Distributed source simulation is the task where two (or more) parties share some correlated randomness and use local operations and no communication to convert this into some target correlation. Wyner’s seminal result showed that asymptotically the rate of uniform shared randomness needed for this task is given by a mutual information induced measure, now referred to as Wyner’s common information. This asymptotic result was extended by Hayashi in the quantum setting to separable states, the largest class of states for which this task can be performed to vanishing error. In this work we characterize this task in a near-tight manner in the one-shot setting using the smooth entropy framework. We do this by introducing one-shot operational quantities and correlation measures that characterize them. We establish asymptotic equipartition properties for our correlation measures thereby recovering the previous vanishing-error asymptotic results. In doing so, we consider technical points in one-shot network information theory and provide methods for cardinality bounds in the smooth entropy calculus. We also introduce entangled state versions of the distributed source simulation task and determine bounds in this setting via quantum embezzling. This provides a strong characterization of this network task in the one-shot, quantum regime.
Distributed source simulation is the task where two (or more) parties share some correlated randomness and use local operations and no communication to convert this into some target correlation. Wyner’s seminal result showed that asymptotically the rate of uniform shared randomness needed for this task is given by a mutual information induced measure, now referred to as Wyner’s common information. This asymptotic result was extended by Hayashi in the quantum setting to separable states, the largest class of states for which this task can be performed to vanishing error. In this work we characterize this task in a near-tight manner in the one-shot setting using the smooth entropy framework. We do this by introducing one-shot operational quantities and correlation measures that characterize them. We establish asymptotic equipartition properties for our correlation measures thereby recovering the previous vanishing-error asymptotic results. In doing so, we consider technical points in one-shot network information theory and provide methods for cardinality bounds in the smooth entropy calculus. We also introduce entangled state versions of the distributed source simulation task and determine bounds in this setting via quantum embezzling. This provides a strong characterization of this network task in the one-shot, quantum regime.
Distributed source simulation is the task where two (or more) parties share some correlated randomness and use local operations and no communication to convert this into some target correlation. Wyner’s seminal result showed that asymptotically the rate of uniform shared randomness needed for this task is given by a mutual information induced measure, now referred to as Wyner’s common information. In this work we characterize the quantum version of this task in the one-shot setting using the smooth entropy framework and one-shot operational quantities. We further establish asymptotic equipartition properties for our correlation measures. We also introduce entanglement versions of the distributed source simulation task and determine bounds in this setting via quantum embezzling.
The novel ways in which correlations may be encoded non-locally in quantum mechanics have led to the promise of new technologies. This, in turn, has led to the first attempts at the development of quantum networks. Nonetheless, our understanding of quantum correlations and the limits of how we may process them over a quantum network remains nascent. This thesis studies the limits of processing quantum correlations between spatially separated parties in various settings and the mathematical tools necessary to determine such limits. To do so, this thesis studies three general problems in quantum network theory. The first problem is the minimum resources to prepare a quantum correlation between non-communicative receivers, which may be viewed as a star network. This study is separated into three pieces. The first piece is a near-tight one-shot quantum generalization of a problem introduced by Wyner, distributed source simulation. Establishing our one-shot bounds builds on recent work on soft-covering without smoothing and relies on establishing a variety of properties for one-shot entropic quantities, as well as improving our understanding of the smooth entropy framework in quantum network information theory. As distributed source simulation cannot be achieved to arbitrary error for sufficiently entangled states, the second piece characterizes the minimal error of converting one entangled state to another using zero communication. This relies on establishing new properties of the fidelity measure. The third piece establishes an entanglement-assisted version of the distributed source simulation task that can be achieved to arbitrary error for arbitrary quantum systems. This result builds on recent work on multipartite state splitting and establishing new properties of one-shot multipartite max mutual information quantities. The second problem is the extraction of perfect classical correlation from a multipartite quantum source where the receivers do not communicate. This may be seen as the reverse process of distributed source simulation. We establish that this is asymptotically possible in quantum mechanics for the first time and establish the asymptotic rate for classical-quantum states. The technical barrier in establishing the asymptotic rate is establishing a limit on how much noiseless correlation can be extracted without communication. Our major technical contribution is establishing such a limit by generalizing a technical lemma of Gács and Körner to classical-quantum states. The final problem is distributed state discrimination of quantum product states. This is motivated by its relevance to both understanding non-locality in quantum mechanics and developing quantum cryptographic schemes. Our study again breaks into three key pieces. First, we study the framework of local operations and simultaneous classical or quantum communication (LOSCC and LOSQC respectively) for product states, which is relevant to quantum cryptography. We establish the first separation between quantum and classical communication in the simultaneous communication model using only product state inputs. We also establish a new uncertainty relation that implies error bounds for discriminating product states using LOSQC. Second, we study state discrimination under local operations and classical communication (LOCC). We develop a new framework for zero-error state discrimination under LOCC. Our framework provides a simple proof of orthogonal product states that cannot be distinguished using LOCC--- a property introduced by Bennett et al. Finally, we study the thermodynamics of zero-error distributed state discrimination in the framework of Faist et al. This allows us to resolve an open problem from the aforementioned paper of Bennett et al.
