Dephrasure Channel: Hybrid Quantum Noise Model

• Dephrasure channel is a qubit quantum channel combining dephasing and erasure noise, characterized by two probabilistic parameters and a flagged convex structure.
• It exhibits superadditivity of coherent information and clear gaps between quantum and private capacities, influencing both theoretical and operational studies.
• Advancements in neural network and permutation-invariant quantum codes, along with experimental implementations, have validated its role in exploring quantum communication limits.

Searching arXiv for recent and foundational papers on the dephrasure channel. The dephrasure channel is a qubit quantum channel that combines dephasing and erasure noise. In the formulation studied by Leditzky, Leung, and Smith, it acts as

$$\mathcal{N}_{p,q}(\rho) = (1-q)\left[(1-p)\rho + p Z\rho Z\right] + q\,\mathrm{Tr}(\rho) |e\rangle\langle e|$$

with $0 \leq p,q \leq 1$, where $Z$ is the Pauli-$Z$ operator and $|e\rangle$ is an orthogonal erasure flag state (Leditzky et al., 2018). The channel is the concatenation of a dephasing channel and an erasure channel, and its simplicity has made it a prominent model for studying coherent-information superadditivity, gaps between quantum and private information measures, exact assisted capacities, and alternative operational quantities such as the communication value (Leditzky et al., 2018, Pirandola et al., 2018, Chitambar et al., 2021).

1. Definition and structural representations

The dephrasure channel is parameterized by a dephasing probability and an erasure probability. In one common convention,

$$\mathcal{N}_{p,q}(\rho) = (1-q)\left[(1-p)\rho + p Z\rho Z\right] + q\,\mathrm{Tr}(\rho) |e\rangle\langle e| ,$$

so that erasure occurs with probability $q$ and dephasing acts on the non-erased branch (Leditzky et al., 2018, Chitambar et al., 2021). A notationally equivalent convention also appears in the literature, with the parameters ordered as an erasure probability $p$ and a dephasing probability $q$: $$\mathcal{E}_{p,q}^{\mathrm{dr}}(\rho) = (1-p)\left[(1-q)\rho + q Z\rho Z\right] + p \mathcal{E}_e(\rho),$$ where $0 \leq p,q \leq 1$0 (Pirandola et al., 2018). These formulas describe the same hybrid noise model up to parameter relabeling.

The channel can be written as a flagged convex mixture

$0 \leq p,q \leq 1$1

where $0 \leq p,q \leq 1$2 is the dephasing channel and $0 \leq p,q \leq 1$3 is the erasure map (Leditzky et al., 2018). This flagged structure is central to several analytical calculations because the erasure outcome is perfectly distinguishable from the non-erasure output.

A Kraus representation given for the same channel family is

$0 \leq p,q \leq 1$4

with

$0 \leq p,q \leq 1$5

in the parameter convention of (Pirandola et al., 2018). The same work gives a Choi-matrix expression,

$0 \leq p,q \leq 1$6

where $0 \leq p,q \leq 1$7 (Pirandola et al., 2018).

The complementary channel is also unusually tractable. In the notation of (Leditzky et al., 2018),

$0 \leq p,q \leq 1$8

with

$0 \leq p,q \leq 1$9

This explicit form is one reason the dephrasure channel has become a useful laboratory for capacity questions (Leditzky et al., 2018).

2. Coherent information, positivity thresholds, and superadditivity

The principal reason for the channel’s prominence is that it exhibits superadditivity of coherent information despite its elementary definition. The one-shot quantum capacity is characterized by coherent information,

$Z$0

and the full quantum capacity is given by the regularized limit

$Z$1

These formulas are used throughout the dephrasure literature (Leditzky et al., 2018, Siddhu et al., 2020, Bhalerao et al., 13 Aug 2025).

