose mean is the second output marginal. The entropy of the mean full output is at most the binary entropy of its first output population plus the entropy of that same marginal. Subtraction leaves one scalar damping objective plus at most (\ln2)\chi(\mathcal N). The minimum-output equality follows from the same one-step estimate, and the capacity equality follows by iterating against \mathcal N^{\otimes n} before regularizing. Section 4 proves the recursion and population convexity, Section 5 carries out the ensemble optimization, and Section 6 derives the arbitrary-partner consequences. Section 7 records the ordinary, unital, and endpoint formulas. Appendix A gives direct Hessian proofs of the scalar concavity and log-determinant convexity, providing an independent analytic route to the same zero-block estimate.

2 An entropy inequality with one zero block

Our first objective is to separate the entropy produced by the block pattern from the entropies within its three nonzero entries. The rectangular formulation is needed because a channel output and its Kraus index space can have different dimensions. The following is the case of the triangular block entropy inequality of Tang, Zhu, Bai, and Wang [18, Theorem 4.2] in which all blocks have the same rectangular shape, after permuting block rows. We give a direct proof in the form needed for the thermal extension.

The scalar matrix \bigl(\begin{smallmatrix}\sqrt x&\sqrt y\\\sqrt z&0\end{smallmatrix}\bigr) has squared singular values with sum 1 and product yz. The term g(yz) is their entropy. To separate this contribution from the entropies inside the three blocks, write the entropy deficit as

Since \ent(AA^*)=x\ent(AA^*/x)-x\ln x for x>0, and similarly for the other blocks, the theorem is equivalent to

We will express \Delta as an integral of log-determinant differences. Joint convexity then permits averaging over coordinate signs, and homogeneous scalar concavity combines the resulting coordinates. This integral and averaging route is also used in [18, Appendix A.2–A.4].

Block norm compression provides a related perspective. King [14, Theorem 1] compared the Schatten norm of a positive two-by-two block matrix with the norm of the scalar matrix of block norms. Audenaert [1, Conjecture 1] studied the corresponding comparison for arbitrary two-row block matrices and proved several special cases. The entropy argument below allows noncommuting blocks and does not require the general norm-compression conjecture.

2.1 From entropy to log determinants

The direct sum of the three individual Gram matrices and the full Gram matrix have equal trace. The following identity converts their entropy difference into an integral, including when their dimensions or ranks differ.

2.2 Joint convexity by a contraction

The log-determinant function used in the averaging step is the following. Its joint convexity follows by specialization and continuity from Hiai’s trace-function theorem [9, preprint, Theorem 5.2]. We give a direct proof: the contraction argument is the specialization of [18, Lemma A.1] to a two-by-two block matrix.

2.3 The scalar deficit and its homogeneity

We next use the same convexity to control the scalar entropy deficits that sign averaging will produce. For x,y,z\ge0, define

Here m^2\ge4yz, and \lambda_\pm are nonnegative with sum m and product yz. Thus F is continuous on the nonnegative octant, with the usual convention at zero. On the probability simplex it is exactly the deficit in our target: F(x,y,z)=H(x,y,z)-g(yz).

2.4 Sign averaging and the block estimate

The two convexity consequences are now in place. The Schur complement will put the matrix deficit into the form involving G_Z; averaging over signs will replace its matrix arguments by diagonal entries; and Lemma 2.4 will combine the resulting scalar deficits.

2.5 A scalar convexity fact for the channel application

The scalar deficit estimate completes the zero-block theorem. The later channel argument also needs a different property of the same entropy function: its convexity as a function of the square root of the determinant. We record it separately.

The function \kappa\mapsto g(\kappa^2/4)/\ln2 is the entropy–concurrence function of Wootters [19, Eq. (8)]. Its monotonicity and convexity are the scalar properties just proved. We use these properties, rather than any additivity statement about entanglement of formation.

3 A thermal block entropy inequality

The zero-block theorem does not apply directly when thermal excitation and decay both occur. We will express the resulting state as a convex mixture of two states to which it does apply, preserving the same scalar entropy term in each. We first recall two elementary entropy facts, including their proofs to fix the directions of the inequalities used below. For a state P,

Indeed, its diagonal is obtained from its eigenvalue vector by the doubly stochastic matrix of squared absolute unitary entries. Concavity of -s\ln s, applied row by row and then summed, gives (3.1). Entropy is also concave on states:

To see this, choose a basis diagonalizing the mixture, apply classical entropy concavity in that basis, and then use (3.1) on each P_i.

The next result is the extension needed for thermal channels.

