Quantum Internet relies on quantum entanglement as a fundamental resource for secure and efficient quantum communication, reshaping data transmission. In this context, entanglement distillation emerges as a crucial process that plays a pivotal role in realizing the full potential of the quantum internet. Nevertheless, it remains challenging to accurately estimate the distillable entanglement and its closely related essential quantity, the quantum capacity. In this work, we consider a general resource measure known as the reverse divergence of resources which quantifies the minimum divergence between a target state and the set of free states. Leveraging this measure, we propose efficiently computable upper bounds for both quantities based on the idea that the useless entanglement within a state or a quantum channel does not contribute to the distillable entanglement or the quantum capacity, respectively. Our bounds can be computed via semidefinite programming and have practical applications for purifying maximally entangled states under practical noises, such as depolarizing and amplitude damping noises, leading to improvements in estimating the one-way distillable entanglement. Furthermore, we provide valuable benchmarks for evaluating the quantum capacities of qubit quantum channels, including the Pauli channels and the random mixed unitary channels, which are of great interest for the development of a quantum internet.
Quantum Internet relies on quantum entanglement as a fundamental resource for secure and efficient quantum communication, reshaping data transmission. In this context, entanglement distillation emerges as a crucial process that plays a pivotal role in realizing the full potential of the quantum internet. Nevertheless, it remains challenging to accurately estimate the distillable entanglement and its closely related essential quantity, the quantum capacity. In this work, we consider a general resource measure known as the reverse divergence of resources which quantifies the minimum divergence between a target state and the set of free states. Leveraging this measure, we propose efficiently computable upper bounds for both quantities based on the idea that the useless entanglement within a state or a quantum channel does not contribute to the distillable entanglement or the quantum capacity, respectively. Our bounds can be computed via semidefinite programming and have practical applications for purifying maximally entangled states under practical noises, such as depolarizing and amplitude damping noises, leading to improvements in estimating the one-way distillable entanglement. Furthermore, we provide valuable benchmarks for evaluating the quantum capacities of qubit quantum channels, including the Pauli channels and the random mixed unitary channels, which are of great interest for the development of a quantum internet.
Quantum communication relies on the existence of high quality quantum channels to exchange information. In practice, however, all communication links are affected by noise from the environment. Here we investigate the ability of quantum channels to perform quantum communication tasks by restricting the participants to use only local operations and one-way classical communication (one-way LOCC) along with the available quantum channel. In particular, a channel can be used to distill a highly entangled state between two parties, which further enables quantum or private communication. In this work, we invoke the framework of superchannels to study the distillation of a resourceful quantum state, such as a maximally entangled state or a private state, using multiple instances of a point-to-point quantum channel. We use the idea ofk-extendibility to obtain a semidefinite relaxation of the set of one-way LOCC superchannels and define a class of entanglement measures for quantum channels that decrease monotonically under such superchannels; therefore these measures, dubbed collectively the “unextendible entanglement of a channel”, yield upper bounds on several communication-theoretic quantities of interest in the regimes of resource distillation and zero error.We then generalize the formalism ofk-extendibility to bipartite superchannels, thus obtaining functions that are monotone under two-extendible superchannels. This allows us to analyze probabilistic distillation of ebits or secret key bits from a bipartite state when using a resourceful quantum channel. Moreover, we propose semidefinite programs to evaluate several of these quantities, providing a computationally feasible method of comparison between quantum channels for resource distillation.
We derive general upper bounds on the distillable entanglement of a mixed state under one-way and two-way local operations and classical communication (LOCC). In both cases, the upper bound is based on a convex decomposition of the state into “useful” and “useless” quantum states. By “useful,” we mean a state whose distillable entanglement is non-negative and equal to its coherent information (and thus given by a single-letter, tractable formula). On the other hand, “useless” states are undistillable, i.e., their distillable entanglement is zero. We prove that in both settings, the distillable entanglement is convex on such decompositions. Hence, an upper bound on the distillable entanglement is obtained from the contributions of the useful states alone, being equal to the convex combination of their coherent informations. Optimizing over all such decompositions of the input state yields our upper bound. The useful and useless states are given by degradable and antidegradable states in the one-way LOCC setting, and by maximally correlated and positive partial transpose (PPT) states in the two-way LOCC setting, respectively. We also illustrate how our method can be extended to quantum channels. Interpreting our upper bound as a convex roof extension, we show that it reduces to a particularly simple, non-convex optimization problem for the classes of isotropic states and Werner states. In the one-way LOCC setting, this non-convex optimization yields an upper bound on the quantum capacity of the qubit depolarizing channel that is strictly tighter than previously known bounds for large values of the depolarizing parameter. In the two-way LOCC setting, the non-convex optimization achieves the PPT-relative entropy of entanglement for both isotropic and Werner states.
