Quantum entanglement distillation protocols are LOCC protocols that, over ``imperfect'' EPR pairs shared between Alice and Bob as input, output perfect or near-perfect EPR pairs. In this paper, we study the communication complexity of these protocols, that is, the minimal number of (classical) bits needed for Alice and Bob. To our knowledge, it is the first paper that studies entanglemend distillation protocols in the resource-bounded case. We consider 3 models for imperfect EPR pairs. In the measure-$r$ model, $r$ out of $n$ EPR pairs are measured by an adversary (and thus become ``corrupted''); in the depolarization model, the qubits of Bob underwent a depolarization channel; in the fidelity model, the only information Alice and Bob possess is the fidelity of the shared state. For the first 2 models, we prove tight and almost-tight bounds on the base case, i.e., how well Alice and Bob can do if they are not allowed to communicate. For the fidelity model, we prove a very tight lower bound (up to an additive constant) on the communication complexity. This lower bound implies the optimality of the Random Hash protocol in [ASY02].
Quantum entanglement distillation protocols are LOCC protocols that, over imperfect'' EPR pairs shared between Alice and Bob as input, output perfect or near-perfect EPR pairs. In this paper, we study the communication complexity of these protocols, that is, the minimal number of (classical) bits needed for Alice and Bob. To our knowledge, it is the first paper that studies entanglemend distillation protocols in the resource-bounded case. We consider 3 models for imperfect EPR pairs. In the measure-$r$ model, $r$ out of $n$ EPR pairs are measured by an adversary (and thus become corrupted''); in the depolarization model, the qubits of Bob underwent a depolarization channel; in the fidelity model, the only information Alice and Bob possess is the fidelity of the shared state. For the first 2 models, we prove tight and almost-tight bounds on the base case, i.e., how well Alice and Bob can do if they are not allowed to communicate. For the fidelity model, we prove a very tight lower bound (up to an additive constant) on the communication complexity. This lower bound implies the optimality of the Random Hash protocol in [ASY02].
Quantum entanglement distillation protocols are LOCC protocols between Alice and Bob that convert imperfect EPR pairs, or, in general, partially entangled bipartite states into perfect or near-perfect EPR pairs. The classical communication complexity of these protocols is the minimal amount of classical communication needed for the conversion. In this paper, we focus on the communication complexity of protocols that operate with incomplete information, i.e., where the inputs are mixed states and/or prepared adversarially. We study 3 models of imperfect EPR pairs. In the measure-r model, r out of n EPR pairs are measured by an adversary; in the depolarization model, Bob's share of qubits underwent a depolarization channel; in the fidelity model, the only information Alice and Bob possess is the fidelity of the shared state. For the measure-r model and the depolarization model, we prove tight and almost-tight bounds on the outcome of LOCC protocols that don't use communication. For the fidelity model, we prove a lower bound on the communication complexity that matches the upper bound given by Ambainis, Smith, and Yang [ASY02].
Entanglement is an essential resource for quantum communication and quantum computation, similar to shared random bits in the classical world. Entanglement distillation extracts nearly-perfect entanglement from imperfect entangled state. The classical communication complexity of these protocols is the minimal amount of classical information that needs to be exchanged for the conversion. In this paper, we focus on the communication complexity of protocols that operate with incomplete information, i.e., where the inputs are mixed states and/or prepared adversarially. We consider three models of imperfect entanglement, namely, the bounded measurement model, the depolarization model, and the fidelity model. We describe these models as well as the motivations for studying them. For the bounded measurement model and the depolarization model, we prove tight and almost-tight bounds on the output quality of non-interactive protocols. For the fidelity model, we prove a lower bound that matches the upper bound given by Ambainis et al., and thus completely characterizes communication complexity of entanglement distillation protocols for this model. Our result also suggests the optimality of the BB84 protocol in terms of communication complexity. We emphasize that although some of the results appear intuitively straightforward, their proofs are not. In fact, two novel techniques are developed for proving these results. We believe that these techniques are of independent interests, too.
