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) a single-letter expression for the exact error exponent of quantum state merging in terms of Petz-Rényi conditional entropies, and (ii) regularized expressions for the achievable error exponent 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.
Quantum information decoupling addresses the challenge of eliminating correlations between a local system and its environment through quantum evolution. This task involves transforming a bipartite quantum state ρ AE by applying a unitary operation on system A, followed by a decoupling map T A→C , such that the the resulting system C becomes independent to the environment E. A prominent special case of this theory is standard quantum information decoupling, when T A→C is given by the partial trace over a subsystem of A. As a fundamental structural pillar in quantum information theory, decoupling provides the theoretical foundation for numerous landmark results. Its applications are widespread, ranging from quantum state merging [1-3], quantum channel simulation [4][5][6], entanglement distillation [7,8], to quantum channel coding [9][10][11].
In many physically relevant scenarios, decoupling is considered in the absence of auxiliary resources, where the unitary operation is drawn from the Haar measure and no additional catalytic systems are available. The performance of such a scheme is quantified by a divergence measuring the residual correlations between C and E. A substantial body of work has been devoted to the trace distance and purified distance criteria. In particular, one-shot upper bounds on the decoupling error under the trace distance were derived for general maps T A→C , expressed in terms of smooth conditional min-entropies [12]. While the analysis suffices for proving standard coding theorems, these smooth-entropy-based bounds necessarily involve non-negligible fudge terms. As a consequence, they do not yield exact non-asymptotic error exponent characterizations and are primarily tailored to first-and second-order asymptotics. To address this issue, Cheng et al. [13] recently strengthened the one-shot bound by expressing it in terms of sandwiched conditional Rényi entropies, thereby clarifying the associated achievable error exponents. However, even in this refined form, the analysis remains tied to the trace distance and does not provide a sharp characterization under quantum relative entropy (or purified distance).
Quantum relative entropy plays a central role in quantum information theory. Beyond its operational significance, bounds formulated under relative entropy can be converted into purifieddistance statements via standard entropy-fidelity inequalities, thereby enabling Uhlmann-type arguments that are indispensable across a wide range of fully quantum applications. Consequently, for tasks such as quantum state merging and quantum channel coding, relative entropy thus provides a robust performance criterion. Despite its foundational importance, quantum information decoupling under the relative entropy has remained comparatively unexplored. In particular, general one-shot bounds -analogous to the well-established results for trace distance -have been notably absent from the literature. This gap constitutes a genuine technical bottleneck: Without a sharp one-shot bound formulated directly in terms of the relative entropy, a precise characterization of exact finite-blocklength error exponents remained out of reach.
In this work, we overcome this bottleneck by leveraging a recently developed trace inequality [14] to establish a general one-shot upper bound on the decoupling error measured by the quantum relative entropy. Our bound takes an exact exponential form without the need for smoothing and remains valid for arbitrary blocklengths. Informally speaking, for arbitrary decoupling channels T A→C with Choi state ω AC and bipartite states ρ AE , we prove that
where D(•∥•) is Umegaki’s relative entropy [15], E U(A) denotes the integration over the Haar measure over the unitary group U(A), and H α (A|E) ρ is the sandwiched conditional Rényi entropy defined later in Eq. (16). A key structural feature is that the bound is directly expressed in terms of the exponents, is fully non-asymptotic, and involves no smoothing parameters or additive fudge terms. Consequently, in the n-fold product setting, the induced error exponent is valid for every blocklength n, and is positive whenever the sum-entropy criterion [12], H 1 (A|E) ρ + H 1 (A ′ |C) ω > 0, is satisfied. This stricture is particularly appealing for finite-resource quantum regimes and consequently for applications in physics.
For the standard decoupling scenario, we further establish a lower bound on the one-shot relative entropy decoupling error using the pinching technique combined with an operator inequality. This lower bound is expressed through an information-spectrum quantity, allowing us to obtain an ensemble-tight converse bound that matches the achievable error exponent, provided the subsystem being removed grows at a rate below a certain critical rate. Hence, in the most operationally significant low-cost-rate regime, we determine the exact error exponent of standard quantum decoupling under the relative entropy criterion. Whi
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