Sending quantum information reliably over long distances is a central challenge in quantum technology in general, and in quantum optics in particular, since most quantum communication relies on optical fibres or free-space links. Here, we address this problem by shifting the focus from the quantity of information sent to the quality of the transmission, i.e. the rate of decay of the transmission error with respect to the number of channel uses. For the general class of teleportation-simulable channels, which includes all channels arising in quantum optical communication, we prove that the single-letter reverse relative entropy of entanglement of the Choi state upper bounds the error exponent of two-way assisted quantum communication - paralleling the celebrated capacity bound of [Pirandola et al., Nat. Comm. (2017)] in terms of the regularised relative entropy of entanglement. Remarkably, for Gaussian channels our bound can be computed efficiently through a convex program with simple constraints involving only finite-dimensional covariance matrices. As a prototypical application, we derive closed-form analytical expressions of our upper bound as well as random-coding-based lower bounds for several one-mode Gaussian channels. Extending recent work [Lami et al., arXiv:2408.07067 (2024)] to infinite-dimensional systems, we further endow the reverse relative entropy of entanglement with an exact operational interpretation in entanglement testing, and show that it characterises the rate of entanglement distillation under non-entangling operations. These findings offer a new perspective on entanglement as a resource and sharpen the theoretical benchmarks for future quantum optical networks.
The quantum capacity of a noisy quantum channel determines the maximal rate at which we can code reliably over asymptotically many uses of the channel, and it characterizes the channel's ultimate ability to transmit quantum information coherently. In this paper, we derive single-letter upper bounds on the quantum and private capacities of quantum channels. The quantum capacity of a quantum channel is always no larger than the quantum capacity of its extended channels, since the extensions of the channel can be considered as assistance from the environment. By optimizing the parametrized extended channels with specific structures such as the flag structure, we obtain new upper bounds on the quantum capacity of the original quantum channel. Furthermore, we extend our approach to estimating the fundamental limits of private communication and one-way entanglement distillation. As notable applications, we establish improved upper bounds to the quantum and private capacities for fundamental quantum channels of interest in quantum information, some of which are also the sources of noise in superconducting quantum computing. In particular, our upper bounds on the quantum capacities of the depolarizing channel and the generalized amplitude damping channel are strictly better than previously best-known bounds for certain regimes.
We derive an upper bound on the one-way distillable entanglement of bipartite quantum states. To this end, we revisit the notion of degradable, conjugate degradable, and antidegrad-able bipartite quantum states [1]. We prove that for degradable and conjugate degradable states the one-way distillable entanglement is equal to the coherent information, and thus given by a single-letter formula. Furthermore, it is well-known that the one-way distillable entanglement of antidegradable states is zero. We use these results to derive an upper bound for arbitrary bipartite quantum states, which is based on a convex decomposition of a bipartite state into degradable and antidegradable states. This upper bound is always at least as good an upper bound as the entanglement of formation. Applying our bound to the qubit depolarizing channel, we obtain an upper bound on its quantum capacity that is strictly better than previously known bounds in the high noise regime. We also transfer the concept of approximate degradability [2] to quantum states and show that this yields another easily computable upper bound on the one-way distillable entanglement. Moreover, both methods of obtaining upper bounds on the one-way distillable entanglement can be combined into a generalized one.
Quantum communication and computation heavily rely on entanglement distillation protocols. There is a plethora of distillation protocols for Pauli channels and also for some non-Pauli channels. However, an effort to relate the effectiveness of these protocols for different types of noise has been missing. For most quantum channels, the gap between the existing lower and upper bounds on distillation rates is substantial, and improvements of achievable rates have been stagnant for decades. In this work, we improve the best known distillation lower bounds, for both the amplitude damping and depolarizing channels. We build on a key observation that distillation protocols reshape several uses of a very noisy channel into a better effective channel. We apply this channel processing in an adaptive and recurrent manner. For the amplitude damping channel, our suggested protocol reshapes the channel into an erasure channel, achieving rates exceeding the best known lower bound given by the channel’s reverse coherent information. For the depolarizing channel, we introduce the Greedy recurrence protocol with proven performance guarantees and construct a combined protocol improving upon previously known distillation rates. Improved bounds on attainable distillation rates give insights for both practical implementations and theoretical understanding of quantum information processing.