Classical-to-Quantum Encoding Algorithms

• Classical-to-quantum encoding algorithms are methods that translate classical data into quantum states using resource constraints like dephasing and twirling.
• They employ rigorous resource theory techniques and entropic measures, such as hypothesis testing relative entropy, to quantify message capacity in both one-shot and asymptotic regimes.
• This framework underpins protocols in quantum coherence, super-dense coding, and thermodynamic work extraction, providing operational benchmarks for quantum information processing.

Classical-to-Quantum Encoding Algorithms

Classical-to-quantum encoding algorithms constitute the foundational translation step for exploiting quantum computational resources using classically specified information. These algorithms convert classical data—be it messages, vectors, matrices, or more structured datasets—into quantum states or quantum operations, forming the quantum input required for computation, communication, sensing, or learning tasks. The landscape of encoding strategies is broad, encompassing mathematically rigorous architectures, resource theory-inspired mechanisms, and high-efficiency circuit-level constructions, with performance metrics often driven by entropic quantities, gate complexity, and physical resource constraints.

1. Resource-Constrained Encoding Operations

A rigorous framework for classical-to-quantum encoding arises when encoding operations are constrained by a resource destroying map ℵ (Korzekwa et al., 2019). Such encodings allow only the resourceful (i.e., non-free) aspects of an initial quantum state ρ to carry classical information. Formally, a family of encoding channels {𝒞ₘ} is permissible if it meets:

• Resource non-generating: 𝒞ₘ ∘ ℵ = ℵ.
• Non-activating: ℵ ∘ 𝒞ₘ = ℵ.

These constraints ensure that classical messages can be encoded only in degrees of freedom associated with a chosen resource—e.g., coherence, asymmetry, thermodynamic nonequilibrium—by excluding operations that could either create or activate the resource in free states or hide information outside the resource subspace. The canonical example is encoding into quantum coherences via unitaries diagonal in a reference basis with the resource destroying map being the dephasing channel Δ (see also Sec. 5). For group-structured resources, such as asymmetry with respect to a subgroup G of unitaries, the twirling channel

$$\mathcal{N}(\rho) = \frac{1}{|G|} \sum_{g \in G} U_g \rho U_g^\dagger$$

provides the resource destroying map, where allowed encodings employ elements of G.

2. Resource Destroying Maps and Encoding Classes

Resource destroying maps (RDMs) are completely positive, trace-preserving, and idempotent channels that erase the resource of interest:

• For coherence: the dephasing channel Δ acting as

$$\Delta(\rho) = \sum_{k} |k\rangle\langle k| \rho |k\rangle\langle k|$$

annihilates all off-diagonal terms.

• For asymmetry: twirling over a symmetry group projects onto the invariant subspace.
• For thermodynamics: the thermalization channel $\mathcal{N}(\rho) = \gamma$ enforces Gibbs preservation.

The choice of RDM specifies the resource and, correspondingly, the space of allowed encoding operations. Encoding maps that satisfy the RDM-induced constraints ensure that classical information, after resource erasure, becomes inaccessible, enforcing an operational demarcation between “resourceful” and “free” components within the quantum state.

3. Quantitative Capacity: Information Spectrum and Hypothesis Testing Relative Entropy

The information-carrying capacity of a resourceful quantum state ρ under resource-constrained encoding is bounded using the information spectrum relative entropy $D_s^\delta(\rho \| \sigma)$ and the hypothesis testing relative entropy $D_H^\epsilon(\rho \| \sigma)$—non-commutative generalizations of statistical distances:

$$D_s^\delta(\rho \| \sigma) = \sup \{ K \in \mathbb{R} \mid \operatorname{Tr}\left[\rho\, \Pi\{\rho \leq 2^K\sigma\}\right] \leq \delta \}$$

with $\Pi$ being the projector onto the non-negative eigenspace of $2^K\sigma - \rho$. The one-shot (single-use) upper bound on the maximal number of distinguishable messages M is

$$\log M(\rho, \epsilon) \leq D_H^\epsilon(\rho \| \mathcal{N}(\rho)).$$

A matching lower bound is obtained via randomized (uniformly random codebooks) constructions and “pretty good” measurement decoders. In the asymptotic regime, as the number of copies N increases,

$$\frac{1}{N} D_s^{\epsilon \pm \delta}(\rho^{\otimes N} \| \mathcal{N}(\rho)^{\otimes N}) \simeq D(\rho \| \mathcal{N}(\rho)) + \frac{\Phi^{-1}(\epsilon)}{\sqrt{N} \sqrt{V(\rho \| \mathcal{N}(\rho))}}$$

where $D(\cdot\|\cdot)$ is the quantum relative entropy and $V(\cdot\|\cdot)$ its variance, giving both the first-order encoding rate and the finite-size correction.

