Hadamard Channels: Quantum Communication

• Hadamard channels are quantum channels whose complementary channels are entanglement-breaking, ensuring additivity and tractable single-letter formulas for various capacities.
• Their degradability enables precise characterization of trade-off regions for simultaneous classical and quantum information transmission, outperforming naive time-sharing strategies.
• Physically significant examples, including generalized dephasing, universal cloning, and Unruh channels, illustrate their impact on optimizing quantum communication protocols.

A Hadamard channel is a class of quantum channels whose complementary channel is entanglement-breaking. This structural property has profound implications for their information-theoretic capacities: all relevant capacity formulas—classical, quantum, and entanglement-assisted—reduce to single-letter optimization, making them tractable. Their explicit structure also enables precise analysis of trade-off regions for the simultaneous transmission of classical and quantum information, with or without entanglement assistance. Hadamard channels encompass physically significant examples such as generalized dephasing, universal cloning, and Unruh channels. They also admit a strong converse property for classical communication, meaning transmission at rates above capacity results in exponentially vanishing decoding success probability.

1. Mathematical Definition and Structure

A quantum channel $\mathcal{N}:A' \to B$ admits a Kraus decomposition: $$\mathcal{N}(\rho) = \sum_j N_j\,\rho\,N_j^\dagger,\quad \sum_j N_j^\dagger N_j = I,$$ and can always be written via an isometric extension $U_\mathcal{N}:A' \to B\otimes E$ as

$$\mathcal{N}(\rho) = \mathrm{Tr}_E \left[ U_{\mathcal{N}}\,\rho\,U_{\mathcal{N}}^\dagger\right].$$

The complementary channel is defined as

$$\mathcal{N}^c(\rho) = \mathrm{Tr}_B[U_\mathcal{N}\,\rho\,U_\mathcal{N}^\dagger].$$

$\mathcal{N}$ is called a Hadamard channel if its complementary channel $\mathcal{N}^c$ is entanglement-breaking (EB).

An EB channel $\mathcal{M}$ has the property that for all input–reference states $\rho_{AR}$, $(\mathcal{M}\otimes\mathrm{id}_R)(\rho_{AR})$ is separable. Equivalently, $$\mathcal{N}(\rho) = \sum_j N_j\,\rho\,N_j^\dagger,\quad \sum_j N_j^\dagger N_j = I,$$0 admits a measure-and-prepare structure or a Kraus decomposition with rank-one operators. In a suitable orthonormal basis $$\mathcal{N}(\rho) = \sum_j N_j\,\rho\,N_j^\dagger,\quad \sum_j N_j^\dagger N_j = I,$$1 on $$\mathcal{N}(\rho) = \sum_j N_j\,\rho\,N_j^\dagger,\quad \sum_j N_j^\dagger N_j = I,$$2, the Hadamard channel acts as a Schur (entrywise) multiplier: $$\mathcal{N}(\rho) = \sum_j N_j\,\rho\,N_j^\dagger,\quad \sum_j N_j^\dagger N_j = I,$$3 where $$\mathcal{N}(\rho) = \sum_j N_j\,\rho\,N_j^\dagger,\quad \sum_j N_j^\dagger N_j = I,$$4 is a positive semidefinite matrix and $$\mathcal{N}(\rho) = \sum_j N_j\,\rho\,N_j^\dagger,\quad \sum_j N_j^\dagger N_j = I,$$5 denotes Hadamard (Schur) product (Wilde et al., 2013).

Since measure-and-prepare channels are degradable, any Hadamard channel is degradable: the receiver $$\mathcal{N}(\rho) = \sum_j N_j\,\rho\,N_j^\dagger,\quad \sum_j N_j^\dagger N_j = I,$$6 can simulate the complementary channel by measurements in the defining basis and a preparation step (Bradler et al., 2010).

2. Capacity Formulas and Single-Letterization

For general channels, classical capacity $$\mathcal{N}(\rho) = \sum_j N_j\,\rho\,N_j^\dagger,\quad \sum_j N_j^\dagger N_j = I,$$7, quantum capacity $$\mathcal{N}(\rho) = \sum_j N_j\,\rho\,N_j^\dagger,\quad \sum_j N_j^\dagger N_j = I,$$8, and capacity regions are given by regularized (multi-letter) optimizations due to possible nonadditivity. However, the Hadamard property ensures additivity for key information quantities.

