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arXiv:2607.26123

Quantum Information Decoupling Beyond Finite Dimensions

Hayata Yamasaki

quant-phSubmitted Jul 28, 2026• Announced Jul 30, 2026• 0 citations

Abstract

Quantum information decoupling is a pillar of quantum information theory, underlying quantum communication, error correction, and information recovery. However, its existing formulations rely on random unitary operations tied to finite-dimensional assumptions on quantum systems. Here we establish a decoupling framework for arbitrary separable, possibly infinite-dimensional systems. For finite-dimensional inputs and arbitrary separable reference/output systems, we derive error bounds on one-shot decoupling for completely positive maps in terms of sandwiched Rényi conditional entropies. For partial-trace decoupling with finite-dimensional references, the error exponent is optimal up to the known critical rate. To handle infinite-dimensional inputs, we assume finite entropy of the manipulated system. We construct finite-rank projections on independent and identically distributed (IID) states to restrict randomization to a projected finite-dimensional subspace, while ensuring high success probability by including auxiliary components in the discarded subsystem at asymptotically negligible cost. This leads to an infinite-dimensional IID partial-trace decoupling protocol achieving optimal asymptotic dimension rates. As an application, we construct an infinite-dimensional quantum-state-merging protocol, a mother protocol of quantum information theory. Under finite entropy of Alice's marginal, it achieves the same quantum-communication and total-cost rates as in finite dimensions, governed by mutual information and conditional entropy. These results show that the operational interpretation of entropic quantities through these achievable rates constitutes a universal principle beyond finite dimensions. More broadly, our framework provides foundational tools for exploring quantum information regardless of whether we model the physical world using finite- or infinite-dimensional spaces.

Comments: 109 pages, no figure

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Quantum information decoupling is a pillar of quantum information theory, underlying quantum communication, error correction, and information recovery. However, its existing formulations rely on random unitary operations tied to finite-dimensional assumptions on quantum systems. Here we establish a decoupling framework for arbitrary separable, possibly infinite-dimensional systems. For finite-dimensional inputs and arbitrary separable reference/output systems, we derive error bounds on one-shot decoupling for completely positive maps in terms of sandwiched Rényi conditional entropies. For partial-trace decoupling with finite-dimensional references, the error exponent is optimal up to the known critical rate. To handle infinite-dimensional inputs, we assume finite entropy of the manipulated system. We construct finite-rank projections on independent and identically distributed (IID) states to restrict randomization to a projected finite-dimensional subspace, while ensuring high success probability by including auxiliary components in the discarded subsystem at asymptotically negligible cost. This leads to an infinite-dimensional IID partial-trace decoupling protocol achieving optimal asymptotic dimension rates. As an application, we construct an infinite-dimensional quantum-state-merging protocol, a mother protocol of quantum information theory. Under finite entropy of Alice's marginal, it achieves the same quantum-communication and total-cost rates as in finite dimensions, governed by mutual information and conditional entropy. These results show that the operational interpretation of entropic quantities through these achievable rates constitutes a universal principle beyond finite dimensions. More broadly, our framework provides foundational tools for exploring quantum information regardless of whether we model the physical world using finite- or infinite-dimensional spaces.

Quantum information decoupling is a fundamental primitive in quantum information theory, underlying various applications in quantum physics. We prove a novel one-shot decoupling theorem formulated in terms of quantum relative entropy distance, with the decoupling error bounded by two sandwiched Rényi conditional entropies. In the asymptotic i.i.d. setting of standard information decoupling via partial trace, we show that this bound is ensemble-tight in quantum relative entropy distance and thereby yields a characterization of the associated decoupling error exponent in the low-cost-rate regime. Leveraging this framework, we derive several operational applications formulated in terms of purified distance: (i) single-letter expressions for the exact error exponents of quantum state merging in the low-entanglement-cost and high-entanglement-distillation-rate regimes, in terms of Petz-Rényi conditional entropies, and (ii) regularized expressions for achievable error exponents of entanglement distillation and quantum channel coding in terms of Petz-Rényi coherent informations. We further prove that these achievable bounds are tight for maximally correlated states and generalized dephasing channels, respectively, for the high distillation-rate/coding-rate regimes.