4 From the thermal inequality to channel outputs

We first turn the thermal matrix inequality into an entropy bound for \A_{\gamma,\nu}\otimes\mathcal N, with any finite-dimensional partner channel \mathcal N. Its two matrices will encode the outputs of conditional input branches; a common Kraus index will preserve their cross operator. We then iterate this bound for repeated damping uses, retaining the input population at every site.

4.1 The one-qubit entropy and its convexity

Fix \gamma,\nu\in[0,1] and abbreviate \A=\A_{\gamma,\nu}. Put

Then

The population transition matrix is \bigl(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\bigr) and the coherence multiplier is k. A Kraus representation is

Direct multiplication gives \A(P)=\sum_{i=0}^3L_iPL_i^* and \sum_{i=0}^3L_i^*L_i=I, including all parameter endpoints.

For p\in[0,1], define

Any pure qubit state of excitation probability p has coherence modulus \sqrt{p(1-p)}. Hence v(p) is its output determinant and e(p) its output entropy. In particular, v(p)\in[0,1/4]. Expanding (4.1) in (4.4) gives (1.7). If \gamma>0 and 0<\nu<1, then v(p)\ge\gamma\nu(1-\nu)>0 for every p. Every pure input therefore has a positive definite output. A mixed input is a convex combination of pure inputs, so its output is positive definite as well. Thus these thermal parameters lie outside the pure-output class treated by Tang, Zhu, Bai, and Wang [18, Corollary 5.1 and Lemma 5.2].

4.2 A single damping use with an arbitrary partner

The following estimate is the channel form of Theorem 3.1. Its scalar term is the pure one-qubit entropy e(p), and the remaining terms are the output entropies of the two conditional branches.

4.3 Iteration over a block

We now take the partner to be the channel on the remaining qubits. Convexity of e will combine the conditional populations at each later site into that site’s original population.

The same population bound holds for mixed inputs. This form will let us bound every signal in a Holevo ensemble directly.

5 The ensemble bound and its attainment

We now turn the population-dependent entropy bound into the full Holevo assertion. This step is essential: an unconstrained minimum-output-entropy bound alone would not give the result.

6 Additivity with a finite-dimensional partner

The branch estimate in Lemma 4.2 applies to every finite-dimensional partner. We now use it to prove the three assertions of Corollary 1.2. For the Holevo bound, the key point is that the conditional partner outputs form an ensemble whose mean is exactly the partner marginal of the mean full output.

7 Ordinary damping, the unital case, and endpoints

The population convention makes several special cases transparent. Throughout this section, capacities and Holevo information are in bits, while h and g remain in nats.

Tang, Zhu, Bai, and Wang [18, Corollary 5.1 and Theorem 5.5] established the ordinary-damping capacity and its additivity with arbitrary finite-dimensional partners. The following specialization recovers their capacity formula in the present parameters.

The scalar maximum and the opposite-phase attaining ensemble agree with Giovannetti–Fazio [6, Section III A, Eqs. (42)–(43)], whose transmissivity is 1-\gamma in our convention. That one-use optimization becomes the unrestricted classical capacity through the finite-power identity.

Let \mathsf X=\bigl(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\bigr). Direct substitution in (1.2) gives

Unitary conjugations preserve the Holevo optimization, so the capacity is invariant under \nu\mapsto1-\nu. In particular, \nu=1 has the ordinary-channel capacity as well, with the signal population replaced by 1-p.

The next formula also follows from King’s general unital-qubit theorem [13]. We recover it directly from the present scalar optimization.

At \gamma=0, the channel is the identity, v_{\gamma,\nu}=0, and the capacity is one bit per use. At \gamma=1, it replaces every input by \diag(1-\nu,\nu), so its capacity is zero: the two terms in the objective both equal h(\nu). These conclusions include all four corners of the parameter square.

A Alternative Hessian proofs

The main proof obtains scalar concavity and matrix log-determinant convexity from the same contraction argument. Here we give direct Hessian proofs of the two analytic statements. They provide an alternative route to Lemmas 2.4 and 2.3; the entropy integral, Schur complement, sign averaging, and rectangular embedding in Section 2 are unchanged.

A.1 The Hessian of the scalar deficit

Recall the function F from (2.6). Its homogeneity follows by the scaling cancellation already proved in Lemma 2.4. We verify concavity directly on the probability simplex, where F(x,y,z)=H(x,y,z)-g(yz), using the derivatives (2.13). For 0<u<1/4, write t=\sqrt{1-4u} and L=\atanh(t)/t.

A.2 The Hessian of the log-determinant difference

The contraction proof avoids choosing singular-vector coordinates. The following calculation instead diagonalizes the fixed matrix and expresses the second derivative as a sum of nonnegative squares. It proves Lemma 2.3 directly for the same domain, including singular Z.