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arXiv:2608.28500

Classical Commitment over Quantum Channels with Limited Entanglement Assistance

Remi A. Chou

quant-phcs.ITSubmitted Aug 28, 2026• Announced Aug 31, 2026• 0 citations
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Abstract

We study classical string commitment over quantum channels with limited preshared entanglement. For noninteractive protocols, we determine the commitment capacity of a class of channels with input dimension $d$ that, at each use, sample a pair of classical random variables $(F,Z)$, apply one of the $d^2$ Heisenberg--Weyl operators indexed by $Z$ to the input, and deliver the transformed quantum system together with $F$ to the receiver. If $E$ is the available entanglement rate in bits per channel use, then the capacity is $\min\{H(Z|F),\log_2d+E\}$. This class of channels encompasses quantum erasure and depolarizing channels, as well as families of Pauli channels. Additionally, for interactive protocols, we show that the commitment rate cannot exceed $\log_2d+E$ bits per channel use, so that when $H(Z|F)\geq\log_2d+E$, interactive communication does not increase the capacity. As a consequence, for interactive protocols, we determine the capacity of the quantum erasure channel.

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We study classical string commitment over quantum channels with limited preshared entanglement. For noninteractive protocols, we determine the commitment capacity of a class of channels with input dimension ddd that, at each use, sample a pair of classical random variables (F,Z)(F,Z)(F,Z), apply one of the d2d^2d2 Heisenberg--Weyl operators indexed by ZZZ to the input, and deliver the transformed quantum system together with FFF to the receiver. If EEE is the available entanglement rate in bits per channel use, then the capacity is min⁡{H(Z∣F),log⁡2d+E}\min\{H(Z|F),\log_2d+E\}min{H(Z∣F),log2​d+E}. This class of channels encompasses quantum erasure and depolarizing channels, as well as families of Pauli channels. Additionally, for interactive protocols, we show that the commitment rate cannot exceed log⁡2d+E\log_2d+Elog2​d+E bits per channel use, so that when H(Z∣F)≥log⁡2d+EH(Z|F)\geq\log_2d+EH(Z∣F)≥log2​d+E, interactive communication does not increase the capacity. As a consequence, for interactive protocols, we determine the capacity of the quantum erasure channel.
The optimal rate of reliable communication over a quantum channel can be enhanced by preshared entanglement. Whereas the enhancement may be unbounded in infinite-dimensional settings even when the input power is constrained, a long-standing conjecture asserts that the ratio between the entanglement-assisted and unassisted classical capacities is bounded in finite-dimensional settings [Bennett et al., IEEE Trans. Inf. Theory 48, 2637 (2002)IETTAW0018-944810.1109/TIT.2002.802612]. In this Letter, we prove this conjecture by showing that their ratio is upper bounded by o(d^{2}), where d is the input dimension of the channel. An application to quantum communication with noisy encoders and decoders is given.
Quantum communications promises reliable transmission of quantum information, efficient distribution of entanglement and generation of completely secure keys. For all these tasks, we need to determine the optimal point-to-point rates that are achievable by two remote parties at the ends of a quantum channel, without restrictions on their local operations and classical communication, which can be unlimited and two-way. These two-way assisted capacities represent the ultimate rates that are reachable without quantum repeaters. Here, by constructing an upper bound based on the relative entropy of entanglement and devising a dimension-independent technique dubbed 'teleportation stretching', we establish these capacities for many fundamental channels, namely bosonic lossy channels, quantum-limited amplifiers, dephasing and erasure channels in arbitrary dimension. In particular, we exactly determine the fundamental rate-loss tradeoff affecting any protocol of quantum key distribution. Our findings set the limits of point-to-point quantum communications and provide precise and general benchmarks for quantum repeaters.
Current advancements in communication equipment demand the investigation of classical information transfer over quantum channels, by encompassing realistic scenarios in finite dimensions. To address this issue, we develop a framework for analyzing classical capacities of quantum channels where the set of states used for encoding information is restricted based on various physical properties. Specifically, we provide expressions for the classical capacities of noiseless and noisy quantum channels when the average energy of the encoded ensemble or the energy of each of the constituent states in the ensemble is bounded. In the case of qubit energy-preserving dephasing channels, we demonstrate that a nonuniform probability distribution based on the energy constraint maximizes capacity, while we derive the compact form of the capacity for equiprobable messages. We suggest an energy-constrained dense coding (DC) protocol that we prove to be optimal in the two-qubit situation and obtain a closed-form expression for the DC capacity. Additionally, we demonstrate a no-go result, which states that when the dimension of the sender and the receiver is two, no energy-preserving operation can offer any quantum advantage for energy-constrained entanglement-assisted capacity. We exhibit that, in the energy-constrained situation, classical-quantum noisy channels can show improved capabilities under entanglement assistance, a phenomenon that is unattainable in the unrestricted scenario.
We study four well-known capacities of a two-parameter family of qubit Pauli channels. These are the channels that are covariant under the SO(2)\text{SO}(2)SO(2) group and contain the depolarizing channel as a special case. We find exact expressions for the classical capacity and entanglement-assisted capacities, and analytically determine the regions where the quantum capacity of the channel vanishes. We then use a flag extension to find upper bound for the quantum capacity and private capacity of these channels in the entire region of parameter space and also obtain the lower bound for the quantum capacity by calculating the single-shot quantum capacity numerically. In conjunction with previous results on depolarizing channels, our result is one step forward for determining the capacities of the full Pauli channel.
We study commitment scheme for classical-quantum channels. To accomplish this we define various notions of commitment capacity for these channels and prove matching upper and lower bound on it in terms of the conditional entropy. Our achievability (lower bound) proof is quantum generalisation of the work of one of the authors (arXiv:2103.11548) which studied the problem of secure list decoding and its application to bit-string commitment. The techniques we use in the proof of converse (upper bound) is similar in spirit to the techniques introduced by Winter, Nascimento and Imai (Cryptography and Coding 2003) to prove upper bound on the commitment capacity of classical channels. However, generalisation of this technique to the quantum case is not so straightforward and requires some new constructions, which can be of independent interest.
We study quantum-state oblivious transfer over the quantum erasure channel under information-theoretic security guarantees. Specifically, Alice holds two quantum systems, Bob wishes to recover one of them, and the parties have access to independent channel uses and authenticated public classical communication, with no initial shared entanglement. We consider both honest-but-curious and malicious behavior. We establish finite-blocklength converse and achievability bounds, which match in the honest-but-curious setting. The malicious setting construction combines interactive hashing with tests of whether Bob received selected channel outputs, and distributes each input across several qubits so that erasures conceal an entire input from a dishonest receiver, even when the inputs are entangled with an external reference. We also derive fixed-error asymptotic expansions and show that both malicious and honest-but-curious settings have the same capacity.