4. Regimes: One-Shot vs. Asymptotic Encoding Capacity

The framework supports both one-shot and i.i.d. encoding scenarios:

• In the one-shot case, randomized codebooks and measurement strategies using entropic quantities (e.g., collision relative entropy) characterize message distinguishability at a fixed error probability.
• In the asymptotic regime, resource monotones such as the relative entropy of coherence, $R_\Delta(\rho) = D(\rho \| \Delta(\rho))$, and other variants (e.g., relative entropy to the thermal state) govern the optimal encoding rate per copy. Higher-order corrections are precisely quantified by the relative entropy variance.

The following table summarizes key quantitative expressions:

Regime	Encoding Rate/Bound	Governing Quantity
One-shot upper bound	$\log M \leq D_H^\epsilon(\rho\|\mathcal{N}(\rho))$	Hypothesis testing rel. entropy
One-shot lower bound	$$\log M \geq D_s^{\epsilon-\delta}(\rho\|\mathcal{N}(\rho))$$	Information spectrum rel. entropy
Asymptotic	$R(\rho) = D(\rho\|\mathcal{N}(\rho))$	Resource monotone (e.g., coherence)
Asymptotic finite size	$$R(\rho,N,\epsilon) \approx D(\rho\|\mathcal{N}(\rho)) + \Phi^{-1}(\epsilon)/[\sqrt{N}\sqrt{V(\rho\|\mathcal{N}(\rho))}]$$	Rel. entropy & variance

5. Operational Scenarios: Coherence, Super-Dense Coding, and Thermodynamics

The general methodology is adaptable to numerous canonical communication protocols:

• Quantum coherence encoding: With $\mathcal{N} = \Delta$, the encoding capacity becomes $S(\Delta(\rho)) - S(\rho)$, i.e., the relative entropy of coherence. Only the non-diagonal elements (coherences) of ρ are used for message storage, and the encoding map does not disturb population distributions.
• Super-dense coding: For bipartite resource states (e.g., entangled pairs), local unital encoding channels act on subsystem A. The classical channel capacity recovers the optimal rate for maximally entangled states, with resource monotones quantifying the extent of capacity enhancement.
• Thermodynamics: With $\mathcal{N}(\rho) = \gamma$ (thermalization to a Gibbs state), only Gibbs-preserving maps are allowed, and the encoding rate is bounded by $D(\rho \| \gamma)$, which is related to available free energy. This links resource-theoretic capacities to fundamental operational limits in thermodynamic work extraction and information storage.
• Extended settings: Collective encoding (across multiple copies), permutation invariance (important for multipartite/global resource scenarios), and protocol variants for shared reference frames and privacy can all be described under the same entropic resource-governed formalism.

6. Resource Monotones as Operational Capacities

A central conceptual contribution is the endowment of resource monotones with operational significance in classical message encoding. For example:

• Relative entropy of coherence $R_\Delta(\rho)$ quantifies the maximal number of classical messages storable solely in coherence.
• Purity $P(\rho) = \log d - S(\rho)$ appears as an encoding capacity for operations constrained by purity-preserving maps.
• Quantum asymmetry and thermodynamic free energy emerge in scenarios where symmetry or thermal equilibrium impose the encoding constraints.

These monotones quantify not only the distinguishable message count but also bear on the efficiency of practical algorithms for state preparation, error correction, and quantum communication protocols.

7. Unification, Generalizations, and Limitations

The framework of resource-constrained classical-to-quantum encoding provides a unified treatment for a variety of quantum information processing scenarios by connecting constrained channel sets, resource theory, and entropic operational quantities (Korzekwa et al., 2019). It exposes the fundamental trade-off between the structure of the allowed encoding maps (imposed by the resource destroying map) and the entropic resource contents of the carrier state. Limitations include:

• Explicit code construction and decoder design in practical scenarios may require knowledge of the resource structure and group symmetries for the system.
• The generality of the analysis depends on the commutation properties and idempotency of the resource destroying map; exotic resources or non-idempotent maps may fall outside the present theory.

A plausible implication is that as resource theories are further developed and new operational tasks arise, this encoding formalism will provide the essential entropic benchmarks and capacity results for any resource-constrained quantum communication or computation paradigm.