Leditzky, Leung, and Smith showed that the dephrasure channel displays a “pronounced superadditivity of coherent information” and “nonadditivity of coherent information at the two-letter level” (Leditzky et al., 2018). For $Z$2, they consider repetition-type inputs such as

$Z$3

and identify parameters $Z$4 for which

$Z$5

This provides a direct violation of additivity at low blocklength (Leditzky et al., 2018).

A later analysis using a gluing framework gives an explicit positivity threshold for the single-letter coherent information. In the notation of generalized erasure channels, the positivity boundary is

$Z$6

and for fixed $Z$7, $Z$8 is zero for $Z$9 and positive for $Z$0 (Siddhu et al., 2020). The same work gives a second threshold

$Z$1

which marks a transition in the location of the optimal input state (Siddhu et al., 2020).

The gluing analysis also reports a striking perturbative phenomenon: when $Z$2, $Z$3 is zero for $Z$4, but for any positive $Z$5, however small, it becomes strictly positive for all $Z$6 (Siddhu et al., 2020). The source explicitly notes that this effect lacks an intuitive explanation. This suggests that the interplay between dephasing and erasure is not captured by simple interpolation from the pure-erasure endpoint.

A recent perturbative study further enlarges the region in which capacity separations are known. It defines

$Z$7

and states that $Z$8 throughout this region (Wu et al., 22 Jul 2025). The same work proves that the complementary channel also has positive one-shot quantum capacity throughout $Z$9, which yields a strict one-shot private-capacity gap, discussed below (Wu et al., 22 Jul 2025).

3. Private information, complementary channels, and assisted capacities

Beyond coherent information, the dephrasure channel exhibits substantial separation between quantum and private communication quantities. The original analysis reports a “big gap between single-letter coherent and private informations” (Leditzky et al., 2018). In particular, the single-letter private information can “greatly exceed” the single-letter coherent information in regions where the latter is small or vanishes (Leditzky et al., 2018).

The later perturbative work sharpens this observation. For all $|e\rangle$0, it states

$|e\rangle$1

and relates the gap to positivity of the complementary channel’s one-shot quantum capacity (Wu et al., 22 Jul 2025). In the same parameter region, the complementary channel satisfies

$|e\rangle$2

which broadens earlier evidence that the complement of the dephrasure channel can remain quantum-capacity-positive across the regime of interest (Wu et al., 22 Jul 2025). The 2018 work had already emphasized that the complementary channel “always has positive quantum capacity for $|e\rangle$3” (Leditzky et al., 2018).

The channel is also notable because certain two-way assisted capacities are exactly known. Using conditional channel simulation, Pirandola, Laurenza, Ottaviani, and Banchi established that

$|e\rangle$4

where $|e\rangle$5 is the binary Shannon entropy (Pirandola et al., 2018). Here $|e\rangle$6 denotes the two-way quantum capacity, $|e\rangle$7 the two-way assisted entanglement distribution capacity, $|e\rangle$8 the two-way private capacity, and $|e\rangle$9 the secret-key capacity.

The proof depends on the decomposition

$$\mathcal{N}_{p,q}(\rho) = (1-q)\left[(1-p)\rho + p Z\rho Z\right] + q\,\mathrm{Tr}(\rho) |e\rangle\langle e| ,$$0

together with the fact that the dephasing and erasure components are teleportation covariant but not jointly so (Pirandola et al., 2018). Standard Choi-state simulation therefore fails for the average channel, and the conditional channel simulation framework is used instead. An upper bound is obtained from the relative entropy of entanglement of a control-program state, while achievability follows from measuring the erasure flag and postselecting onto the non-erasure branch (Pirandola et al., 2018).

The same work also reports explicit formulas for reverse coherent information and coherent information in its parameter convention: $$\mathcal{N}_{p,q}(\rho) = (1-q)\left[(1-p)\rho + p Z\rho Z\right] + q\,\mathrm{Tr}(\rho) |e\rangle\langle e| ,$$1 and

$$\mathcal{N}_{p,q}(\rho) = (1-q)\left[(1-p)\rho + p Z\rho Z\right] + q\,\mathrm{Tr}(\rho) |e\rangle\langle e| ,$$2

These expressions underscore that the exact two-way capacities exceed the reverse coherent information unless $$\mathcal{N}_{p,q}(\rho) = (1-q)\left[(1-p)\rho + p Z\rho Z\right] + q\,\mathrm{Tr}(\rho) |e\rangle\langle e| ,$$3 (Pirandola et al., 2018).