The coherent information is additive on degradable channels, giving a single-letter formula for $$\mathcal{N}(\rho) = \sum_j N_j\,\rho\,N_j^\dagger,\quad \sum_j N_j^\dagger N_j = I,$$9. For trade-off capacities between classical, quantum, and entanglement-assisted communications (the $U_\mathcal{N}:A' \to B\otimes E$0 region), mutual information rates are also additive as the complementary is EB. Explicitly, for any Hadamard $U_\mathcal{N}:A' \to B\otimes E$1, arbitrary $U_\mathcal{N}:A' \to B\otimes E$2, and $U_\mathcal{N}:A' \to B\otimes E$3: $U_\mathcal{N}:A' \to B\otimes E$4 and similarly for the entanglement-assisted (CE) objective (Bradler et al., 2010).

Thus, for all these channels, the one-shot (single-use) regions coincide with their regularizations, and all capacities of interest single-letterize. This is a key distinguishing feature for Hadamard channels, not generic among quantum channels.

3. Trade-off Capacity Regions: The Main Theorem

For a Hadamard channel $U_\mathcal{N}:A' \to B\otimes E$5, define the joint state

$U_\mathcal{N}:A' \to B\otimes E$6

where $U_\mathcal{N}:A' \to B\otimes E$7 is an ensemble of purifications and $U_\mathcal{N}:A' \to B\otimes E$8 is a classical label.

The achievable region of triplets $U_\mathcal{N}:A' \to B\otimes E$9 (classical, quantum, and consumed entanglement rates, respectively) is given by the constraints: $$\mathcal{N}(\rho) = \mathrm{Tr}_E \left[ U_{\mathcal{N}}\,\rho\,U_{\mathcal{N}}^\dagger\right].$$0 where the union over all one-shot states $$\mathcal{N}(\rho) = \mathrm{Tr}_E \left[ U_{\mathcal{N}}\,\rho\,U_{\mathcal{N}}^\dagger\right].$$1 gives the complete region; no regularization (multi-use) is required (Bradler et al., 2010).

4. Explicit Examples: Dephasing, Cloning, and Unruh Channels

Three physically relevant Hadamard channel families admit closed-form boundary formulas for the $$\mathcal{N}(\rho) = \mathrm{Tr}_E \left[ U_{\mathcal{N}}\,\rho\,U_{\mathcal{N}}^\dagger\right].$$2, $$\mathcal{N}(\rho) = \mathrm{Tr}_E \left[ U_{\mathcal{N}}\,\rho\,U_{\mathcal{N}}^\dagger\right].$$3, and $$\mathcal{N}(\rho) = \mathrm{Tr}_E \left[ U_{\mathcal{N}}\,\rho\,U_{\mathcal{N}}^\dagger\right].$$4 regions.

a) Generalized dephasing channel $$\mathcal{N}(\rho) = \mathrm{Tr}_E \left[ U_{\mathcal{N}}\,\rho\,U_{\mathcal{N}}^\dagger\right].$$5: $$\mathcal{N}(\rho) = \mathrm{Tr}_E \left[ U_{\mathcal{N}}\,\rho\,U_{\mathcal{N}}^\dagger\right].$$6 with parameter $$\mathcal{N}(\rho) = \mathrm{Tr}_E \left[ U_{\mathcal{N}}\,\rho\,U_{\mathcal{N}}^\dagger\right].$$7: $$\mathcal{N}(\rho) = \mathrm{Tr}_E \left[ U_{\mathcal{N}}\,\rho\,U_{\mathcal{N}}^\dagger\right].$$8 For the CE curve: $$\mathcal{N}(\rho) = \mathrm{Tr}_E \left[ U_{\mathcal{N}}\,\rho\,U_{\mathcal{N}}^\dagger\right].$$9

b) Universal $$\mathcal{N}^c(\rho) = \mathrm{Tr}_B[U_\mathcal{N}\,\rho\,U_\mathcal{N}^\dagger].$$0 cloning channel $$\mathcal{N}^c(\rho) = \mathrm{Tr}_B[U_\mathcal{N}\,\rho\,U_\mathcal{N}^\dagger].$$1:

Define $$\mathcal{N}^c(\rho) = \mathrm{Tr}_B[U_\mathcal{N}\,\rho\,U_\mathcal{N}^\dagger].$$2, and

$$\mathcal{N}^c(\rho) = \mathrm{Tr}_B[U_\mathcal{N}\,\rho\,U_\mathcal{N}^\dagger].$$3

