We study entanglement distillation and dilution of states and channels in the single-shot regime. With the help of a recently introduced conversion distance, we provide compact closed-form expressions for the dilution and distillation of pure states and show how this can be used to efficiently calculate these quantities on multiple copies of pure states. These closed-form expressions also allow us to obtain second-order asymptotics. We then prove that the ε-single-shot entanglement cost of mixed states is given exactly in terms of an expression containing a suitably smoothed version of the conditional max-entropy. For pure states, this expression reduces to the smoothed max-entropy of the reduced state, for which we provide a closed-form expression. Analogously, we provide a closed-form expression for the smoothed min-entropy and connect it to the ε-single-shot distillable entanglement. Based on these results, we bound the single-shot entanglement cost of channels. We then turn to the one-way entanglement distillation of states and channels and provide bounds in terms of a quantity we denote coherent information of entanglement.
We study entanglement distillation and dilution of states and channels in the single-shot regime. With the help of a recently introduced conversion distance, we provide compact closed-form expressions for the dilution and distillation of pure states and show how this can be used to efficiently calculate these quantities on multiple copies of pure states. These closed-form expressions also allow us to obtain second-order asymptotics. We then prove that the ε-single-shot entanglement cost of mixed states is given exactly in terms of an expression containing a suitably smoothed version of the conditional max-entropy. For pure states, this expression reduces to the smoothed max-entropy of the reduced state, for which we provide a closed-form expression. Analogously, we provide a closed-form expression for the smoothed min-entropy and connect it to the ε-single-shot distillable entanglement. Based on these results, we bound the single-shot entanglement cost of channels. We then turn to the one-way entanglement distillation of states and channels and provide bounds in terms of a quantity we denote coherent information of entanglement.
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.
The information-theoretic approach to entanglement typically relies on arbitrarily many input copies and unboundedly complex operations; however, the available resources are constrained in reality. This motivates the study of entanglement theory with limited resources. One of its central questions is which quantities characterize entanglement manipulations, and so far, only the idealized case of pure states has been well explored. In this work, we address entanglement distillation from mixed states and identify the Rényi coherent information as the quantity determining the optimal rate in the worst case under two distinct resource constraints. We first characterize the sample-efficient regime, in which unbounded computational power is allowed. Our universal protocol achieves this rate without knowledge of the input state under certain polynomial rank constraints. This rate is optimal among universal protocols: we construct states whose distillable entanglement is nearly extensive, yet from which no universal sample-efficient protocol attains a rate above their Rényi coherent information. We then push this characterization to the computationally efficient frontier: the same bound holds for any efficient protocol even when the input state is known, and another universal protocol achieves this rate efficiently under additional polynomial constraints on the marginal ranks. Overall, our results give the Rényi coherent information a novel operational meaning in two computational settings: universal distillation with limited copies and efficient distillation even with knowledge of the input state.
We study entanglement distillation under operations that remain completely positive under partial-transpose conjugation, a.k.a. PPT channels, in the standard quantum Shannon theory regime of asymptotically vanishing but non-zero error. Our main results are: (a) a regularised formula for the PPT distillable entanglement in terms of a measured-relative-entropy-like quantity; and (b) two different single-letter converses, one based on operator quadratic forms and the other on Hirschman's strengthening of the Hadamard three-line theorem. As an immediate application of (b), we show that the PPT distillable entanglement can be strictly smaller than the regularised Rains bound, thereby resolving an open problem in the theory of entanglement manipulation [Regula et al., NJP 21:103017, 2019]. A gap appears already for 3×3 Werner states: at antisymmetric weight 25/26, a certified upper bound of 0.62107 ebits lies below the (regularised) Rains bound, which equals 2625log25−log23≈0.64766 ebits. On a different note, (a) implies a faithful lower bound on the PPT distillable entanglement in terms of the entanglement negativity, which provides a quantitative counterpart to the qualitatively known fact that any NPT state is PPT distillable [Eggeling et al., PRL 87:257902, 2001].
How much entanglement can two distant laboratories share when their connection is noisy? We relate this question to deciding which of two quantum hypotheses is present. Our one-shot bounds apply to entanglement generation over a quantum channel and to one-way entanglement distillation from a shared state. We use Cheng's trace inequality to bound the infidelity of a Petz decoder in terms of a binary hypothesis testing cost. Applying the inequality before averaging over codes reduces the error analysis to computing a second moment of random subspaces. The proof covers every integer target dimension and treats rank-deficient states explicitly. A corollary expresses the achievable entanglement through hypothesis-testing coherent information, with a single logarithmic error slack and exact integer rounding. We also prove an error-preserving conversion from entanglement generation codes to entanglement transmission codes: the target dimension decreases by only a constant factor at fixed error, and the decoder is the original decoder followed by local processing. Consequently, for both entanglement generation and entanglement transmission, arbitrary pure reference-input states yield the same coherent-information rate, second-order Gaussian correction at the original error tolerance, and explicit achievable logarithmic remainder. For commuting testing hypotheses, the logarithmic remainder improves to a constant. These results give a direct route from binary distinguishability to explicit recovery guarantees and finite-resource quantum communication rates.
Quantum entanglement is an essential resource for implementing practical quantum devices. However, for the resource to be useful in large-scale applications, it must be accessible to parties with limited computational resources. Computational entanglement measures quantify the usefulness of entanglement in the presence of limited computational resources in the practical setting of non-asymptotic manipulations. In this paper, we build a framework of non-asymptotic efficient entanglement manipulations by systematically analyzing a wide range of properties of two recently introduced computational entanglement measures: the computational one-shot distillable entanglement and cost. To do so, we adapt definitions of some common properties of scalar entanglement measures to the case in which only lower or upper function bounds are available. Using the developed framework, we then proceed to investigate the mathematical behavior of the newly introduced entanglement measures under probabilistic state mixing and tensor-product operations. Furthermore, we analyze how the values of both measures change under local unitary and LOCC transformations. For this, we derive an explicit quantitative connection between the geometry of projective-net-generated families of states and their computational one-shot distillable entanglement, and use this connection to establish an unconditional separation between computational and information-theoretic one-shot distillable entanglement. Furthermore, we investigate the robustness of efficiently accessible entanglement against noise and derive the first quantitative continuity bound on computational distillable entanglement. Finally, as an application of the developed "dictionary" of properties, we establish a fundamental relation between computational entanglement measures that we call the "Second Law of Efficient Entanglement Manipulation".
Exact entanglement distillation converts a noisy bipartite state into a maximally entangled state with zero error. Under completely PPT-preserving operations, the additive min-Rains relative entropy provides a single-letter upper bound on the regularized distillation rate. An interesting problem in entanglement theory dating back to 2016 has been whether this bound is always tight. Here we resolve this question in the negative. The key is a tensor-stable rigidity absent from the min-Rains relaxation: every feasible exact-distillation effect must act as the identity on the support of the input state. We convert this constraint into a new single-letter upper bound using a generally non-Hermitian, range-supported witness. For a rank-three subspace, we show this upper bound lies strictly below the min-Rains relative entropy for every state with that support. Thus, the constraint discarded by the min-Rains relaxation remains relevant under arbitrary tensor powers. Our result rules out the min-Rains relative entropy as a closed-form formula for exact PPT distillable entanglement and reveals the subtle asymptotic structure of exact entanglement manipulation under PPT operations.