4. Communication value and classical behavior

A distinct operational quantity attached to the dephrasure channel is the communication value $$\mathcal{N}_{p,q}(\rho) = (1-q)\left[(1-p)\rho + p Z\rho Z\right] + q\,\mathrm{Tr}(\rho) |e\rangle\langle e| ,$$4, defined as the optimal success probability of transmitting a uniformly random classical message over a channel (Chitambar et al., 2021). For the dephrasure channel, the paper “The Communication Value of a Quantum Channel” gives the exact formula

$$\mathcal{N}_{p,q}(\rho) = (1-q)\left[(1-p)\rho + p Z\rho Z\right] + q\,\mathrm{Tr}(\rho) |e\rangle\langle e| ,$$5

independent of the dephasing parameter $$\mathcal{N}_{p,q}(\rho) = (1-q)\left[(1-p)\rho + p Z\rho Z\right] + q\,\mathrm{Tr}(\rho) |e\rangle\langle e| ,$$6 (Chitambar et al., 2021).

The optimal protocol is explicitly classical. The sender uses the computational basis $$\mathcal{N}_{p,q}(\rho) = (1-q)\left[(1-p)\rho + p Z\rho Z\right] + q\,\mathrm{Tr}(\rho) |e\rangle\langle e| ,$$7, and the receiver decodes with the projective measurement

$$\mathcal{N}_{p,q}(\rho) = (1-q)\left[(1-p)\rho + p Z\rho Z\right] + q\,\mathrm{Tr}(\rho) |e\rangle\langle e| ,$$8

which succeeds with probability $$\mathcal{N}_{p,q}(\rho) = (1-q)\left[(1-p)\rho + p Z\rho Z\right] + q\,\mathrm{Tr}(\rho) |e\rangle\langle e| ,$$9 for either input, yielding a total communication value of $q$0 (Chitambar et al., 2021).

The same paper provides an entropic characterization,

$q$1

where $q$2 is the Choi matrix and the conditional min-entropy is taken over the cone of separable operators (Chitambar et al., 2021). It also proves

$q$3

with $q$4 the channel’s max-Holevo information (Chitambar et al., 2021). Therefore, for the dephrasure channel,

$q$5

A salient contrast with coherent information emerges in the behavior under parallel tensor products. The same source states that

$q$6

Thus the communication value is multiplicative for the dephrasure channel, even though coherent information for the same channel is superadditive (Chitambar et al., 2021). This divergence isolates two different operational regimes: classical-message guessing, where the channel behaves in an erasure-limited and effectively classical manner, and quantum transmission, where entanglement across channel uses is advantageous.

The paper additionally studies a PPT relaxation of communication value. For the dephrasure family,

$q$7

because the optimal encoding and measurement are diagonal and “PPT/SEP positivity coincide” (Chitambar et al., 2021). The agreement is reported both analytically and numerically.

5. Coding constructions for quantum communication

The dephrasure channel has become a benchmark for explicit code design because its nonadditivity is strong enough to expose the limitations of simple ansätze. Early analyses already identified weighted repetition codes as effective probes. A standard family is

$q$8

with $q$9 optimized for coherent information (Bausch et al., 2018). For these codes, the coherent information admits the explicit formula

$p$0

where $p$1 is the binary entropy and

$p$2

This formula appears in the neural-network code study (Bausch et al., 2018).