. This generalizes a foundational result in quantum information theory that was previously formulated only for operations and states on finite-dimensional systems. The extension to infinite-dimensional systems is nontrivial because the conventional tools for establishing both the direct and converse bounds, i.e., strong typicality, monotonicity, and asymptotic continuity, are no longer directly applicable. To address this problem, we construct a new entanglement dilution protocol for infinite-dimensional states implementable by local operations and a finite amount of one-way classical communication (one-way LOCC), using weak and strong typicality multiple times. We also prove the optimality of this protocol among all protocols, even under infinite-dimensional separable operations, by developing an argument based on alternative forms of monotonicity and asymptotic continuity of the entanglement of formation for infinite-dimensional states. Along the way, we derive a new integral representation for the quantum entropy of infinite-dimensional states, which we believe to be of independent interest. Our results allow us to fully characterize an important operational entanglement measure-the entanglement cost-for all infinite-dimensional physical systems.

Single-shot quantum information theory is governed not only by entropy exponents, but also by the finite-resource constants that multiply them. These constants directly affect the quantitative performance of decoupling, covering, convex-splitting, position-based decoding, and one-shot communication protocols, yet they are often inherited from nonoptimal scalar estimates or from classical-to-quantum lifting arguments that introduce additional losses. In this work we show that the operator layer-cake representation provides a mechanism for lifting sharp scalar inequalities to the noncommutative setting without loss. Using an iterative Riemann--Stieltjes integration-by-parts method, we derive sharp quantum trace inequalities that tighten several standard single-shot bounds. For a logarithmic trace inequality recently introduced by Cheng \emph{et al.}\ and subsequently used in quantum covering and decoupling problems, we determine the exact optimal prefactor, replacing the previously known constant by a smaller Lambert-

W

constant and proving universal optimality for positive operators. We also completely characterize the threshold behavior that appears under normalization to quantum states. In addition, we establish optimal two-sided collision-divergence inequalities, which lead to improved position-based decoding and single-shot classical communication bounds. These results show that several finite-resource bounds in single-shot quantum information can be tightened, and that within the layer-cake Rényi-divergence framework the resulting constants are genuine optimality barriers rather than artifacts of the proof.

Correlations across channel uses can improve quantum communication rates, making optimization over arbitrarily large blocks a central difficulty in determining quantum capacity. We show how full permutation invariance limits this advantage for every finite-dimensional memoryless channel. The optimized coherent information per use converges to the single-use maximum and consequently attains its supremum at a finite block length. Thus finite-block superadditivity remains possible, while the asymptotic optimization reduces to a single channel use. We also establish a strong converse for pure-source entanglement generation with permutation-invariant inputs: at any fixed rate above the single-use coherent-information maximum, the fidelity tends to zero. Beyond these communication bounds, we identify a common structure underlying completely bounded norms and the sandwiched Rényi channel quantities governing discrimination against a replacer and entanglement-assisted communication. De Finetti reduction and permutation covariance yield systematic alternative proofs of their known multiplicativity and additivity results within a shared variational framework. We further show that, for permutation-invariant inputs, correlations cannot sustain an asymptotic reduction in output entropy per use below the single-use minimum. This entropy limit holds for Rényi orders greater than one and for the von Neumann entropy, extending the role of symmetry from communication bounds to the control of output entropy.

In this paper, we consider the standard quantum information decoupling, in which Alice aims to decouple her system from the environment by discarding some of her systems. To achieve an ε-decoupling with trace distance as the error criterion, we establish a one-shot characterization for the largest size of the remainder system in terms of the conditional (1 − ε)-hypothesis-testing entropy. When the underlying system is independent and identically prepared, our result leads to the matched second-order rate as well as the moderate deviation rate.

We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering state-of-the-art asymptotic i.i.d. error exponents. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.