Abstract Entanglement is an essential resource for quantum com-munication and quantum computation, similar to shared random bits in the classical world. Entanglement distil-lation extracts nearly-perfect entanglement from imperfect entangled state. The classical communication complexity ofthese protocols is the minimal amount of classical information that needs to be exchanged for the conversion. In thispaper, we focus on the communication complexity of protocols that operate with incomplete information, i.e., wherethe inputs are mixed states and/or prepared adversarially. We consider three models of imperfect entanglement,namely, the bounded measurement model, the depolarization model, and the fidelity model. We describe there mod-els as well as the motivations for studying them. For the bounded measurement model and the depolarization model,we prove tight and almost-tight bounds on the output quality of non-interactive protocols. For the fidelity model weprove a lower bound that matches the upper bound given by Ambainis et al., and thus completely characterizes commu-nication complexity of entanglement distillation protocols for this model. Our result also suggests the optimality ofthe BB84 protocol in terms of communication complexity. We emphasize that although some of the results appearintuitively straightforward, their proofs are not. In fact, two novel techniques are developed for proving these results.We believe that these techniques are of independent interests, too.
A powerful operational paradigm for distributed quantum information processing involves manipulating pre-shared entanglement by local operations and classical communication (LOCC). The LOCC round complexity of a given task describes how many rounds of classical communication are needed to complete the task. Despite some results separating one-round versus two-round protocols, very little is known about higher round complexities. In this paper, we revisit the task of one-shot random-party entanglement distillation as a way to highlight some interesting features of LOCC round complexity. We first show that for random-party distillation in three qubits, the number of communication rounds needed in an optimal protocol depends on the entanglement measure used; for the same fixed state some entanglement measures need only two rounds to maximize whereas others need an unbounded number of rounds. In doing so, we construct a family of LOCC instruments that require an unbounded number of rounds to implement. We then prove explicit tight lower bounds on the LOCC round number as a function of distillation success probability. Our calculations show that the original W-state random distillation protocol by Fortescue and Lo is essentially optimal in terms of round complexity.
We study and solve the problem of distilling a secret key from quantum states representing correlation between two parties (Alice and Bob) and an eavesdropper (Eve) via one–way public discussion: we prove a coding theorem to achieve the ‘wire–tapper’ bound, the difference of the mutual information Alice–Bob and that of Alice–Eve, for so–called classical–quantum–quantum–correlations, via one–way public communication. This result yields information–theoretic formulae for the distillable secret key, giving ‘ultimate’ key rate bounds if Eve is assumed to possess a purification of Alice and Bob's joint state. Specializing our protocol somewhat and making it coherent leads us to a protocol of entanglement distillation via one–way LOCC (local operations and classical communication) which is asymptotically optimal: in fact we prove the so–called ‘hashing inequality’, which says that the coherent information (i.e. the negative conditional von Neumann entropy) is an achievable Einstein–Podolsky–Rosen rate. This result is known to imply a whole set of distillation and capacity formulae, which we briefly review.
It is crucial to share maximally entangled pairs in quantum information processing. However, unwanted interactions with the environment can reduce the entanglement fidelity of the produced pairs. Entanglement distillation is a method to overcome this problem by creating higher-fidelity entangled pairs from a more significant number of lower-fidelity pairs. This study uses Werner states and imperfect gates to analyse the performance of BBPSSW, DEJMPS, 3-to-1, and EPL protocols for entanglement distillation. Specifically, this study focuses on how the increase in fidelity depends on the input pair and gate fidelities. The results are obtained through numerical simulation using the NetSquid simulator. After characterising the numerical errors, the study demonstrates that all four protocols increase the pair fidelity for a gate higher than 0.96. The study also reveals that the higher the success fidelity, the lower the probability, as predicted theoretically. Additionally, based on a list of requirements and hardware performance, we developed a simple recipe for selecting the most suitable protocol. The simulation results highlight the practical limitations of entanglement distillation protocols, indicating the need for further investigation. This includes the development of a theoretical model for the EPL protocol implemented with Werner states, incorporating noisy measurements in the simulated protocols and increasing the number of simulations for higher accuracy.