A major subsequent development was the use of neural network states as variational ansätze for quantum codes. Bausch and Leditzky state that neural network states yield quantum codes with high coherent information for the dephrasure channel and that these codes “outperform all other known codes for these channels” (Bausch et al., 2018). For the dephrasure channel, the paper uses a feed-forward network with four hidden layers, each of width $p$3, $p$4 activation in the first layer, $p$5 in the rest, and a polar output for amplitude encoding (Bausch et al., 2018). Optimization is performed by Particle Swarm Optimization followed by pattern search (Bausch et al., 2018).

For $p$6, the authors report that neural network codes for $p$7 outperform the best weighted repetition codes across relevant intervals of $p$8, with stronger improvements at larger blocklength (Bausch et al., 2018). One explicit example given is $p$9 with $q$0, where the best neural network code achieves $q$1 (Bausch et al., 2018).

More recently, permutation-invariant codes have pushed the achievable rates further. The 2025 work on permutation-invariant codes develops a representation-theoretic method for evaluating coherent information for symmetric input states of the form

$q$2

using Schur–Weyl duality and block diagonalization in irreducible-representation sectors (Bhalerao et al., 13 Aug 2025). For mixtures of pure i.i.d. states, it gives the simplified coherent-information formula

$q$3

(Bhalerao et al., 13 Aug 2025).

Applied to the dephrasure channel, the paper reports that permutation-invariant optimization yields higher achievable rates in “mid-noise” regimes than previously known weighted repetition codes and neural network codes, particularly for larger blocklengths (Bhalerao et al., 13 Aug 2025). For $q$4, the optimized permutation-invariant codes achieve positive coherent information beyond regions accessible to neural-network codes (Bhalerao et al., 13 Aug 2025). The best codes found are described as convex combinations of two non-orthogonal pure i.i.d. code states, interpreted in the paper as non-orthogonal repetition codes (Bhalerao et al., 13 Aug 2025). The authors also state that these codes do not exceed known analytic thresholds for the existence of positive quantum capacity, but they do expand the numerically certified region of positive achievable rates (Bhalerao et al., 13 Aug 2025).

6. Experimental realization and broader significance

The dephrasure channel has also been realized experimentally. An optical implementation reported in 2020 constructs a dephrasure channel with both dephasing and erasure noise and studies coherent-information superadditivity using up to three channel uses (Yu et al., 2020). In that work the channel is written as

$q$5

with $q$6 for the experiments (Yu et al., 2020).

The experiment uses spontaneous parametric down-conversion to generate a four-photon GHZ state, beam displacers and waveplate groups for controlled dephasing, and path splitting with attenuation for erasure (Yu et al., 2020). Coherent information is evaluated from tomography of both channel and complementary outputs (Yu et al., 2020). The study reports parameter regions where $q$7 but $q$8, and even where $q$9 but $$\mathcal{E}_{p,q}^{\mathrm{dr}}(\rho) = (1-p)\left[(1-q)\rho + q Z\rho Z\right] + p \mathcal{E}_e(\rho),$$0, thereby directly demonstrating that finite-use coherent information can fail to detect positive quantum capacity (Yu et al., 2020).

These results reinforce the role of the dephrasure channel as a testing ground for nonadditivity phenomena. The original theoretical work emphasized its “clean form,” large gap between coherent and private information, and positive quantum capacity of complementary channels (Leditzky et al., 2018). Subsequent studies have used it to compare one-shot and regularized capacities, exact assisted capacities, classical-message transmission metrics, numerical optimization methods, representation-theoretic algorithms, and laboratory implementations (Pirandola et al., 2018, Chitambar et al., 2021, Bausch et al., 2018, Bhalerao et al., 13 Aug 2025, Yu et al., 2020).

A plausible implication is that the dephrasure channel serves not merely as a special example but as a structurally minimal model in which several otherwise separate quantum Shannon-theoretic effects become simultaneously visible: superadditivity of coherent information, strict private-versus-quantum capacity gaps, exact two-way-assisted formulas, and a sharply classical behavior for communication value. That combination explains why it continues to be used as a benchmark family in both analytic and computational studies of quantum channel capacity.