Then

$$\mathcal{N}^c(\rho) = \mathrm{Tr}_B[U_\mathcal{N}\,\rho\,U_\mathcal{N}^\dagger].$$4

where $$\mathcal{N}^c(\rho) = \mathrm{Tr}_B[U_\mathcal{N}\,\rho\,U_\mathcal{N}^\dagger].$$5.

c) Unruh channel $$\mathcal{N}^c(\rho) = \mathrm{Tr}_B[U_\mathcal{N}\,\rho\,U_\mathcal{N}^\dagger].$$6:

With acceleration parameter $$\mathcal{N}^c(\rho) = \mathrm{Tr}_B[U_\mathcal{N}\,\rho\,U_\mathcal{N}^\dagger].$$7, blocks decompose as $$\mathcal{N}^c(\rho) = \mathrm{Tr}_B[U_\mathcal{N}\,\rho\,U_\mathcal{N}^\dagger].$$8 cloners with weight $$\mathcal{N}^c(\rho) = \mathrm{Tr}_B[U_\mathcal{N}\,\rho\,U_\mathcal{N}^\dagger].$$9, $\mathcal{N}$0, and analogous $\mathcal{N}$1, $\mathcal{N}$2. Explicit trade-off curves are sums over $\mathcal{N}$3 weighted by $\mathcal{N}$4, involving entropy functions of the normalized $\mathcal{N}$5 and $\mathcal{N}$6 (Bradler et al., 2010).

5. Optimal Coding vs. Time-Sharing and Figures of Merit

A naive time-sharing protocol divides $\mathcal{N}$7 channel uses into blocks dedicated solely to classical or quantum communication, yielding only the convex hull of $\mathcal{N}$8 and $\mathcal{N}$9. In contrast, the classically-enhanced father protocol allows simultaneous transmission of both types, encoding into entangled codewords and fully exploiting the channel's capabilities.

For Hadamard channels, the optimal protocol combines the father code, super-dense coding (exchanging $\mathcal{N}^c$0 for additional $\mathcal{N}^c$1), and entanglement distribution (trading $\mathcal{N}^c$2 for $\mathcal{N}^c$3). This optimal protocol attains the boundary of all capacity regions, outperforming time-sharing.

The improvement is quantified using the relative gain: $\mathcal{N}^c$4 and analogously for $\mathcal{N}^c$5. For generalized dephasing, cloning, and Unruh channels, $\mathcal{N}^c$6 and $\mathcal{N}^c$7 except in the classical or noiseless limit where time-sharing becomes optimal (Bradler et al., 2010).

6. Classical Capacity and Strong Converse

The Holevo (classical) capacity for a quantum channel $\mathcal{N}^c$8 is

$\mathcal{N}^c$9

with $\mathcal{M}$0. For Hadamard (as well as entanglement-breaking) channels, $\mathcal{M}$1 is additive: $\mathcal{M}$2 Thus, the one-shot expression gives the actual classical capacity (Wilde et al., 2013).

A strong converse holds: when transmitting at rates $\mathcal{M}$3, the average success probability for decoding $\mathcal{M}$4-block codes decays exponentially, i.e.,

$\mathcal{M}$5

This follows from subadditivity of sandwiched Rényi divergences for Hadamard channels and provides a sharp operational threshold for classical information transmission (Wilde et al., 2013).

7. Connections, Interpretation, and Significance

Hadamard channels, as a natural complement to entanglement-breaking channels, are a distinguished degradable family with explicitly tractable capacity regions for hybrid classical-quantum-communication scenarios. Physically, they model prominent noise processes including dephasing (coherence loss), universal cloning (stimulated emission), and the Unruh effect in relativistic quantum information.

Their structural features—degradability, entanglement-breaking complementarity, and Schur multiplier form—directly yield single-letterized capacity regions, bypassing the generic intractabilities of regularized formulas. For all canonical communication tasks (classical, quantum, entanglement-assisted), optimal codes for Hadamard channels achieve provable gains over time-sharing, quantifiable by the relative-area metric.

The strong converse for classical capacity further distinguishes this class, providing a rigorous threshold in the sense of Shannon theory—the ultimate relevance of Hadamard channels is thus both physical (as concrete models) and foundational (as a paradigm for tractable, optimally codable quantum communication) (Bradler et al., 2010, Wilde et al., 2013).