Recently, Cheng et al generalized quantum state splitting (QSS) to the multipartite setting for applications in quantum network information theory. Furthermore, Cheng and Gao recently noted that unipartite convex splitting reduces to soft covering for classical-quantum states. Soft covering is a common primitive in (classical, network) information theory that does not seem to need an extension to the multipartite setting, e.g. when establishing rates for many party distributed source simulation (DSS). This suggests a gap in our understanding of how these tasks are related. In this work we clarify this relation. We first identify “incoherent” QSS without entanglement assistance with DSS in the bipartite setting. Motivated by this, we identify “coherent” DSS as a special case of multipartite QSS. We then establish a one-shot rate region for multipartite QSS in terms of smooth multipartite mutual information quantities, which in turn implies a one-shot rate region for coherent DSS. Lastly, we establish a strong asymptotic equipartition property for mutual information quantities to establish a strong converse rate region for multipartite QSS and coherent DSS.
Three distributed parties, two transmitters (Txs) and a receiver (Rx), hold one component each of a tripartite quantum state ρA1A2C. The goal is to simulate the action of a separable instrument acting on the A1 and A2 components, with the Rx recovering the classical outcome. To enable this, each Tx k can transfer bits on a noiseless bit pipe and share randomness at rates Rk and Ck, respectively, with the Rx. Undertaking a Shannon-theoretic study, we characterize two new sets of inner bounds. The first set, derived for the one-shot regime, is based on instrument simulation protocols built using unstructured IID codes, while the second set, derived for the asymptotic regime, relies on coset codes and new decoding POVMs. The first set of bounds recovers current known inner bounds for instrument and measurement simulation in all previously studied scenarios. Our protocols are based on likelihood POVMs, and our analysis leverages Sen's smooth multiparty covering and simultaneous decoding, while handling the distributed-component scenario via a compatible operator sliding trick.
We consider the problem of shared randomness-assisted multiple access channel (MAC) simulation for product inputs and characterize the one-shot communication cost region via almost-matching inner and outer bounds in terms of the smooth max-information of the channel, featuring auxiliary random variables of bounded size. The achievability relies on a rejection-sampling algorithm to simulate an auxiliary channel between each sender and the decoder, and producing the final output based on the output of these intermediate channels. The converse follows via information-spectrum based arguments. To bound the cardinality of the auxiliary random variables, we employ the perturbation method from [Anantharamet al., IEEE Trans. Inf. Theory (2019)] in the one-shot setting. For the asymptotic setting and vanishing errors, our result expands to a tight single-letter rate characterization and consequently extends a special case of the simulation results of [Kurriet al., IEEE Trans. Inf. Theory (2022)] for fixed, independent and identically distributed (iid) product inputs to universal simulation for any product inputs. We broaden our discussion into the quantum realm by studying feedback simulation of quantum-to-classical (QC) MACs with product measurements [Atifet al., IEEE Trans. Inf. Theory (2022)]. For fixed product inputs and with shared randomness assistance, we give a quasi tight one-shot communication cost region with corresponding single-letter asymptotic iid expansion.
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.